# Normalized solutions for $p$-Laplacian equation with critical Sobolev
exponent and mixed nonlinearities

Shengbing Deng111Corresponding author. 222E-mail address<EMAIL_ADDRESS>(S. Deng<EMAIL_ADDRESS>(Q. Wu). and Qiaoran Wu
School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
P.R. China

###### Abstract

In this paper, we consider the existence and multiplicity of normalized
solutions for the following $p$-Laplacian critical equation

$\displaystyle\left\\{\begin{array}[]{ll}-\Delta_{p}u=\lambda\lvert
u\rvert^{p-2}u+\mu\lvert u\rvert^{q-2}u+\lvert u\rvert^{p^{*}-1}u&\mbox{in}\
\mathbb{R}^{N},\\\ \int_{\mathbb{R}^{N}}\lvert
u\rvert^{p}dx=a^{p},\end{array}\right.$

where $1<p<N$, $2<q<p^{*}=\frac{Np}{N-p}$, $a>0$, $\mu\in\mathbb{R}$ and
$\lambda\in\mathbb{R}$ is a Lagrange multiplier. Using concentration
compactness lemma, Schwarz rearrangement, Ekeland variational principle and
mini-max theorems, we obtain several existence results under $\mu>0$ and other
assumptions. We also analyze the asymptotic behavior of there solutions as
$\mu\rightarrow 0$ and $\mu$ goes to its upper bound. Moreover, we show the
nonexistence result for $\mu<0$ and get that the $p$-Laplacian equation has
infinitely solutions by genus theory when $p<q<p+\frac{p^{2}}{N}$.

_Keywords:_ Normalized solutions, $p$-Laplacian equation; Sobolev critical
nonlinearities; Infinitely solutions.

_2020 Mathematics Subject Classification:_ 35B33, 35J62, 35J92.

##  1 Introduction

In this paper, we consider the following $p$-Laplacian equation

$-\Delta_{p}u=\lambda\lvert u\rvert^{p-2}u+\mu\lvert u\rvert^{q-2}u+\lvert
u\rvert^{p^{*}-2}u\quad\mbox{in}\ \mathbb{R}^{N},$ (1.1)

where $1<p<N$, $2<q<p^{*}=\frac{Np}{N-p}$, $\lambda,\mu\in\mathbb{R}$ and
$\Delta_{p}=\mbox{div}(\lvert\nabla u\rvert^{p-2}\nabla u)$ is the
$p$-Laplacian operator.

If $p=2$, then equation (1.1) can be derived from the time-dependent equation
as

$i\psi_{t}+\Delta\psi+\mu\lvert\psi\rvert^{q-2}+\lvert\psi\rvert^{2^{*}-2}\psi=0\quad\mbox{in}\
\mathbb{R}_{+}\times\mathbb{R}^{N},$ (1.2)

when we look for the stationary waves of the form $\psi(t,x)=e^{-i\lambda
t}u(x)$. Equation (1.2) can represent both the famous Schrödinger equation
which describes the laws of particle motion[13, 26, 27], and the Bose-Einstein
condensates[8, 30, 37]. Consider the following equation

$-\Delta u=\lambda u+f(u)\quad\mbox{in}\ \mathbb{R}^{N},$ (1.3)

A direct way for studying the existence of solutions for (1.3) is to find the
critical points for the following functional

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx-\int_{\mathbb{R}^{N}}F(u)dx,$

where $F(s)=\int_{0}^{s}f(t)dt$. In this case, particular attention is devoted
to least action solutions, namely solutions minimizing $I(u)$ among all non-
trivial solutions. Here we refer the readers to [2, 7]. Another possible
approach is to give a prescribed $L^{2}$ mass of $u$, that is, consider the
constraint

$\int_{\mathbb{R}^{N}}\lvert u\rvert^{2}dx=a^{2},$ (1.4)

where $a>0$ is a constant. Then, the corresponding functional of (1.3) is

$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\int_{\mathbb{R}^{N}}F(u)dx,$

and $\lambda$ appears as a Lagrange multiplier. The solutions of (1.3) which
has prescribed mass called normalized solutions.

When we consider the normalized solutions of (1.3), a new critical exponent
$2+\frac{4}{N}$ appears which is called $L^{2}$-critical exponent. This
constant can be derived from the Gagliardo-Nirenberg inequality: for every
$p<q<p^{*}$, there exist an optimal constant $C_{N,p,q}>0$ such that

$\lVert u\rVert_{q}^{q}\leqslant C_{N,p,q}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}}\lVert u\rVert_{p}^{q(1-\gamma_{q})}\quad\forall u\in
W^{1,p}(\mathbb{R}^{N}),$

where

$\gamma_{q}:=\frac{N(q-p)}{pq}.$

By [1, section 1], $C_{N,p,q}$ can be attained for some $\psi_{0}\in
W^{1,p}(\mathbb{R}^{N})$ which satisfies

$-\Delta_{p}u+\lvert u\rvert^{p-2}u=\beta\lvert u\rvert^{q-2}$

for some $\beta>0$. Moreover, $\psi_{0}$ can be chosen non-negative, radially
symmetric, radially non-increasing and tends to $0$ as $\lvert
x\rvert\rightarrow+\infty$. If the nonlinearities of (1.3) are pure
$L^{2}$-subcritical terms, for example $f(u)=\lvert u\rvert^{q-2}u$ with
$2<q<2+\frac{4}{N}$, then by Gagliardo-Nirenberg inequality, it is not
difficult to prove that $J(u)$ is bounded from below if $u$ satisfies (1.4),
and we can find a global minimum solution of (1.3). Here we refer the readers
to [22, 25]. If the nonlinearities of (1.3) are pure $L^{2}$-supercritical
terms, for example $f(u)=\lvert u\rvert^{q-2}u$ with $2+\frac{4}{N}<q<2^{*}$,
then $J(u)$ is unbounded from below if $u$ satisfies (1.4). The first result
to deal with $L^{2}$-supercritical case was studied by Jeanjean[15]. He proved
that problem (1.3) has a mountain-pass type solution under suitable
assumptions. Compared with the pure $L^{2}$-subcritical or
$L^{2}$-supercritical case, the mixed nonlinearities terms are more
complicated, Soave [26] considered the following problem

$\left\\{\begin{array}[]{ll}-\Delta u=\lambda u+\mu\lvert
u\rvert^{q-2}u+\lvert u\rvert^{p-2}u&\mbox{in}\ \mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}\lvert u\rvert^{2}dx=a^{2},\end{array}\right.$ (1.5)

where $N\geqslant 3$, $\mu>0$, $2<q\leqslant 2+\frac{4}{N}\leqslant p<2^{*}$,
and analyzed the existence, asymptotic behavior and stability of solutions.
All the references listed above are considered in Sobolev subcritical case,
the first result of the normalized solutions for Sobolev critical case was
studied by Soave [27], that is, Soave studied the following problem

$\left\\{\begin{array}[]{ll}-\Delta u=\lambda u+\mu\lvert
u\rvert^{q-2}u+\lvert u\rvert^{2^{*}-2}u&\mbox{in}\ \mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}\lvert u\rvert^{2}dx=a^{2},\end{array}\right.$ (1.6)

where $N\geqslant 3$, $\mu>0$, $2<q<p^{*}$ and analyzed the existence,
nonexistence, asymptotic behavior and stability of solutions. We refer to [16,
17, 34] and references therein for the existence of normalized solutions for
the mixed nonlinearities.

If $p\neq 2$, there are few papers on the normalized solution of $p$-Laplacian
equation. Wang et al. [33] considered

$\left\\{\begin{array}[]{ll}-\Delta_{p}u+\lvert u\rvert^{p-2}u=\mu u+\lvert
u\rvert^{s-2}u&\mbox{in}\ \mathbb{R}^{N},\\\ \int_{\mathbb{R}^{N}}\lvert
u\rvert^{2}dx=\rho,\end{array}\right.$

where $1<p<N$, $\mu\in\mathbb{R}$, $s\in(\frac{N+2}{N}p,p^{*})$, they
considered the $L^{2}$ constraint, by the Gagliardo-Nirenberg inequality, the
$L^{2}$-critical exponent should be $\frac{N+2}{N}p$. Moreover, we know
$L^{2}(\mathbb{R}^{N})\not\subset W^{1,p}(\mathbb{R}^{N})$, so the work space
is $W^{1,p}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N})$ which is a Hilbert
space. The work space is Hilbert space is very important for [33]. [38] is the
first paper to study the $p$-Laplacian equation with $L^{p}$ constraint:

$\left\\{\begin{array}[]{ll}-\Delta_{p}u=\lambda\lvert
u\rvert^{p-2}u+\mu\lvert u\rvert^{q-2}u+g(u)&\mbox{in}\ \mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}\lvert u\rvert^{p}dx=a^{p},\end{array}\right.$ (1.7)

where $g\in C(\mathbb{R},\mathbb{R})$ and there exist
$p+\frac{p^{2}}{N}<\alpha\leqslant\beta<p^{*}$ such that for all
$s\in\mathbb{R}$, there is

$0<\alpha G(s)s\leqslant g(s)s\leqslant\beta G(s),\quad
G(s)=\int_{0}^{s}g(t)dt.$

A simple example is $g(s)=\lvert s\rvert^{r-2}s$ with
$p+\frac{p^{2}}{N}<r<p^{*}$. Moreover, Wang and Sun [32] considered both the
$L^{2}$ constraint and the $L^{p}$ constraint for the following problem

$\left\\{\begin{array}[]{ll}-\Delta_{p}u+V(x)\lvert
u\rvert^{p-2}u=\lambda\lvert u\rvert^{r-2}u+\lvert u\rvert^{q-2}u&\mbox{in}\
\mathbb{R}^{N}\\\ \int_{\mathbb{R}^{N}}\lvert
u\rvert^{r}dx=c,\end{array}\right.$

where $1<p<N$, $\lambda\in\mathbb{R}$, $r=p$ or $2$, $p<q<p^{*}$ and $V(x)$ is
a trapping potential satisfies

$V(x)\in C(\mathbb{R}^{N}),\quad\lim_{\lvert
x\rvert\rightarrow+\infty}V(x)=+\infty\quad\mbox{and}\quad\inf_{x\in\mathbb{R}^{N}}V(x)=0.$

In this work, we study the existence of normalized solutions for (1.1) by
fixing $L^{p}$-norm of $u$. Let

$S_{a}:=\\{u\in W^{1,p}(\mathbb{R}^{N}):\lVert u\rVert_{p}^{p}=a^{p}\\},$

where $a>0$ is a constant. Following [27, definition 1], we give the
definition of ground state as follows.

######  Definition 1.1.

We say $u$ is a ground state of (1.1) on $S_{a}$, if $u$ is a solution to
(1.1) and have minimal energy among all the solutions which belongs to
$S_{a}$, that is

$dE_{\mu}|_{S_{a}}(u)=0,\quad\mbox{{\rm and}}\quad
E_{\mu}(u)=\inf\big{\\{}E_{\mu}(v):dE_{\mu}|_{S_{a}}(v)=0,u\in
S_{a}\big{\\}}.$

Since $u\in S_{a}$, there are some difficulties to observe the structure of
$E_{\mu}(u)$ directly. A possible approach is to consider an auxiliary
function

$\Psi_{u}^{\mu}(s):=E_{\mu}(s\star u)=\frac{1}{p}e^{ps}\lVert\nabla
u\rVert_{p}^{p}-\frac{\mu}{q}e^{q\gamma_{q}s}\lVert
u\rVert_{q}^{q}-\frac{1}{p^{*}}e^{p^{*}s}\lVert u\rVert_{p^{*}}^{p^{*}},$

where

$s\star u:=e^{\frac{Ns}{p}}u(e^{s}\cdot).$

It is clear that $s\star u\in S_{a}$ for all $s\in\mathbb{R}$ if $u\in S_{a}$.
Thus, we can investigate the structure of $\Psi_{u}^{\mu}$ to speculate the
structure of $E_{\mu}|_{S_{a}}$.

Assume that $u$ is a critical point of $E_{\mu}|_{S_{a}}$. Then $0$ may be a
critical point of $\Psi_{u}^{\mu}$. If $0$ is the critical point of
$\Psi_{u}^{\mu}$, we have $(\Psi_{u}^{\mu})^{\prime}(0)=0$, that is

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}.$ (1.8)

In fact, by the Pohozaev indentity of (1.1), all critical point of $E_{\mu}$
satisfies (1.8)(see Proposition 2.1). Therefore, if we consider such a
manifold

$\mathcal{P}_{a,\mu}=\\{u\in S_{a}:P_{\mu}(u)=0\\},$

where

$P_{\mu}(u)=\lVert\nabla u\rVert_{p}^{p}-\mu\gamma_{q}\lVert
u\rVert_{q}^{q}-\lVert u\rVert_{p^{*}}^{p^{*}},$

we know all critical points of $E_{\mu}|_{S_{a}}$ belong to
$\mathcal{P}_{a,\mu}$ and $s\star u\in\mathcal{P}_{a,\mu}$ if and only if
$(\Psi_{u}^{\mu}(s))^{\prime}=0$. The manifold $\mathcal{P}_{a,\mu}$ is always
called Pohozaev manifold.

We divde $\mathcal{P}_{a,\mu}$ into three parts

$\displaystyle\mathcal{P}_{a,\mu}^{+}$
$\displaystyle=\big{\\{}u\in\mathcal{P}_{a,\mu}:(\Psi_{u}^{\mu})^{\prime\prime}(0)>0\big{\\}}=\big{\\{}u\in\mathcal{P}_{a,\mu}:p\lVert\nabla
u\rVert_{p}^{p}>\mu q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}+p^{*}\lVert
u\rVert_{p^{*}}^{p^{*}}\big{\\}},$ $\displaystyle\mathcal{P}_{a,\mu}^{0}$
$\displaystyle=\big{\\{}u\in\mathcal{P}_{a,\mu}:(\Psi_{u}^{\mu})^{\prime\prime}(0)=0\big{\\}}=\big{\\{}u\in\mathcal{P}_{a,\mu}:p\lVert\nabla
u\rVert_{p}^{p}=\mu q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}+p^{*}\lVert
u\rVert_{p^{*}}^{p^{*}}\big{\\}},$

and

$\displaystyle\mathcal{P}_{a,\mu}^{-}$
$\displaystyle=\big{\\{}u\in\mathcal{P}_{a,\mu}:(\Psi_{u}^{\mu})^{\prime\prime}(0)<0\big{\\}}=\big{\\{}u\in\mathcal{P}_{a,\mu}:p\lVert\nabla
u\rVert_{p}^{p}<\mu q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}+p^{*}\lvert
u\rVert_{p^{*}}^{p^{*}}\big{\\}}.$

Define

$m(a,\mu)=\inf_{u\in\mathcal{P}_{a,\mu}}E_{\mu}(u),\quad
m^{\pm}(a,\mu)=\inf_{u\in\mathcal{P}_{a,\mu}^{\pm}}E_{\mu}(u),$

and

$m_{r}(a,\mu)=\inf_{u\in\mathcal{P}_{a,\mu}\cap
W_{rad}^{1,p}(\mathbb{R}^{N})}E_{\mu}(u),\quad
m_{r}^{\pm}(a,\mu)=\inf_{u\in\mathcal{P}_{a,\mu}^{\pm}\cap
W_{rad}^{1,p}(\mathbb{R}^{N})}E_{\mu}(u).$

Obviously, by Definition 1.1, if we can prove $u$ is a critical point of
$E_{\mu}|_{S_{a}}$ and $E_{\mu}(u)=m(a,\mu)$, then $u$ is a ground state of
(1.1).

Although we give the method to observe the structure of $E_{\mu}$ on $S_{a}$,
the difficulty that arises is how do we get the compactness of PS(Palais-
Smale) sequence. In [38, lemma 2.9], the authors considered the equation (1.7)
and proved the compactness of PS sequence, but the nonlinearities is Sobolev
subcritical. When $p=2$, in [27, proposition 3.1], the author proved a
compactness lemma of PS sequence for (1.1): Assume that $\\{u_{n}\\}\in
S_{a,r}=S_{a}\cap H^{1}(\mathbb{R}^{N})$ is a PS sequence of
$E_{\mu}|_{S_{a}}$ at level $c$. Furthermore we assume that
$P_{\mu}(u_{n})\rightarrow 0$. Then, one of the following alternatives holds:
up to a subsequence, either $u_{n}\rightharpoonup u$ in
$H^{1}(\mathbb{R}^{N})$ but not strongly, and

$E_{\mu}(u)\leqslant c-\frac{1}{N}S^{\frac{N}{2}};$

or $u_{n}\rightarrow u$ in $H^{1}(\mathbb{R}^{N})$. However, the proof of [27,
proposition 3.1] need to use the Brézis-Lieb lemma[5] and the linearity of
Laplace operator. By using concentration compactness lemma(see [28, section 4]
or [21, lemma 1.1]) and referring to the idea of [14], we prove a compactness
result of PS sequence similar to [27, proposition 3.1]. Therefore, our main
goal is to obtain the PS sequence and exclude the case of weak convergence,
thereby obtaining strong convergence.

Now, we can state the existence results. Even though $\lVert
u\rVert_{p^{*}}^{p^{*}}$ is always a $L^{p}$-supercritical term, there are
some differences in the existence results when $\lVert u\rVert_{q}^{q}$ is a
$L^{p}$-subcritical, critical or supercritical term. Therefore, we will state
the existence results separately when $q<(=,>)p+\frac{p^{2}}{N}$.

If $p<q<p+\frac{p^{2}}{N}$. Since $q\gamma_{q}<p$, the function
$\Psi_{u}^{\mu}$ may have two critical points on $\mathbb{R}$(such as
$f(s)=50e^{2s}-50e^{s}-e^{6s}$), one is a local minimum point and the other is
global maximum point, we note them as $s_{u}$ and $t_{u}$ respectively.
Moreover, it is not difficult to prove that $s_{u}\star
u\in\mathcal{P}_{a,\mu}^{+}$ and $t_{u}\star u\in\mathcal{P}_{a,\mu}^{-}$. Of
course, $\Psi_{u}^{\mu}$ may not have any critical points on $\mathbb{R}$(such
as $f(s)=50e^{2s}-200e^{s}-e^{6s}$). Therefore, it is natural to speculate
$E_{\mu}$ has two critical points on $S_{a}$ under appropriate assumptions,
one is a local minimizer and is also a minimizer of $E_{\mu}$ on
$\mathcal{P}_{a,\mu}^{+}$, the other is mountain-pass type critical point and
is also a minimizer of $E_{\mu}$ on $\mathcal{P}_{a,\mu}^{+}$.

Let

$C^{\prime}=\bigg{(}\frac{p^{*}S^{p^{*}/p}(p-q\gamma_{q})}{p(p^{*}-q\gamma_{q})}\bigg{)}^{\frac{p-q\gamma_{q}}{p^{*}-p}}\frac{q(p^{*}-p)}{pC_{N,q}^{q}(p^{*}-q\gamma_{q})},$

and

$C^{\prime\prime}=\frac{pp^{*}}{N\gamma_{q}C_{N,q}^{q}(p^{*}-p)}\bigg{(}\frac{q\gamma_{q}S^{N/p}}{p-q\gamma_{q}}\bigg{)}^{\frac{p-q\gamma_{q}}{p}},$

where $S$ is the optimal constant of Sobolev inequality

$S\lVert u\rVert_{p^{*}}^{p}\leqslant\lVert\nabla u\rVert_{p}^{p}\quad\forall
u\in D^{1,p}(\mathbb{R}^{N}).$

Define

$\alpha(N,p,q):=\min\\{C^{\prime},C^{\prime\prime}\\}.$ (1.9)

Then, the existence result of local minimizer for $p<q<\frac{p^{2}}{N}$ can be
stated as follows.

######  Theorem 1.1.

Let $N\geqslant 2$, $1<p<N$, $p<q<p+\frac{p^{2}}{N}$, and $a,\mu>0$. Assume
that

$\mu a^{q(1-\gamma_{q})}<\alpha(N,p,q),$ (1.10)

then $E_{\mu}|_{S_{a}}$ has a ground state $u_{a,\mu}^{+}$ which is positive,
radially symmetric, radially non-increasing, and solves (1.1) for some
$\lambda_{a,\mu}^{+}<0$. Moreover,

$E_{\mu}(u_{a,\mu}^{+})=m(a,\mu)=m^{+}(a,\mu)<0,$

and $u_{a,\mu}^{+}$ is a local minimizer of $E_{\mu}$ on the set

$A_{k}:=\big{\\{}u\in S_{a}:\lVert\nabla u\rVert_{p}\leqslant k\big{\\}},$

for a suitable $k>0$ sufficiently small. Any other ground state of
$E_{\mu}|_{S_{a}}$ is a local minimizer of $E_{\mu}|_{A_{k}}$.

In addition to guaranteeing that $\Psi_{u}^{\mu}$ has two critical points on
$\mathbb{R}$(in fact, this conclusion can be obtained by $\mu
a^{q(1-\gamma_{q})}<C^{\prime}$), another important reason we assume (1.10) is
to ensure the convergence of the PS sequence by using compactness lemma we
have obtained.

The existence result of mountain-pass type solution for
$p<q<p+\frac{p^{2}}{N}$ can be stated as follows.

######  Theorem 1.2.

Let $N\geqslant 2$, $1<p<N$, $p<q<p+\frac{p^{2}}{N}$, and $a,\mu>0$ satisfies
(1.10). Further we assume that $N\geqslant p^{2}$ or $N<p^{2}<9$. Then
$E_{\mu}|_{S_{a}}$ has a critical point of mountain-pass type $u_{a,\mu}^{-}$
which is positive, radially symmetric, radially non-increasing, and solves
(1.1) for some $\lambda_{a,\mu}^{-}<0$. Moreover,
$E_{\mu}(u_{a,\mu}^{-})=m^{-}(a,\mu)$ and

$0<m^{-}(a,\mu)<m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}}.$

In order to use compactness lemma to obtain the convergence of PS sequences,
the strict inequality $m^{-}(a,\mu)<m^{+}(a,\mu)+\frac{1}{N}$ is a crucial
step in our proof. Here we refer to the ideas of [10, 34] to prove the strict
inequality. However, there are some difficulties to obtain the inequality for
$N<p^{2}$ and $p\geqslant 3$, we don’t know whether the result hold.

For $q\geqslant p+\frac{p^{2}}{N}$. Since $q\gamma_{q}\geqslant p$ and hence
$\Psi_{u}^{\mu}(s)$ has unique critical point $t_{\mu}$ of mountain-pass type
under suitable assumptions of $\mu$ if $q=p+\frac{p^{2}}{N}$. Moreover,
$t_{u}\star u\in\mathcal{P}_{a,\mu}^{-}$. Therefore, it is natural to
speculate that $E_{\mu}|_{S_{a}}$ has a critical point of mountain-pass type
which is also a minimizer of $E_{\mu}$ on $\mathcal{P}_{a,\mu}^{-}$.

######  Theorem 1.3.

Let $N\geqslant p^{\frac{3}{2}}$, $1<p<N$, $p+\frac{p^{2}}{N}\leqslant
q<p^{*}$, and $a,\mu>0$. Further we assume that

$\mu a^{\frac{p^{2}}{N}}<\frac{q}{pC_{N,p,q}^{q}},$ (1.11)

if $q=p+\frac{p^{2}}{N}$. Then $E_{\mu}|_{S_{a}}$ has a ground state
$u_{a,\mu}^{-}$ which is positive, radially symmetric, radially non-
increasing, and solves (1.1) for some $\lambda_{a,\mu}^{-}<0$. Moreover,
$u_{a,\mu}^{-}$ is a critical point of mountain pass-type and

$E_{\mu}(u_{a,\mu}^{-})=m^{-}(a,\mu)=m(a,\mu)\in\Big{(}0,\frac{1}{N}S^{\frac{N}{p}}\Big{)}.$

Similar to Theorem 1.2, to obtain the convergence of PS sequence, the strict
inequality $m^{-}(a,\mu)<\frac{1}{N}S^{\frac{N}{p}}$ plays an important role.
However, obtaining this result seems somewhat difficult for
$N<p^{\frac{3}{2}}$ by classical method(see [6, 27]).

Now, we start to analyze the asymptotic behavior of $u_{a,\mu}^{\pm}$ as
$\mu\rightarrow 0$ and $\mu$ goes it’s upper bound. To state these results,
let us first introduce some necessary notations. Through a scaling, we know
the equation

$-\Delta_{p}u+u^{p-1}=u^{q-1}\quad\mbox{in}\ \mathbb{R}^{N},$ (1.12)

has a radial solution $u$ which is non-negative. Similar to [10, theorem A.1],
we can prove $u\in L_{loc}^{\infty}(\mathbb{R}^{N})$. Then, by regularity
result[31], we know $u\in C_{loc}^{1,\alpha}$ for some $\alpha\in(0,1)$. Thus,
by [24], (1.12) has unique radial “ground state” $\phi_{0}$. Here, in [24],
the meaning of “ground state” is a non-negative non-trivial $C^{1}$
distribution solution of (1.12).

The asymptotic result as $\mu\rightarrow 0$ can be stated as follows.

######  Theorem 1.4.

Let $N\geqslant 2$, $1<p<N$, $p<q<p^{*}$, $a>0$ and $\mu>0$ sufficiently
small. Let $u_{a,\mu}^{+}$ be the local minimizer which is obtained by Theorem
1.1 and $u_{a,\mu}^{-}$ be the mountain-pass solution which is obtained by
Theorem 1.2 and 1.3. Then,

(1) We have
$\sigma_{0}^{\frac{1}{p-q}}\mu^{-\frac{N}{p(p-q\gamma_{q})}}u_{a,\mu}^{+}\Big{(}\sigma_{0}^{-\frac{1}{p}}\mu^{-\frac{1}{p-q\gamma_{q}}}\cdot\Big{)}\rightarrow\phi_{0}$
in $W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$, where $\phi_{0}$ is the
unique radial “ground state” solution of (1.12) and
$\sigma_{0}=\Big{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\Big{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

(2) For $N\leqslant p^{2}$, there exists $\sigma_{\mu}>0$ such that
$w_{\mu}=\sigma_{\mu}^{\frac{N-p}{p}}u_{\mu}^{-}(\sigma_{\mu}\cdot)\rightarrow
U_{\varepsilon_{0}}$ in $D^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$ for
some $\varepsilon_{0}>0$, where $U_{\varepsilon_{0}}$ is given by (2.1) and
$\sigma_{\mu}\rightarrow 0$ as $\mu\rightarrow 0$.

(3) For $N>p^{2}$, we have $u_{a,\mu}^{-}\rightarrow U_{\varepsilon_{0}}$ in
$W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$, where $U_{\varepsilon_{0}}$
satisfies $\lVert U_{\varepsilon_{0}}\rVert_{p}^{p}=a^{p}$.

In fact, for $p<q<p+\frac{p^{2}}{N}$, we can prove $\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}\rightarrow 0$ and $m^{+}(a,\mu)\rightarrow 0$ as
$\mu\rightarrow 0$. Thus, $\\{u_{a,\mu}^{+}\\}$ does not convergence strongly
in $W^{1,p}(\mathbb{R}^{N})$. For $p+\frac{p^{2}}{N}<q<p^{*}$, we can prove

$\lVert\nabla u_{a,\mu}^{-}\rVert_{p}^{p},\ \ \lVert
u_{a,\mu}^{-}\rVert_{p^{*}}^{p^{*}}\rightarrow S^{\frac{N}{p}}\quad and\quad
m^{-}\rightarrow\frac{1}{N}S^{\frac{N}{p}}$

as $\mu\rightarrow 0$ which implies $\\{u_{a,\mu}^{-}\\}$ is a minimizing
sequence of minimizing problem

$S=\inf_{u\in D^{1,p}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\lVert\nabla
u\rVert_{p}^{p}}{\lVert u\rVert_{p^{*}}^{p^{*}}}.$

When $N>p^{2}$, we can obtain strongly convergence result. But if $N\leqslant
p^{2}$, since $U_{\varepsilon}\notin L^{p}(\mathbb{R}^{N})$ for all
$\varepsilon>0$, it is posible to obtain $u_{a,\mu}^{-}\rightarrow
U_{\varepsilon_{0}}$ in $W^{1,p}(\mathbb{R}^{N})$.

Next we analyze the asymptotic behavior as $\mu$ goes its upper bound. Let

$\bar{\alpha}=\frac{1}{pa^{p^{2}/N}C_{N,p,q}^{q}},$

when $q=p+\frac{p^{2}}{N}$. We have following asymptotic result.

######  Theorem 1.5.

Let $N\geqslant 2$, $1<p<N$, $p+\frac{p^{2}}{N}\leqslant q<p^{*}$, and
$a,\mu>0$ satisfies (1.11) if $q=p+\frac{p^{2}}{N}$. Let $u_{a,\mu}^{+}$ be
the mountain-pass solution which is obtained by Theorem 1.3. Then,

(1) For $q=p+\frac{p^{2}}{N}$, we have
$(\bar{\alpha}\sigma_{0})^{\frac{1}{p-q}}s_{\mu}^{\frac{N}{p}}u_{\mu}^{-}\Big{(}\sigma_{0}^{-\frac{1}{p}}s_{\mu}\cdot\Big{)}\rightarrow\phi_{0}$
in $W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow\bar{\alpha}$, where $\phi_{0}$
is the unique radial ”ground state” solution of (1.12),
$s_{\mu}=(\bar{\alpha}-\mu)^{-(N-p)/p^{2}}$ and
$\sigma_{0}=\bar{\alpha}^{\frac{p^{2}}{N(q-p)=p^{2}}}\bigg{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\bigg{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

(2) For $p+\frac{p^{2}}{N}<q<p^{*}$, we have
$\sigma_{0}^{\frac{1}{p-q}}\mu^{-\frac{N}{p(p-q\gamma_{q})}}u_{a,\mu}^{+}\Big{(}\sigma_{0}^{-\frac{1}{p}}\mu^{-\frac{1}{p-q\gamma_{q}}}\cdot\Big{)}\rightarrow\phi_{0}$
in $W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow+\infty$, where $\phi_{0}$ is
the unique radial ”ground state” solution of (1.12) and
$\sigma_{0}=\Big{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\Big{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

For $q=p+\frac{p^{2}}{N}$, we can prove $m_{a,\mu}^{-}=0$ when
$\mu\geq\bar{\alpha}$(see Lemma 7.7) and hence $u_{a,\mu}^{-}$ does not exist.
By Theorem 1.3, we know $u_{a,\mu}^{-}$ exist when $\mu<\bar{\alpha}$. Thus,
$\bar{\alpha}$ is the sharp constant such that $u_{a,\mu}^{-}$ exist and we
can analyze the asymptotic behavior as $\mu\rightarrow\bar{\alpha}$. However,
For $p<q<p+\frac{p^{2}}{N}$, we can not claim that $\alpha(N,p,q)$ which is
given by (1.9) is the sharp constant such that $u_{a,\mu}^{\pm}$ exist. From
our proof, it seems that $\alpha(N,p,q)$ is not optimal(see Section 4). Thus,
it is impossible to study the asymptotic behavior as $\mu$ goes it’s upper
when $p<q<p+\frac{p^{2}}{N}$.

Finally, we want to investigate the nonexistence and multiplicity of equation
(1.1). The nonexistence result is attained for $\mu<0$.

######  Theorem 1.6.

Let $N\geqslant 2$, $1<p<N$, $p<q<p^{*}$, $a>0$ and $\mu<0$. Then

(1) If $u$ is a critical point of $E_{\mu}|_{S_{a}}$(not necessary positive),
then then associated Lagrange multiplier $\lambda$ is positive, and
$E_{\mu}(u)>S^{\frac{N}{p}}/N$.

(2) The problem $-\Delta_{p}u=\lambda u^{p-1}+\mu u^{q-1}+u^{p^{*}-1},\quad
u>0\quad in\ \mathbb{R}^{N}$ has no solution $u\in W^{1,p}(\mathbb{R}^{N})$
for any $\mu<0$.

The reason why we speculate equation (1.1) has no solution is due to Theorem
1.6(1). In the previous existence results, $c<\frac{1}{N}S^{\frac{N}{p}}$ is
crucial to obtain the convergence of $\mbox{(PS)}_{c}$ sequence. Moreover,
there is an example such that a $\mbox{(PS)}_{c}$ sequence with
$c=\frac{1}{N}S^{\frac{N}{p}}$ without any convergent subsequence(see [4]).
Thus, it is natural to guess that equation (1.1) has no solution.

We use genus theory to prove multiplicity result. One of the crucial problem
is that the functional should bounded from below when we use genus theory, but
due to $\lVert u\rVert_{p^{*}}^{p^{*}}$ is a $L^{2}$-supercritical term,
$E_{\mu}|_{S_{a}}(u)$ does not bounded from below. To overcome this problem,
we introduce a truncation function to complete the proof.

Let

$S_{a,r}=S_{a}\cap W_{rad}^{1,p}(\mathbb{R}^{N}).$

The multiplicity result can be stated as follows.

######  Theorem 1.7.

Let $N\geqslant 3$, $2<p<N$, $p<q<p+\frac{p^{2}}{N}$ and $a,\mu>0$ satisfies
(1.10). Then equation $(\ref{equation})$ has infinitely many solutions on
$S_{a,r}$ at negative levels.

In Theorem 1.7 we assume that $p>2$, since the quantitative deformation
lemma[35, lemma 5.15] will be used in the proof and it requires $\lVert
u\rVert_{p}^{p}\in C^{2}(L^{p}(\mathbb{R}^{N},\mathbb{R}))$. Therefore, we
need $p>2$.

Notations. Throughout this paper, $C$ is indiscriminately used to denote
various absolutely positive constants. $a\sim b$ means that there exist $C>1$
such that $C^{-1}a\leqslant b\leqslant Ca$.

##  2 Prelimilaries

In this section, we collect some results which will be used in the rest of the
paper. Firstly, Let us recall the Sobolev inequality.

######  Lemma 2.1.

For every $N\geqslant 2$ and $1<p<N$, there exists an optimal constant $S$
depends on $N$ and $p$ such that

$S\lVert u\rVert_{p^{*}}^{p}\leqslant\lVert\nabla u\rVert_{p}^{p}\quad\forall
u\in D^{1,p}(\mathbb{R}^{N}),$

where $D^{1,p}(\mathbb{R}^{N})$ denotes the completion of
$C_{c}^{\infty}(\mathbb{R}^{N})$ with respect to the norm $\lVert
u\rVert_{D^{1,p}}:=\lVert\nabla u\rVert_{p}^{p}$.

It is well knwon[29] that the optimal constant is attained by

$U_{\varepsilon,y}=d_{N,p}\varepsilon^{\frac{N-p}{p(p-1)}}\Big{(}\varepsilon^{\frac{p}{p-1}}+\lvert
x-y\rvert^{\frac{p}{p-1}}\Big{)}^{\frac{p-N}{p}},$ (2.1)

where $\varepsilon>0$, $y\in\mathbb{R}^{N}$ and $d_{N,p}>0$ depends on $N$ and
$p$ such that $U_{\varepsilon,y}$ satisfies

$-\Delta_{p}u=u^{p*-1},\quad u>0\quad\mbox{in}\ \mathbb{R}^{N},$

and hence

$\lVert\nabla U_{\varepsilon,y}\rVert_{p}^{p}=\lVert
U_{\varepsilon,y}\rVert_{p^{*}}^{p^{*}}=S^{\frac{N}{p}}.$

If $y=0$, we set $U_{\varepsilon}=U_{\varepsilon,0}$.

Next, we introduce the Pohozaev identity for $p$-Laplacian.

######  Lemma 2.2.

[19] Assume that $N\geqslant 2$, $1<p<N$, $f\in C(\mathbb{R},\mathbb{R})$ such
that $f(0)=0$ and let $u$ be a local weak solution of

$-\Delta_{p}u=f(u)\quad\mbox{in}\ D^{\prime}(\mathbb{R}^{N}),$

where $D(\mathbb{R}^{N})=C_{c}^{\infty}(\mathbb{R}^{N})$, and
$D^{\prime}(\mathbb{R}^{N})$ is the dual space of $D(\mathbb{R})$. Suppose
that

$u\in L_{loc}^{\infty}(\mathbb{R}^{N}),\quad\nabla u\in
L^{p}(\mathbb{R}^{N}),\quad\mbox{and}\quad F(u)\in L^{1}(\mathbb{R}^{N}).$

Then $u$ satisfies

$(N-p)\lVert\nabla u\rVert_{p}^{p}=Np\int_{\mathbb{R}^{N}}F(u)dx.$

By the Pohozaev identity, we can prove that all critical points belong to the
Pohozaev manifold.

######  Proposition 2.1.

Assume that $u\in S_{a}$ is a solution to (1.1), then
$u\in\mathcal{P}_{a,\mu}$.

###### Proof.

Similar to [10, lemma A1], we can prove that $u\in
L_{loc}^{\infty}(\mathbb{R}^{N})$. It is clear that

$\nabla u\in L^{p}(\mathbb{R}^{N})\quad\mbox{and}\quad
F(u)=\frac{\lambda}{p}\lvert u\rvert^{p}+\frac{\mu}{q}\lvert
u\rvert^{q}+\frac{1}{p^{*}}\lvert u\rvert^{p^{*}}\in L^{1}(\mathbb{R}^{N}).$

Thus, by Lemma 2.2, we have

$(N-p)\lVert\nabla u\rVert_{p}^{p}=\lambda Na^{p}+\frac{\mu Np}{q}\lVert
u\rVert_{q}^{q}+(N-p)\lVert u\rVert_{p^{*}}^{p^{*}}.$ (2.2)

Using the euqation (1.1), we have

$\lVert\nabla u\rVert_{p}^{p}=\lambda a^{p}+\mu\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{P^{*}},$

which together with (2.2), implies

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}.$

∎

##  3 Compactness of PS sequence

In this section, we prove a compactness lemma of PS sequence under suitable
assumptions. This is a crucial result to obtain the existence of critical
point for $E_{\mu}|_{S_{a}}$.

######  Proposition 3.1.

Let $N\geqslant 2$, $p<q<p^{*}$, and $a,\mu>0$. Let $\\{u_{n}\\}\subset
S_{a,r}$ be a PS sequence for $E_{\mu}|_{S_{a}}$ at level $c$, with

$c<\frac{1}{N}S^{\frac{N}{p}}\quad\mbox{and}\quad c\neq 0.$

Suppose in addition that $P_{\mu}(u_{n})\rightarrow 0$ as
$n\rightarrow\infty$. Then, one of the following two conclusions is true: (i)
either up to a sequence $u_{n}\rightharpoonup u$ in $W^{1,p}(\mathbb{R}^{N})$
but not strongly, where $u\not\equiv 0$ is a solution to (1.1) for some
$\lambda<0$, and $E_{\mu}(u)<c-\frac{1}{N}S^{\frac{N}{p}};$ (ii) or up to a
subsequence $u_{n}\rightarrow u$ in $W^{1,p}(\mathbb{R}^{N})$, $E_{\mu}(u)=m$,
and $u$ solves (1.1) for some $\lambda<0$.

Since the proof of Proposition 3.1 is relatively long, We will divide the
proof into some lemmas.

######  Lemma 3.1.

$\\{u_{n}\\}$ is bounded in $W^{1,p}(\mathbb{R}^{N})$.

###### Proof.

At first, we assume that $p<q<p+\frac{p^{2}}{N}$, so that $q\gamma_{q}<p$.
Since $P_{\mu}(u_{n})\rightarrow 0$, by the Gagliardo-Nirenberg inequality, we
have

$\displaystyle E_{\mu}(u_{n})$ $\displaystyle=\frac{1}{N}\lVert\nabla
u_{n}\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u_{n}\rVert_{q}^{q}+o_{n}(1)$ $\displaystyle\geqslant\frac{1}{N}\lVert\nabla
u_{n}\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}C_{N,p,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u_{n}\rVert_{q}^{q\gamma_{q}}+o_{n}(1).$

Then, using $E_{\mu}(u_{n})\rightarrow c$ as $n\rightarrow\infty$, we deduce
that $\\{u_{n}\\}$ is bounded in $W^{1,p}(\mathbb{R}^{N})$.

Now, let $q=p+\frac{p^{2}}{N}$, so that $q\gamma_{q}=p$. Then
$P_{\mu}(u_{n})\rightarrow 0$ gives

$E_{\mu}(u_{n})=\frac{1}{N}\lVert u_{n}\rVert_{p^{*}}^{p^{*}}+o_{n}(1),$

which implies $\\{u_{n}\\}$ is bounded in $L^{p^{*}}(\mathbb{R}^{N})$. By the
Hölder inequality

$\lVert u_{n}\rVert_{q}^{q}\leqslant\lVert
u_{n}\rVert_{p^{*}}^{q\gamma_{q}}\lVert
u_{n}\rVert_{p}^{q(1-\gamma_{q})}=a^{q(1-\gamma_{q})}\lVert
u_{n}\rVert_{p^{*}}^{q\gamma_{q}},$

we obtain $\\{u_{n}\\}$ is bounded in $L^{q}(\mathbb{R}^{N})$. Using again
that $P_{\mu}(u_{n})\rightarrow 0$, we know $\\{u_{n}\\}$ is bounded in
$W^{1,p}(\mathbb{R}^{N})$.

Finally, let $p+\frac{p^{2}}{N}<q<p^{*}$, so that $q\gamma_{q}>p$. Since
$P_{\mu}(u_{n})\rightarrow 0$, we have

$E_{\mu}(u_{n})=\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u_{n}\rVert_{q}^{q}+\frac{1}{N}\lVert u_{n}\rVert_{p^{*}}^{p^{*}}+o_{n}(1),$

and the cofficient of $\lVert u_{n}\rVert_{q}^{q}$ is positive. Therefore,
$\\{\lVert u_{n}\rVert_{q}\\}$ and $\\{\lVert u_{n}\rVert_{p^{*}}\\}$ are both
bounded which implies $\\{\lVert\nabla u_{n}\rVert_{p}\\}$ is bounded, since
$P_{\mu}(u_{n})\rightarrow 0$. ∎

Now, we can state the concentration compactness lemma of $\\{u_{n}\\}$, the
proof can be found in [28, section 4] and [21, lemma 1.1].

######  Lemma 3.2.

Suppose $u_{n}\rightharpoonup u$ in $W^{1,p}(\mathbb{R}^{N})$ and
$\lvert\nabla u_{n}\rvert^{p}\rightharpoonup\kappa,\lvert
u_{n}\rvert^{p^{*}}\rightharpoonup\nu$ in the sense of measures where $\kappa$
and $\nu$ are bounded non-negative measures on $\mathbb{R}^{N}$. Then, for
some at most countable set $J$, we have

$\kappa\geqslant\lvert\nabla u\rvert^{p}+\sum_{j\in
J}\kappa_{j}\delta_{x_{j}},\quad\nu=\lvert u\rvert^{p^{*}}+\sum_{j\in
J}\nu_{j}\delta_{x_{j}},$

where $\kappa_{j},\nu_{j}>0$ satisfies
$S\nu_{j}^{\frac{p}{p^{*}}}\leqslant\kappa_{j}$ for all $j\in J$.

######  Lemma 3.3.

There is

$\lim_{n\rightarrow\infty}\lVert u_{n}\rVert_{p^{*}}^{p^{*}}=\lVert
u\rVert_{p^{*}}^{p^{*}}+\sum_{j\in J}\nu_{j}.$ (3.1)

###### Proof.

For every $R>0$, let $\varphi_{R}\in C_{c}^{\infty}(\mathbb{R}^{N})$ be such
that

$0\leqslant\varphi_{R}\leqslant 1,\quad\varphi_{R}=1\ \mbox{for}\ \lvert
x\rvert\leqslant R,\quad\mbox{and}\quad\varphi_{R}=0\ \mbox{for}\ \lvert
x\rvert\geqslant R+1.$

Since $\lvert u_{n}\rvert^{p^{*}}\rightharpoonup\nu$, we have

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}dx$
$\displaystyle=\lim_{n\rightarrow\infty}\Big{(}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}\varphi_{R}dx+\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}(1-\varphi_{R})dx\Big{)}$
$\displaystyle=\int_{\mathbb{R}^{N}}\lvert
u\rvert^{p^{*}}\varphi_{R}dx+\sum_{j\in
J}\varphi_{R}(x_{j})\nu_{j}+\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}(1-\varphi_{R})dx.$

Let $R\rightarrow+\infty$, by the Lebesgue dominated convergence theorem, we
obtain

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}dx=\int_{\mathbb{R}^{N}}\lvert u\rvert^{p^{*}}dx+\sum_{j\in
J}\nu_{j}+\lim_{R\rightarrow+\infty}\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}(1-\varphi_{R})dx.$

Now, we prove that

$\lim_{R\rightarrow+\infty}\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}(1-\varphi_{R})dx=0,$ (3.2)

which leads to (3.1). Since $\\{u_{n}\\}$ is bounded in
$W_{rad}^{1,p}(\mathbb{R}^{N})$, we have

$\lvert u_{n}(x)\rvert\leqslant C\lvert x\rvert^{\frac{1-N}{p}}\quad\mbox{a.e.
in}\ \mathbb{R}^{N},$

where $C>0$ is a constant independent of $n$. It follows that

$\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{p^{*}}(1-\varphi_{R})dx\leqslant\int_{\lvert x\rvert\geqslant
R}\lvert u_{n}\rvert^{p^{*}}dx\leqslant CR^{\frac{N(1-p)}{N-p}},$

which implies (3.2) holds. ∎

Proof of Proposition 3.1 Since $\\{u_{n}\\}$ is a bounded PS sequence for
$E_{\mu}|_{S_{a}}$, there exists $\\{\lambda_{n}\\}\in\mathbb{R}$ such that
for every $\psi\in W^{1,p}(\mathbb{R}^{N})$,

$\int_{\mathbb{R}^{N}}\Big{(}\lvert\nabla u_{n}\rvert^{p-2}\nabla
u_{n}\cdot\nabla\psi-\lambda_{n}\lvert u_{n}\rvert^{p-2}u_{n}\psi-\mu\lvert
u_{n}\rvert^{q-2}u_{n}\psi-\lvert
u_{n}\rvert^{p^{*}-2}u_{n}\psi\Big{)}=o_{n}(1)\lVert\psi\rVert_{W^{1,p}}$
(3.3)

as $n\rightarrow\infty$. Choosing $\psi=u_{n}$, we deduce that
$\\{\lambda_{n}\\}$ is bounded as well, and hence, up to a subsequence
$\lambda_{n}\rightarrow\lambda\in\mathbb{R}$. Using the fact that
$P_{\mu}(u_{n})\rightarrow 0$ and $\gamma_{q}<1$, we know that

$\displaystyle\lambda a^{p}$
$\displaystyle=\lim_{n\rightarrow\infty}\lambda_{n}\lVert
u_{n}\rVert_{p}^{p}=\lim_{n\rightarrow\infty}\Big{(}\lVert\nabla
u_{n}\rVert_{p}^{p}-\mu\lVert u_{n}\rVert_{q}^{q}-\lVert
u_{n}\rVert_{p^{*}}^{p^{*}}\Big{)}$
$\displaystyle=\lim_{n\rightarrow\infty}\mu(\gamma_{q}-1)\lVert
u_{n}\rVert_{q}^{q}=\mu(\gamma_{q}-1)\lVert u\rVert_{q}^{q}\leqslant 0,$ (3.4)

with $\lambda=0$ is and only if $u\equiv 0$.

We consider $\varphi_{\varepsilon}\in C_{c}^{\infty}(\mathbb{R}^{N})$ such
that

$0\leqslant\varphi_{\varepsilon}\leqslant 1,\quad\varphi_{\varepsilon}=1\
\mbox{in}\ B_{\varepsilon}(x_{j}),\quad\varphi_{\varepsilon}=0\ \mbox{in}\
B_{2\varepsilon}(x_{j}),\quad\mbox{and}\quad\lvert\nabla\varphi_{\varepsilon}\rvert\leqslant\frac{2}{\varepsilon}.$

It is clear that the sequence $\\{\varphi_{\varepsilon}u_{n}\\}$ is bounded in
$W^{1,p}(\mathbb{R}^{N})$, then, testing (3.3) with
$\psi=\varphi_{\varepsilon}u_{n}$, we obtain

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert\nabla
u_{n}\rvert^{p-2}u_{n}\nabla u_{n}\cdot\nabla\varphi_{\varepsilon}dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\Big{(}\lambda_{n}\lvert
u_{n}\rvert^{p}\varphi_{\varepsilon}+\mu\lvert
u_{n}\rvert^{q}\varphi_{\varepsilon}+\lvert
u_{n}\rvert^{p^{*}}\varphi_{\varepsilon}-\lvert\nabla
u_{n}\rvert^{p}\varphi_{\varepsilon}\Big{)}dx$ $\displaystyle=$
$\displaystyle\lambda\int_{\mathbb{R}^{N}}\lvert
u\rvert^{p}\varphi_{\varepsilon}dx+\mu\int_{\mathbb{R}^{N}}\lvert
u\rvert^{q}\varphi_{\varepsilon}dx+\int_{\mathbb{R}^{N}}\varphi_{\varepsilon}d\nu-\int_{\mathbb{R}^{N}}\varphi_{\varepsilon}d\mu.$
(3.5)

By the Hölder inequality,

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{p-2}u_{n}\nabla u_{n}\cdot\nabla\varphi_{\varepsilon}dx\Big{|}$
$\displaystyle\leqslant\frac{2}{\varepsilon}\int_{B_{2\varepsilon}(x_{j})}\lvert\nabla
u\rvert^{p-1}\lvert u_{n}\rvert dx$
$\displaystyle\leqslant\frac{2}{\varepsilon}\Big{(}\int_{B_{2\varepsilon}(x_{j})}\lvert\nabla
u_{u}\rvert^{p}dx\Big{)}^{\frac{p-1}{p}}\Big{(}\int_{B_{2\varepsilon}(x_{j})}\lvert
u_{n}\rvert^{p}dx\Big{)}^{\frac{1}{p}}$
$\displaystyle\leqslant\frac{C}{\varepsilon}\lVert
u_{n}\rVert_{L^{p}(B_{2\varepsilon}(x_{j}))},$

where $C>0$ is a constant independent of $n$. Thus, using the Hölder
inequality again, we have

$\lim_{n\rightarrow\infty}\Big{|}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{p-2}u_{n}\nabla
u_{n}\cdot\nabla\varphi_{\varepsilon}dx\Big{|}\leqslant\frac{C}{\varepsilon}\lVert
u\rVert_{L^{p}(B_{2\varepsilon}(x_{j}))}\leqslant C\lVert
u\rVert_{L^{p^{*}}(B_{2\varepsilon}(x_{j}))},$

which implies

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{p-2}u_{n}\nabla u_{n}\cdot\nabla\varphi_{\varepsilon}dx\rightarrow 0$

as $\varepsilon\rightarrow 0$.

If $J\neq\emptyset$. Let $\varepsilon\rightarrow 0$ on both sides of (3), we
obtain $\nu_{j}=\mu_{j}$. By Lemma 3.2, since $\mu_{j}\geqslant
S\nu_{j}^{p/p^{*}}$, we have $\nu_{j}\geqslant S^{\frac{N}{p}}$. Therefore, by
Lemma 3.3,

$c=\lim_{n\rightarrow\infty}E_{\mu}(u_{n})\geqslant
E_{\mu}(u)+\Big{(}\frac{1}{p}-\frac{1}{p^{*}}\Big{)}\sum_{k\in
J}\nu_{k}\geqslant E_{\mu}(u)+\frac{1}{N}S^{\frac{N}{p}}.$

Since $m<S^{N/p}/N$, so $E_{\mu}(u)<0$ which implies $u\not\equiv 0$.
Following the idea of [36, lemma 2.2], we can prove

$\lvert\nabla u_{n}\rvert^{p-2}\nabla u_{n}\rightharpoonup\lvert\nabla
u\rvert^{p-2}\nabla u\ in\ \big{(}L^{p}(\mathbb{R}^{N})\big{)}^{*}.$

Thus, passing to the limit in (3.3) by weak convergence, we know $u$ is a
solution to (1.1). Now, case (i) in the proposition 3.1 holds.

If instead $J=\emptyset$, then the Brézis-Lieb Lemma [5] and (3.1) implies
$u_{n}\rightarrow u$ in $L^{p^{*}}(\mathbb{R}^{N})$. Now, we prove
$u\not\equiv 0$ and hence by (3), we know $\lambda<0$. Suppose by
contradiction that $u\equiv 0$. Then, by $P_{\mu}(u_{n})\rightarrow 0$,

$c=\lim_{n\rightarrow\infty}E_{\mu}(u_{n})=\lim_{n\rightarrow\infty}\bigg{(}\frac{1}{N}\lVert
u_{n}\rVert_{p^{*}}^{p^{*}}+\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u_{n}\rVert_{q}^{q}\bigg{)}=0,$

which contradicts our assumptions. Let
$T:W^{1,p}(\mathbb{R}^{N})\rightarrow\big{(}W^{1,p}(\mathbb{R}^{N})\big{)}^{*}$
be the mapping given by

$<Tu,v>=\int_{\mathbb{R}^{N}}\Big{(}\lvert\nabla u\rvert^{p-2}\nabla
u\cdot\nabla v-\lambda\lvert u\rvert^{p-2}uv\Big{)}dx.$

Then, slightly modifying the proof in [14, Lemma 3.6], we can derive that
$u_{n}\rightarrow u$ in $W^{1,p}(\mathbb{R}^{N})$, and hence case (ii) in the
proposition 3.1 holds.$\hfill\qed$

##  4 Existence result to the case $p<q<p+\frac{p^{2}}{N}$

In this section, we prove that under assumption (1.10), $E_{\mu}|_{S_{a}}$ has
two critical points, one is a local minimizer and the other is a mountain-pass
type solution. In order to prove this result, we need some properties of
$E_{\mu}$ and $\mathcal{P}_{a,\mu}$ by analyzing the structure of
$\Psi_{u}^{\mu}$.

###  4.1 Some properties of $E_{\mu}$ and $\mathcal{P}_{a,\mu}$

For Pohozaev manifold $\mathcal{P}_{a,\mu}$, we have the following properties.

######  Lemma 4.1.

$\mathcal{P}_{a,\mu}^{0}=\emptyset$, and $\mathcal{P}_{a,\mu}$ is a smooth
manifold of codimension $2$ in $W^{1,p}(\mathbb{R}^{N})$.

###### Proof.

Let us assume that there exists $u\in\mathcal{P}_{a,\mu}^{0}$. Then, combining
$P_{\mu}(u)=0$ and $\big{(}\Psi_{u}^{\mu}\big{)}^{\prime\prime}(0)=0$, we
deduce that

$\mu\gamma_{q}(p-q\gamma_{q})\lVert u\rVert_{q}=(p^{*}-p)\lVert
u\rVert_{p^{*}}^{p^{*}}.$

Using this equation in $P_{\mu}(u)=0$, we obtain

$\lVert\nabla u\rVert_{p}^{p}=\frac{p^{*}-q\gamma_{q}}{p-q\gamma_{q}}\lVert
u\rVert_{p^{*}}^{p^{*}}\leqslant\frac{S^{p^{*}/p}(p^{*}-q\gamma_{q})}{p-q\gamma_{q}}\lVert\nabla
u\rVert_{p}^{p^{*}},$ (4.1)

and

$\lVert\nabla
u\rVert_{p}^{p}=\frac{\mu\gamma_{q}(p^{*}-q\gamma_{q})}{p^{*}-p}\lVert
u\rVert_{q}^{q}\leqslant\frac{\mu\gamma_{q}(p^{*}-q\gamma_{q})}{p^{*}-p}a^{q(1-\gamma_{q})}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}}.$ (4.2)

From (4.1) and (4.2), we infer that

$\bigg{(}\frac{p-q\gamma_{q}}{S^{p^{*}/p}(p^{*}-q\gamma_{q})}\bigg{)}^{\frac{1}{p^{*}-p}}\leqslant\bigg{(}\frac{\mu\gamma_{q}(p^{*}-q\gamma_{q})}{p^{*}-p}C_{N,q}^{q}a^{q(1-\gamma_{q})}\bigg{)}^{\frac{1}{p-q\gamma_{q}}},$

that is

$\mu
a^{q(1-q\gamma_{q})}\geqslant\bigg{(}\frac{p-q\gamma_{q}}{S^{p^{*}/p}(p^{*}-q\gamma_{q})}\bigg{)}^{\frac{p-q\gamma_{q}}{p^{*}-p}}\frac{p^{*}-p}{C_{N,q}^{q}\gamma_{q}(p^{*}-q\gamma_{q})}.$
(4.3)

We can check that this is contradicts with (1.10): it is sufficient to verify
that the right hand side in (1.10) is less than or equal to the right hand
side in (4.3), and this is equivalent to

$\bigg{(}\frac{q\gamma_{q}}{p}\bigg{)}^{p^{*}-p}\bigg{(}\frac{p^{*}}{p}\bigg{)}^{p-q\gamma_{q}}\leqslant
1.$ (4.4)

Since the function $\varphi(t)=\log t/(t-1)$ is monotone decreasing on
$(0,+\infty)$, we have

$\varphi\bigg{(}\frac{q\gamma_{q}}{p}\bigg{)}\leqslant\varphi\bigg{(}\frac{p^{*}}{p}\bigg{)},$

that is (4.4).

Now, we can check that $\mathcal{P}_{a,\mu}$ is a smooth manifold of
codimension $2$ in $W^{1,p}(\mathbb{R}^{N})$. Recall the definition of
$\mathcal{P}_{a,\mu}$, since $P_{\mu}(u)$ and $G(u)=\lVert
u\rVert_{p}^{p}-a^{p}$ are of class $C^{1}$ in $W^{1,p}(\mathbb{R}^{N})$, so
we just show that the differential
$\big{(}dG(u),dP_{\mu}(u)\big{)}:W^{1,p}(\mathbb{R}^{N})\longmapsto\mathbb{R}^{2}$
is surjective for every $u\in\mathcal{P}_{a,\mu}$. To this end, we prove that
for every $u\in\mathcal{P}_{a,\mu}$ there exists $\varphi\in T_{u}S_{a}$ such
that $dP_{\mu}(u)[\varphi]\neq 0$. Once such $\varphi$ exist, the system

$\left\\{\begin{array}[]{l}dG(u)[\alpha\varphi+\theta u]=x,\\\
dP_{\mu}(u)[\alpha\varphi+\theta
u]=y,\end{array}\right.\quad\Longleftrightarrow\quad\left\\{\begin{array}[]{l}a^{p}\theta=x,\\\
\alpha dP_{\mu}(u)[\varphi]+\theta dP_{\mu}(u)[u]=y,\end{array}\right.$

is solvable with respect to $\alpha$ and $\theta$, for every
$(x,y)\in\mathbb{R}^{2}$, and hence the surjectivity is proved. Suppose by
contradiction that for $u\in\mathcal{P}_{a,\mu}$ such that
$dP_{\mu}(u)[\varphi]=0$ for every $\varphi\in T_{u}S_{a}$. Then $u$ is a
constrained critical point for the functional $P_{\mu}|_{S_{a}}$, and hence by
the Lagrange multipliers rule there exists $\nu\in\mathbb{R}$ such that

$-\Delta_{p}u=\nu\lvert u\rvert^{p-2}u+\frac{\mu q\gamma_{q}}{p}\lvert
u\rvert^{q-2}u+\frac{p^{*}}{p}\lvert u\rvert^{p^{*}-2}u,\quad in\
\mathbb{R}^{N}.$

But, by the Pohozaev identity, this implies that

$p\lVert\nabla u\rVert_{p}^{p}=\mu q\gamma_{q}^{2}\lVert
u\rVert_{q}^{q}+p^{*}\lVert u\rVert_{P^{*}}^{p^{*}},$

that is $u\in\mathcal{P}_{a,\mu}^{0}$, a contradiction. ∎

By the Hölder inequality, we have

$E_{\mu}(u)\geqslant\frac{1}{p}\lVert\nabla
u\rVert_{p}^{p}-\frac{\mu}{q}C_{N,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}}-\frac{1}{p^{*}S^{p*/p}}\lVert\nabla
u\rVert_{p}^{p^{*}}=h(\lVert\nabla u\rVert_{p}),$ (4.5)

where

$h(t)=\frac{1}{p}t^{p}-\frac{\mu}{q}C_{N,q}^{q}a^{q(1-\gamma_{q})}t^{q\gamma_{q}}-\frac{1}{p^{*}S^{p*/p}}t^{p^{*}}.$

######  Lemma 4.2.

Under assumption (1.10), the function $h$ has exactly two critical points, one
is a local minimum at negative level, and the other is a global maximum at
positive level. In addition, there exists $R_{1}>R_{0}>0$ such that
$h(R_{0})=h(R_{1})=0$, and $h(t)>0$ if and only if $t\in(R_{0},R(1))$.

###### Proof.

For every $t>0$, we know $h(t)>0$ if and only if

$\varphi(t)>\frac{\mu}{q}C_{N,q}^{q}a^{q(1-\gamma_{q})},\quad
with\quad\varphi(t)=\frac{1}{p}t^{p-q\gamma_{q}}-\frac{1}{pS^{p^{*}/p}}t^{p^{*}-q\gamma_{q}}.$

We can check that $\varphi$ has a unique critical point on $(0,+\infty)$,
which is a global maximum point at positive level. The critical point is

$\bar{t}=\bigg{(}\frac{p^{*}S^{p^{*}/p}(p-q\gamma_{q})}{p(p^{*}-q\gamma_{q})}\bigg{)}^{\frac{1}{p^{*}-p}},$

and the maximum level is

$\varphi(\bar{t})=\frac{p^{*}-p}{p(p^{*}-q\gamma_{q})}\bigg{(}\frac{p^{*}S^{p^{*}/p}(p-q\gamma_{q})}{p(p^{*}-q\gamma_{q})}\bigg{)}^{\frac{p-q\gamma_{q}}{p^{*}-p}}.$

Therefore, $h$ is positive on $(R_{0},R_{1})$ if and only if

$\varphi(\bar{t})>\frac{\mu}{q}C_{N,q}^{q}a^{q(1-\gamma_{q})},$

that is $\mu a^{q(1-\gamma_{q})}<C^{\prime}$. It follows that $h$ has a global
maximum at positive level on $(R_{0},R_{1})$. Moreover, since
$h(0^{+})=0^{-}$, there exists a local minimum point at negative level in
$(0,R_{0})$. The fact that $h$ has no other critical points can be derived
from $h^{\prime}(t)=0$ with only two zeros. ∎

Using the properties of $h(t)$, we can analyze the structure of
$\Psi_{u}^{\mu}$ and $E_{\mu}$.

######  Lemma 4.3.

For every $u\in S_{a}$, the function $\Psi_{u}^{\mu}$ has exactly two critical
points $s_{u}<t_{u}$ and two zeros $c_{u}<d_{u}$, with
$s_{u}<c_{u}<t_{u}<d_{u}$. Moreover,

(i) $s_{u}\star u\in\mathcal{P}_{a,\mu}^{+},t_{u}\star
u\in\mathcal{P}_{a,\mu}^{-}$, and if $s\star u\in\mathcal{P}_{a,\mu}$, then
either $s=s_{u}$ or $s=t_{u}$.

(ii) $\lVert\nabla(s\star u)\rVert_{p}\leqslant R_{0}$ for every $s\leqslant
c_{u}$, and $E_{\mu}(s_{u}\star u)=\min\Big{\\{}E_{\mu}(s\star
u):s\in\mathbb{R},\ \lVert\nabla(s\star u)\rVert_{p}\leqslant
R_{0}\Big{\\}}<0.$

(iii) We have $E_{\mu}(t_{u}\star u)=\max_{s\in\mathbb{R}}E_{\mu}(s\star
u)>0.$ and if $t_{u}<0$, then $P_{\mu}(u)<0$.

(iv) The maps $u\in S_{a}\longmapsto s_{u}\in\mathbb{R}$ and $u\in
S_{a}\longmapsto t_{u}\in\mathbb{R}$ are of class $C^{1}$.

###### Proof.

By (4.5),

$\Psi_{u}^{\mu}(s)=E_{\mu}(s\star u)\geqslant h(\lVert\nabla(s\star
u)\rVert_{p})=h(e^{s}\lVert\nabla u\rVert_{p}).$

Thus, by Lemma 4.2, $\Psi_{u}^{\mu}(s)$ is positive on
$\big{(}\log(R_{0}/\lVert\nabla u\rVert_{p}),\log(R_{1}/\lVert\nabla
u\rVert_{p})\big{)}$. It is clearly that $\Psi_{u}^{\mu}(-\infty)=0^{-}$ and
$\Psi_{u}^{\mu}(+\infty)=-\infty$, hence $\Psi_{u}^{\mu}$ has at least two
critical points $s_{u}<t_{u}$, with $s_{u}$ is local minimum point on
$(-\infty,\log(R_{0}/\lVert\nabla u\rVert_{p}))$ at negative level, and
$t_{u}$ is global maximum point at positive level. Now, we can check that
there are no other critical points of $\Psi_{u}^{\mu}$. Indeed, the equation
$\big{(}\Psi_{u}^{\mu}\big{)}^{\prime}(s)=0$ has only two zeros. Now, the zero
point theorem implies $\Psi_{u}^{\mu}$ has exactly two zeros $c_{u}<d_{u}$,
with $s_{u}<c_{u}<t_{u}<d_{u}$.

By Lemma 2.1, $s\star u\in\mathcal{P}_{a,\mu}$ if and only if
$\big{(}\Psi_{u}^{\mu}\big{)}^{\prime}(s)=0$, that is either $s=s_{u}$ or
$s=t_{u}$. Since $0$ is a local minimum of $\Psi_{s_{u}\star
u}^{\mu}(s)\big{(}=\Psi_{u}^{\mu}(e^{s_{u}+s})\big{)}$, we have
$\big{(}\Psi_{s_{u}\star u}^{\mu}\big{)}^{\prime\prime}(0)\geqslant 0$, which
implies $s_{u}\star u\in\mathcal{P}_{a,\mu}^{+}\cup\mathcal{P}_{a,\mu}^{0}$.
Lemma 4.1 gives $\mathcal{P}_{a,\mu}^{0}=\emptyset$, hence $s_{u}\star
u\in\mathcal{P}_{a,\mu}^{+}$. In the same way $t_{u}\star
u\in\mathcal{P}_{a,\mu}^{+}$.

Since $\Psi_{u}^{\mu}$ is positive on $\big{(}\log(R_{0}/\lVert\nabla
u\rVert_{p}),\log(R_{1}/\lVert\nabla u\rVert_{p})\big{)}$, we deduce that
$c_{u}<\log(R_{0}/\lVert\nabla u\rVert_{p})$. Thus, $\lVert\nabla(c_{u}\star
u)\rVert_{p}\leqslant R_{0}$ which implies $\lVert\nabla(s\star
u)\rVert_{p}\leqslant R_{0}$ for all $s\leqslant c_{u}$. We know $s_{u}$ is a
local minimum on $\big{(}-\infty,\log(R_{0}/\lVert\nabla u\rVert_{p})\big{)}$
at negative level, so

$\displaystyle E_{\mu}(s_{u}\star u)$
$\displaystyle=\Psi_{u}^{\mu}(s_{u})=\min\Big{\\{}\Psi_{u}^{\mu}(s):s\leqslant\log\big{(}R_{0}/\lVert\nabla
u\rVert_{p}\big{)}\Big{\\}}$ $\displaystyle=\min\Big{\\{}E_{\mu}(s\star
u):\lVert\nabla(s\star u)\rVert_{p}\leqslant R_{0}\Big{\\}}.$

Since $t_{u}$ is a global maximum of $\Psi_{u}^{\mu}$, we conclude that
$\Psi_{u}^{\mu}$ is decreasing on $(t_{u},+\infty)$. Therefore,
$\big{(}\Psi_{u}^{\mu}\big{)}^{\prime}(s)<0$ for all
$s\in(t_{u},+\infty)$(since there are no critical points on
$(t_{u},+\infty)$). If $t_{u}<0$, then
$P_{\mu}(u)=\big{(}\Psi_{u}^{\mu}\big{)}^{\prime}(0)<0$.

It remain to show that $u\longmapsto s_{u}$ and $u\longmapsto t_{u}$ are class
of $C^{1}$. Considering the function
$\Phi(s,u):=\big{(}\Psi_{u}^{\mu}\big{)}^{\prime}(s)$. Since
$\Phi(s_{u},u)=0,\partial_{t}\Phi(s_{u},u)=\big{(}\Psi_{u}^{\mu}\big{)}^{\prime\prime}(s_{u})>0$,
and it is possible to pass with continuity from $\mathcal{P}_{a,\mu}^{+}$ to
$\mathcal{P}_{a,\mu}^{-}$(since $\mathcal{P}_{a,\mu}^{0}=\emptyset$), by the
implicit theorem, we known $u\longmapsto s_{u}$ is of class $C^{1}$. The same
argument proves that $u\longmapsto t_{u}$ is of class $C^{1}$. ∎

###  4.2 Existence of local minimizer

In this subsection, we always assume that the assumptions of Theorem 1.1 hold.
Set

$A_{k}=\big{\\{}u\in S_{a}:\lVert\nabla u\rVert_{p}\leqslant k\big{\\}},$

where $k>0$ is a constant.

######  Lemma 4.4.

$m^{+}(a,\mu)=\inf_{u\in A_{R_{0}}}E_{\mu}(u)<0<m^{-}(a,\mu)$.

###### Proof.

For every $v\in\mathcal{P}_{a,\mu}^{+}$, by Lemma 4.3 (i) and (ii), we know
$s_{v}=0$ and $\lVert\nabla v\rVert_{p}\leqslant R_{0}$. Thus,

$\inf_{u\in A_{R_{0}}}E_{\mu}(u)\leqslant E_{\mu}(v)<0,$

which implies

$\inf_{u\in A_{R_{0}}}E_{\mu}(u)\leqslant m^{+}(a,\mu)<0.$

For every $v\in A_{R_{0}}$, using again Lemma 4.3 (i) and (ii), we have
$s_{v}\star v\in\mathcal{P}_{a,\mu}^{+}$ and $E_{\mu}(s_{v}\star v)\leqslant
E_{\mu}(v)$. Hence,

$m^{+}(a,\mu)\leqslant E_{\mu}(s_{v}\star v)\leqslant E_{\mu}(v),$

which implies

$m^{+}(a,\mu)\leqslant\inf_{u\in A_{R_{0}}}E_{\mu}(u).$

It remain to prove that $m^{-}(a,\mu)>0$. For every
$v\in\mathcal{P}_{a,\mu}^{-}$, by Lemma 4.3 (i) and (iii), $t_{v}=0$ and
$E_{\mu}(v)\geqslant E_{\mu}(s\star v)$ for all $s\in\mathbb{R}$. Now, using
(4.5) and Lemma 4.2, we have

$E_{\mu}(v)\geqslant\max_{s\in\mathbb{R}}E_{\mu}(s\star
v)\geqslant\max_{s\in\mathbb{R}}h(e^{s}\lVert\nabla
v\rVert_{p})=\max_{t>0}h(t)>0,$

which implies

$m^{-}(a,\mu)\geqslant\max_{t>0}h(t)>0.$

∎

By (4.5) and Lemma 4.2,

$\inf_{u\in\partial A_{R_{0}}}E_{\mu}(u)\geqslant\inf_{u\in\partial
A_{R_{0}}}h(\lVert\nabla u\rVert_{p})=0.$

Therefore, if we can find a minimizer for $E_{\mu}|_{A_{R_{0}}}$, it must be a
minimizer for $E_{\mu}|_{S_{a}}$.

Proof of Theorem 1.1: Let $\\{v_{n}\\}$ be a minimizing sequence for
$E_{\mu}|_{A_{R_{0}}}$. It is not restrictive to assume that $v_{n}$ is
radially symmetric and radially decreasing(if this is not the case, we can
replace $v_{n}$ with $\lvert v_{n}\rvert^{*}$, the Schwarz rearrangement of
$\lvert v_{n}\rvert$, and we obtain another minimizing sequence for
$E_{\mu}|_{A_{R_{0}}}$). Furthermore, by Lemma 4.4, we can assume that
$v_{n}\in\mathcal{P}_{a,\mu}^{+}$.

Now, the Ekeland’s variational principle gives a new minimizing sequence
$\\{u_{n}\\}\subset A_{R_{0}}$ which is also a PS sequence for
$E_{\mu}|_{S_{a}}$, with the property that $\lVert
u_{n}-v_{n}\rVert\rightarrow 0$ as $n\rightarrow\infty$. The condition $\lVert
u_{n}-v_{n}\rVert\rightarrow 0$ implies that
$P_{\mu}(u_{n})=P_{\mu}(v_{n})+o(1)\rightarrow 0$ as $n\rightarrow\infty$.
Hence one of the cases in proposition 3.1 holds. If case(i) occurs, that is
$u_{n}\rightharpoonup u$ in $W^{1,p}({\mathbb{R}^{N}})$, where $u$ solves
(1.1) for some $\lambda<0$, and

$E_{\mu}(u)\leqslant
m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}}<-\frac{1}{N}S^{\frac{N}{p}}.$ (4.6)

Since $u$ solves (1.1), by the Pohozaev identity $P_{\mu}(u)=0$. Therefore,
the Gagliardo-Nirenberg inequality implies

$\displaystyle E_{\mu}(u)$ $\displaystyle=\frac{1}{N}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u\rVert_{q}^{q}$ $\displaystyle=\frac{1}{N}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}C_{N,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}},$

where we used the fact that $\lVert u\rVert_{p}\leqslant a$ by the Fatou
lemma. we introduce the function

$\varphi(t)=\frac{1}{N}t^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}C_{N,q}^{q}a^{q(1-\gamma_{q})}t^{q\gamma_{q}},\quad
t>0.$

Then, we can check that $\varphi$ has a unique critical point on
$(0,+\infty)$, which is a global minimum point at negative level. The critical
point is

$\bar{t}=\bigg{(}\frac{\mu\gamma_{q}(p^{*}-q\gamma_{q})NC_{N.q}^{q}}{pp^{*}}\bigg{)}^{\frac{1}{p-q\gamma_{q}}}a^{\frac{q(1-\gamma_{q})}{p-q\gamma_{q}}},$

and the minimum is

$\varphi(\bar{t})=-\frac{p-q\gamma_{q}}{q}(N\gamma_{q})^{\frac{q\gamma_{q}}{p-q\gamma_{q}}}\bigg{(}\mu
a^{q(1-\gamma_{q})}\frac{(p^{*}-q\gamma_{q})C_{N,q}^{q}}{pp^{*}}\bigg{)}^{\frac{p}{p-q\gamma_{q}}}<0.$

By (1.9), since $\mu a^{q(1-\gamma_{q})}<\alpha(N,p,q)\leqslant
C^{\prime\prime}$, we have $\varphi(\bar{t})\geqslant-S^{N/p}/N$ which
contradicts with (4.6). This means that $u_{n}\rightarrow u$ in
$W^{1,p}(\mathbb{R}^{N})$ and $u\in S_{a}$ is a solution to (1.1) for some
$\lambda<0$. Since $u_{n}$ is radially symmetric and radially decreasing, we
know $u$ is radially symmetric and radially non-increasing. Now, by the strong
maximum principle , $u$ is positive.

In fact, $u$ is a ground state, since
$E_{\mu}(u)=\inf_{\mathcal{P}_{a,\mu}}E_{\mu}$, and any other normalized
solution stays on $\mathcal{P}_{a,\mu}$. It remains to show that any other
ground state is a local minimizer for $E_{\mu}$ on $A_{R_{0}}$. Let $v$ be a
critical point of $E_{\mu}|_{S_{a}}$ and $E_{\mu}(v)=m^{+}(a,\mu)$, then
$v\in\mathcal{P}_{a,\mu}$. By Lemma 4.3, we know $s_{v}=0$ and $\lVert\nabla
v\rVert_{p}\leqslant R_{0}$. Therefore, $v$ is a local minimizer for $E_{\mu}$
on $A_{R_{0}}$.$\hfill\qed$

###  4.3 Existence of mountain-pass type solution

In this section, we prove Theorem 1.2. By Lemma 4.3 and Lemma 4.4, we can
construct a minimax structure. Let

$E^{c}:=\Big{\\{}u\in S_{a}:E_{\mu}(u)\leqslant c\Big{\\}}.$

We introduce the minimax class

$\Gamma:=\Big{\\{}\gamma=(\alpha,\theta)\in C([0,1],\mathbb{R}\times
S_{a,r}):\gamma(0)\in(0,\mathcal{P}_{a,\mu}^{+}),\gamma(1)\in(0,E^{2m^{+}(a,\mu)})\Big{\\}},$

with associated minimax level

$\sigma(a,\mu):=\inf_{\gamma\in\Gamma}\max_{(\alpha,\theta)\in\gamma([0,1])}\tilde{E}_{\mu}(\alpha,\theta),$

where

$\tilde{E}_{\mu}(s,u):=E_{\mu}(s*u).$

In order to obtain the compactness of PS sequence by using Proposition 3.1,
the following energy estimates are required.

######  Lemma 4.5.

Let $N\geqslant p^{2}$. Then, we have

$m^{-}(a,\mu)<m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}}.$

###### Proof.

By Appendix A, we have

$\lVert\nabla
u_{\varepsilon}\rVert_{p}^{p}=S^{\frac{N}{p}}+O(\varepsilon^{\frac{N-p}{p-1}}),\quad\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}}=S^{\frac{N}{p}}+O(\varepsilon^{\frac{N}{p-1}}),$

and

$\lVert
u_{\varepsilon}\rVert_{r}^{r}=\left\\{\begin{array}[]{ll}C\varepsilon^{N-\frac{(N-p)r}{p}}+O({\varepsilon^{\frac{(N-p)r}{p(p-1)}}})&N>p^{2}\
or\ p<r<p^{*}\\\
C\varepsilon^{p}\lvert\log\varepsilon\rvert+O(\varepsilon^{p})&N=p^{2}\ and\
r=p,\end{array}\right.$

where $p\leqslant r<p^{*}$.

Let $V_{\varepsilon,\tau}=u_{a,\mu}^{+}+\tau u_{\varepsilon}(\cdot-
x_{\varepsilon})$ and

$W_{\varepsilon,\tau}(x)=\big{(}a^{-1}\lVert
V_{\varepsilon,\tau}\rVert_{p}\big{)}^{\frac{N-p}{p}}V_{\varepsilon,\tau}\big{(}a^{-1}\lVert
V_{\varepsilon,\tau}\rVert_{p}x\big{)},$ (4.7)

where $\lvert x_{\varepsilon}\rvert=\varepsilon^{-1}$. Then, we have

$\lVert W_{\varepsilon,\tau}\rVert_{p}^{p}=a^{p},\quad\lVert\nabla
W_{\varepsilon,\tau}\rVert_{p}^{p}=\lVert\nabla
V_{\varepsilon,\tau}\rVert_{p}^{p},\quad\lVert
W_{\varepsilon,\tau}\rVert_{p^{*}}^{p^{*}}=\lVert
V_{\varepsilon,\tau}\rVert_{p^{*}}^{p^{*}},$

and

$\lVert W_{\varepsilon,\tau}\rVert_{q}^{q}=(a\lVert
V_{\varepsilon,\tau}\rVert_{p}^{-1})^{q(1-\gamma_{q})}\lVert
V_{\varepsilon,\tau}\rVert_{q}^{q}.$

Thus, there exists a unique $t_{\varepsilon,\tau}\in\mathbb{R}$ such that
$t_{\varepsilon,\tau}\star W_{\varepsilon,\tau}\in\mathcal{P}_{a,\mu}^{-}$,
that is

$e^{pt_{\varepsilon,\tau}}\lVert\nabla
W_{\varepsilon,\tau}\rVert_{p}^{p}=\mu\gamma_{q,s}e^{q\gamma_{q}t_{\varepsilon,\tau}}\lVert
W_{\varepsilon,\tau}\rVert_{q}^{q}+e^{p^{*}t_{\varepsilon,\tau}}\lVert
W_{\varepsilon,\tau}\rVert_{p^{*}}^{p^{*}}.$ (4.8)

Since $u_{a,\mu}^{+}\in\mathcal{P}_{a,\mu}^{+}$, we know
$t_{\varepsilon,0}>0$. By (4.8),

$e^{(p^{*}-p)t_{\varepsilon,\tau}}<\frac{\lVert\nabla
W_{\varepsilon,\tau}\rVert_{p}^{p}}{\lVert
W_{\varepsilon,\tau}\rVert_{p^{*}}^{p^{*}}}=\frac{\lVert\nabla
V_{\varepsilon,\tau}\rVert_{p}^{p}}{\lVert
V_{\varepsilon,\tau}\rVert_{p^{*}}^{p^{*}}},$

which implies $t_{\varepsilon,\tau}\rightarrow-\infty$ as
$\tau\rightarrow+\infty$. By Lemma 4.3 , $t_{\varepsilon,\tau}$ is continuous
for $\tau$, hence we can choose a suitable $\tau=\tau_{\varepsilon}>0$ such
that $t_{\varepsilon,\tau_{\varepsilon}}=0$. It follows that

$\displaystyle m^{-}(a,\mu)$ $\displaystyle\leqslant
E_{\mu}(W_{\varepsilon,\tau_{\varepsilon}})=\frac{1}{p}\lVert\nabla
W_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}-\frac{\mu}{q}\lVert
W_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}-\frac{1}{p^{*}}\lVert
W_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}$
$\displaystyle=\frac{1}{p}\lVert\nabla
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}-\frac{\mu}{q}(a\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{-1})^{q(1-\gamma_{q})}\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}-\frac{1}{p^{*}}\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}.$ (4.9)

If $\liminf_{\varepsilon\rightarrow 0}\tau_{\varepsilon}=0$ or
$\limsup_{\varepsilon\rightarrow 0}\tau_{\varepsilon}=+\infty$, then

$m^{-}(a,\mu)\leqslant\liminf_{\varepsilon\rightarrow
0}E_{\mu}(W_{\varepsilon,\tau_{\varepsilon}})\leqslant
E_{\mu}(u_{a,\mu}^{+})=m^{+}(a,\mu),$

a contradiction with Lemma 4.4. Therefore, there exists $t_{2}>t_{1}>0$
independent of $\varepsilon$ such that $\tau_{\varepsilon}\in[t_{1},t_{2}]$.

Now, we estimate $\lVert\nabla
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}$. Using the inequality

$(a+b)^{p}\leqslant a^{p}+b^{p}+C(a^{p-1}b+ab^{p-1})\quad\forall a,b\geqslant
0,$

we have

$\displaystyle\lVert\nabla V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}$
$\displaystyle\leqslant\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}+\tau_{\varepsilon}^{p}\lVert\nabla
u_{\varepsilon}\rVert_{p}^{p}$
$\displaystyle\qquad\qquad+C\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{p-1}\lvert\nabla u_{\varepsilon}\rvert
dx+C\int_{\mathbb{R}^{N}}\lvert\nabla u_{a,\mu}^{+}\rvert\lvert\nabla
u_{\varepsilon}\rvert^{p-1}dx$ $\displaystyle=\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}+S^{\frac{N}{p}}\tau_{\varepsilon}^{p}$
$\displaystyle\qquad\qquad+C\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{p-1}\lvert\nabla u_{\varepsilon}\rvert
dx+C\int_{\mathbb{R}^{N}}\lvert\nabla u_{a,\mu}^{+}\rvert\lvert\nabla
u_{\varepsilon}\rvert^{p-1}dx+O(\varepsilon^{\frac{N-p}{p-1}}).$ (4.10)

By the Hölder inequality,

$\displaystyle\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{p-1}\lvert\nabla u_{\varepsilon}\rvert dx$
$\displaystyle=\int_{B_{2}(x_{\varepsilon})}\lvert\nabla
u_{a,\mu}^{+}\rvert^{p-1}\lvert\nabla u_{\varepsilon}\rvert dx$
$\displaystyle\leqslant\lVert\nabla
u_{a,\mu}^{+}\rVert_{L^{p}(B_{2}(x_{\varepsilon}))}^{p-1}\lVert\nabla
u_{\varepsilon}\rVert_{p}$ $\displaystyle\leqslant C\lVert\nabla
u_{a,\mu}^{+}\rVert_{L^{p}(B_{2}(x_{\varepsilon}))}^{p-1}.$

We know there exists $\lambda_{a,\mu}^{+}<0$ such that

$-\Delta_{p}u_{a,\mu}^{+}=\lambda_{a,\mu}^{+}\lvert
u_{a,\mu}^{+}\rvert^{p-1}+\mu\lvert u_{a,\mu}^{+}\rvert^{q-1}+\lvert
u_{a,\mu}^{+}\rvert^{p^{*}-1}.$

Then, by [11, theorem 8], there exists $a,b>0$ such that

$\lvert\nabla u_{a,\mu}^{+}(x)\rvert\leqslant ae^{-b\lvert x\rvert}$

for $\lvert x\rvert$ sufficiently large. This means

$\lVert\nabla u_{a,\mu}^{+}\rVert_{L^{p}(B_{2}(x_{\varepsilon}))}\leqslant
Ce^{-b\varepsilon^{-1}},$

and hence

$\int_{\mathbb{R}^{N}}\lvert\nabla u_{a,\mu}^{+}\rvert^{p-1}\lvert\nabla
u_{\varepsilon}\rvert dx\leqslant Ce^{e^{-b(p-1)\varepsilon^{-1}}}.$

Similarly, we have

$\int_{\mathbb{R}^{N}}\lvert\nabla u_{a,\mu}^{+}\rvert\lvert\nabla
u_{\varepsilon}\rvert^{p-1}dx\leqslant Ce^{e^{-b\varepsilon^{-1}}}.$

Therefore, by (4.3), we obtain

$\lVert\nabla
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}\leqslant\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}+S^{\frac{N}{p}}\tau_{\varepsilon}^{p}+O\big{(}\varepsilon^{\frac{N-p}{p-1}}\big{)}.$
(4.11)

Next, we estimate $\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}$.
We have

$\displaystyle\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}+\tau_{\varepsilon}u_{\varepsilon}\rvert^{p}$
$\displaystyle=\int_{B_{2}(x_{\varepsilon})}\lvert
u_{a,\mu}^{+}+\tau_{\varepsilon}u_{\varepsilon}\rvert^{p}dx+\int_{B_{2}^{c}(x_{\varepsilon})}\lvert
u_{a,\mu}^{+}\rvert^{p}dx$ $\displaystyle\leqslant
a^{p}+\int_{B_{2}(x_{\varepsilon})}\lvert
u_{a,\mu}^{+}+\tau_{\varepsilon}u_{\varepsilon}\rvert^{p}dx$
$\displaystyle\leqslant a^{p}+C\int_{B_{2}(x_{\varepsilon})}\big{(}\lvert
u_{a,\mu}^{+}\rvert^{p}+\lvert u_{\varepsilon}\rvert^{p}\big{)}dx$
$\displaystyle\leqslant
a^{p}+C(e^{-pb\varepsilon^{-1}}+\varepsilon^{p}\lvert\log\varepsilon\rvert)$
$\displaystyle=a^{p}+O(\varepsilon^{p}\lvert\log\varepsilon\rvert).$

Thus,

$\big{(}a\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{-1}\big{)}^{q(1-\gamma_{q})}\geqslant
1+O(\varepsilon^{p}\lvert\log\varepsilon\rvert).$ (4.12)

It is easy to know that

$\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}\geqslant\lvert
u_{a,\mu}^{+}\rVert_{q}^{q}+\tau_{\varepsilon}^{q}\lvert
u_{\varepsilon}\rvert_{q}^{q}\geqslant\lvert
u_{a,\mu}^{+}\rVert_{q}^{q}+C\varepsilon^{N-\frac{(N-p)q}{p}},$ (4.13)

and

$\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}\geqslant\lVert
u_{a,\mu}^{+}\rVert_{p^{*}}^{p^{*}}+\tau_{\varepsilon}^{p^{*}}\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}}=\lVert
u_{a,\mu}^{+}\rVert_{p^{*}}^{p^{*}}+S^{\frac{N}{p}}\tau_{\varepsilon}^{p^{*}}+O(\varepsilon^{\frac{N}{p-1}}).$
(4.14)

Combining (4.3), (4.11), (4.12) (4.13) and (4.14), we obtain

$\displaystyle m^{-}(a,\mu)$ $\displaystyle\leqslant
m^{+}(a,\mu)+S^{\frac{N}{p}}\Big{(}\frac{1}{p}\tau_{\varepsilon}^{p}-\frac{1}{p^{*}}\tau_{\varepsilon}^{p^{*}}\Big{)}-C\varepsilon^{N-\frac{(N-p)q}{p}}+O(\varepsilon^{p}\lvert\log\varepsilon\rvert)$
$\displaystyle<m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}},$

by taking $\varepsilon$ sufficiently small. ∎

######  Lemma 4.6.

Let $N<p^{2}<9$. Then, we have

$m^{-}(a,\mu)<m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}}.$

###### Proof.

We set $V_{\varepsilon,\tau_{\varepsilon}}=u_{a,\mu}^{+}+\tau u_{\varepsilon}$
and the definition of $W_{\varepsilon,\tau_{\varepsilon}}$ is same to (4.7).
Then, we can choose $\tau=\tau_{\varepsilon}>0$ such that
$W_{\varepsilon,\tau_{\varepsilon}}\in\mathcal{P}_{a,\mu}^{-}$. Moreover,
there exists $t_{2}>t_{1}>0$ independent of $\varepsilon$ such that
$\tau_{\varepsilon}\in[t_{1},t_{2}]$. Therefore,

$m^{-}(a,\mu)\leqslant\frac{1}{p}\lVert\nabla
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}-\frac{\mu}{q}(a\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{-1})^{q(1-\gamma_{q})}\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}-\frac{1}{p^{*}}\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}.$ (4.15)

Now, we estimate $\lVert\nabla
V_{\varepsilon,\tau_{{\varepsilon}}}\rVert_{p}^{p}$ and $\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}$. We can prove that for
$a,b\geqslant 0$, there is

$(a^{2}+b^{2}+2ab\cos\alpha)^{\frac{p}{2}}\leqslant
a^{p}+b^{p}+pa^{p-1}b\cos\alpha+Ca^{\frac{p-1}{2}}b^{\frac{p+1}{2}},$

uniformly in $\alpha$. Thus,

$\displaystyle\lVert\nabla V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{p}$
$\displaystyle=\int_{\mathbb{R}^{N}}\big{(}\lvert\nabla
u_{a,\beta}^{+}\rvert^{2}+\tau_{\varepsilon}^{2}\lvert\nabla
u_{\varepsilon}\rvert^{2}+2\tau_{\varepsilon}\nabla u_{a,\mu}^{+}\cdot\nabla
u_{\varepsilon}\big{)}^{\frac{p}{2}}dx$ $\displaystyle\leqslant\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}+\tau_{\varepsilon}^{p}\lVert
u_{\varepsilon}\rVert_{p}^{p}$
$\displaystyle\qquad\qquad+p\tau_{{\varepsilon}}\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{p-2}\nabla u_{a,\mu}^{+}\cdot\nabla
u_{\varepsilon}dx+C\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{\frac{p-1}{2}}\lvert\nabla
u_{\varepsilon}\rvert^{\frac{p+1}{2}}dx.$ (4.16)

By [31], $\nabla u_{a,\mu}^{+}$ is local Hölder continuous, hence

$\int_{\mathbb{R}^{N}}\lvert\nabla
u_{a,\mu}^{+}\rvert^{\frac{p-1}{2}}\lvert\nabla
u_{\varepsilon}\rvert^{\frac{p+1}{2}}dx\leqslant
C\int_{\mathbb{R}^{N}}\lvert\nabla
u_{\varepsilon}\rvert^{\frac{p+1}{2}}dx=O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}}).$
(4.17)

By (1.1), we know

$\displaystyle\int_{\mathbb{R}^{N}}\lvert\nabla u_{a,\mu}\rvert^{p-2}\nabla
u_{a,\mu}^{+}\cdot\nabla u_{\varepsilon}dx$ $\displaystyle=$
$\displaystyle\lambda_{a,\mu}^{+}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p-1}u_{\varepsilon}dx+\mu\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{q-1}u_{\varepsilon}dx+\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p^{*}-1}u_{\varepsilon}dx.$ (4.18)

From (4.3), (4.17) and (4.3), we obtain

$\begin{split}\lVert\nabla
V_{\varepsilon,\tau_{{\varepsilon}}}\rVert_{p}^{p}&\leqslant\lVert\nabla
u_{a,\mu}^{+}\rVert_{p}^{p}+S^{\frac{N}{p}}\tau_{\varepsilon}^{p}+p\tau_{\varepsilon}\lambda_{a,\mu}^{+}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p-1}u_{\varepsilon}dx\\\
&\qquad\qquad+p\tau_{\varepsilon}\mu\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{q-1}u_{\varepsilon}dx+p\tau_{\varepsilon}\lvert
u_{a,\mu}^{+}\rvert^{p^{*}-1}u_{\varepsilon}dx+O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}}).\end{split}$
(4.19)

In the same way, we have

$\displaystyle\lVert V_{\varepsilon}\rVert_{p}^{p}$
$\displaystyle\leqslant\lVert
u_{a,\mu}^{+}\rVert_{p}^{p}+\tau_{\varepsilon}^{p}\lVert
u_{\varepsilon}\rVert_{p}^{p}+p\tau_{\varepsilon}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert u_{\varepsilon}dx+C\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{\frac{p-1}{2}}\lvert
u_{\varepsilon}\rvert^{\frac{p+1}{2}}dx$ $\displaystyle\leqslant
a^{p}+p\tau_{\varepsilon}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p-1}u_{\varepsilon}dx+O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}}),$

which implies

$\big{(}a\lVert
V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p}^{-1}\big{)}^{q(1-\gamma_{q})}\geqslant
1-\frac{q(1-\gamma_{q})\tau_{\varepsilon}}{a^{p}}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p-1}u_{\varepsilon}dx+O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}}).$
(4.20)

Next, we estimate $\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}$
and $\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}$. For
every $a,b\geqslant 0$, we know

$(a+b)^{r}\geqslant\left\\{\begin{array}[]{ll}a^{r}+b^{r}+r(a^{r-1}b+ab^{r-1})&\forall
r\geqslant 3\\\ a^{r}+ra^{r-1}b&\forall r\geqslant 1.\end{array}\right.$

Thus, we have

$\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{q}^{q}\geqslant\lVert
u_{a,\mu}^{+}\rVert_{q}^{q}+q\tau_{\varepsilon}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{q-1}u_{\varepsilon},$ (4.21)

and

$\displaystyle\lVert V_{\varepsilon,\tau_{\varepsilon}}\rVert_{p^{*}}^{p^{*}}$
$\displaystyle\geqslant\lVert
u_{a,\mu}^{+}\rVert_{p^{*}}^{p^{*}}+\tau_{\varepsilon}^{p^{*}}\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}}$
$\displaystyle\qquad\qquad+p^{*}\tau_{\varepsilon}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p^{*}-1}u_{\varepsilon}dx+p^{*}\tau_{\varepsilon}^{p^{*}-1}\int_{\mathbb{R}^{N}}u_{a,\mu}^{+}\lvert
u_{\varepsilon}\rvert^{p^{*}-1}dx$ $\displaystyle=\lVert
u_{a,\mu}^{+}\rVert_{p^{*}}^{p^{*}}+S^{\frac{N}{p}}\tau_{\varepsilon}^{p^{*}}$
$\displaystyle\qquad\qquad+p^{*}\tau_{\varepsilon}\int_{\mathbb{R}^{N}}\lvert
u_{a,\mu}^{+}\rvert^{p^{*}-1}u_{\varepsilon}dx+p^{*}\tau_{\varepsilon}^{p^{*}-1}\int_{\mathbb{R}^{N}}u_{a,\mu}^{+}\lvert
u_{\varepsilon}\rvert^{p^{*}-1}dx+O(\varepsilon^{\frac{N}{p-1}}).$ (4.22)

Combining (4.6), (4.19), (4.20), (4.21), (4.3) and using
$\lambda_{a,\mu}^{+}a^{p}=\mu(\gamma_{q}-1)\lVert
u_{a,\mu}^{+}\rVert_{q}^{q}$, we obtain

$\displaystyle m^{-}(a,\mu)$ $\displaystyle\leqslant
m^{+}(a,\mu)+S^{\frac{N}{p}}\Big{(}\frac{1}{p}\tau_{\varepsilon}^{p}-\frac{1}{p^{*}}\tau_{\varepsilon}^{p^{*}}\Big{)}-\tau_{\varepsilon}^{p^{*}-1}\int_{\mathbb{R}^{N}}u_{a,\mu}^{+}u_{\varepsilon}^{p^{*}-1}dx+O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}})$
$\displaystyle\leqslant
m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}}-C\varepsilon^{\frac{N-p}{p}}+O(\varepsilon^{\frac{(N-p)(p+1)}{2p(p-1)}})$
$\displaystyle<m^{+}(a,\mu)+\frac{1}{N}S^{\frac{N}{p}},$

by taking $\varepsilon$ sufficiently small. ∎

######  Remark 4.1.

Although Lemma 4.5 and Lemma 4.6 both obtain the same estimates, the method of
proof were slightly different, so we state these two results separately.

For every $0<a<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$, let
$u\in\mathcal{P}_{a,\mu}^{\pm}$, then $u_{b}=\frac{b}{a}u\in S_{b}$ for every
$b>0$. By lemma 4.3, there exists unique $t_{\pm}(b)\in\mathbb{R}$ such that
$t_{\pm}(b)\star u_{b}\in\mathcal{P}_{b,\mu}^{\pm}$ for every
$0<b<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$. Clearly,
$t_{\pm}(a)=0$.

######  Lemma 4.7.

For every $0<a<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$,
$t^{\prime}_{\pm}(a)$ exist and

$t^{\prime}_{\pm}(a)=\frac{\mu q\gamma_{q}\lVert u\rVert_{q}^{q}+p^{*}\lVert
u\rVert_{p^{*}}^{p^{*}}-p\lVert\nabla u\rVert_{p}^{p}}{a\big{(}p\lVert\nabla
u\rVert_{p}^{p}-\mu q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}-p^{*}\lVert
u\rVert_{p^{*}}^{p^{*}}\big{)}}.$ (4.23)

###### Proof.

Since $t_{\pm}(b)\star u_{b}\in\mathcal{P}_{b,\mu}^{\pm}$, we have

$\Big{(}\frac{b}{a}\Big{)}^{p}e^{pt_{\pm}(b)}\lVert\nabla
u\rVert_{p}^{p}=\mu\gamma_{q}\Big{(}\frac{b}{a}\Big{)}^{q}e^{q\gamma_{q}t_{\pm}(b)}\lVert
u\rVert_{q}^{q}+\Big{(}\frac{b}{a}\Big{)}^{p^{*}}e^{p^{*}t_{\pm}(b)}\lVert
u\rVert_{p^{*}}^{p^{*}}.$

Considering the function

$\Phi(b,t)=\Big{(}\frac{b}{a}\Big{)}^{p}e^{pt}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{b}{a}\Big{)}^{q}e^{q\gamma_{q}t}\lVert
u\rVert_{q}^{q}-\Big{(}\frac{b}{a}\Big{)}^{p^{*}}e^{p^{*}t}\lVert
u\rVert_{p^{*}}^{p^{*}},$

then $\Phi(a,0)=0$ and $\Phi(b,t)$ has a continuous derivative in some
neighborhood of $(a,0)$. Moreover, since $u\in\mathcal{P}_{a,\mu}^{\pm}$, we
have

$\partial_{t}\Phi(a,0)=p\lVert\nabla u\rVert_{p}^{p}-\mu q\gamma_{q}^{2}\lVert
u\rVert_{q}^{q}-p^{*}\lVert u\rVert_{p^{*}}^{p^{*}}\neq 0.$

Now, by the implict function theorem, we know $t^{\prime}_{\pm}(a)$ exist and
(4.23) holds. ∎

######  Lemma 4.8.

$m^{\pm}(a,\mu)$ is non-increasing for
$0<a<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$.

###### Proof.

Since

$E_{\mu}\big{(}t_{\pm}(b)\star
u_{b}\big{)}=\frac{1}{p}\Big{(}\frac{b}{a}\Big{)}^{p}e^{pt_{\pm}(b)}\lVert\nabla
u\rVert_{p}^{p}-\frac{\mu}{q}\Big{(}\frac{b}{a}\Big{)}^{q}e^{q\gamma_{q}t_{\pm}(b)}\lVert
u\rVert_{q}^{q}-\frac{1}{p^{*}}\Big{(}\frac{b}{a}\Big{)}^{p^{*}}e^{p^{*}t_{\pm}(b)}\lVert
u\rVert_{p^{*}}^{p^{*}},$

and $u\in\mathcal{P}_{a,\mu}^{\pm}$, we have

$\displaystyle\frac{dE_{\mu}\big{(}t_{\pm}(b)\star u_{b}\big{)}}{db}|_{b=a}$
$\displaystyle=\frac{1}{a}\big{(}\lVert\nabla u\rVert_{p}^{p}-\mu\lVert
u\rVert_{q}^{q}-\lVert u\rVert_{p^{*}}^{p^{*}}\big{)}+\big{(}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\lVert u\rVert_{q}^{q}-\lVert
u\rVert_{p^{*}}^{p^{*}}\big{)}t^{\prime}_{\pm}(a)$
$\displaystyle=\frac{\mu(\gamma_{q}-1)\lVert u\rVert_{q}^{q}}{a}<0,$

which implies $E_{\mu}\big{(}t_{\pm}(b)\star u_{b}\big{)}<E_{\mu}(u)$ for
$a<b<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$. Therefore,
$m^{\pm}(a,\mu)\geqslant m^{\pm}(b,\mu)$ for
$a<b<\big{(}\mu^{-1}\alpha\big{)}^{1/(q(\gamma_{q}-1))}$. ∎

######  Lemma 4.9.

We have $m^{-}(a,\mu)=m_{r}^{-}(a,\mu)=\sigma(a,\mu)$.

###### Proof.

By the definition of $m^{-}(a,\mu)$ and $m_{r}^{-}(a,\mu)$, we have
$m^{-}(a,\mu)\leqslant m_{r}^{-}(a,\mu)$. For every
$u\in\mathcal{P}_{a,\mu}^{-}$, let $v=\lvert u\rvert^{*}$, the Schwarz
rearrangement of $\lvert u\rvert$, then

$E_{\mu}(s\star v)\leqslant E_{\mu}(s\star u)\quad\forall s\in\mathbb{R}.$

Therefore, by lemma 4.3,

$m_{r}^{-}(a,\mu)\leqslant E_{\mu}(t_{v}\star v)\leqslant E_{\mu}(t_{v}\star
u)\leqslant E_{\mu}(u),$

which implies $m_{r}^{-}(a,\mu)\leqslant m^{-}(a,\mu)$.

Next, we prove $m_{r}^{-}(a,\mu)=\sigma(a,\mu)$. For every
$u\in\mathcal{P}_{a,\mu}^{-}\cap S_{a,r}$, choosing $s_{0}$ such that
$E_{\mu}(s_{0}\star u)\leqslant 2m^{+}(a,\mu)$ and defining

$\gamma_{u}:\tau\in[0,1]\longmapsto\big{(}0,((1-\tau)s_{u}+\tau s_{0}\star
u)\big{)}\in\mathbb{R}\times S_{a,r}.$

By lemma 4.3, $\gamma_{u}\in\Gamma$. Thus

$\displaystyle\sigma(a,\mu)$
$\displaystyle\leqslant\max_{\tau\in[0,1]}\tilde{E}_{\mu}\big{(}0,((1-\tau)s_{u}+\tau
s_{0})\star u\big{)}$
$\displaystyle=\max_{\tau\in[0,1]}E_{\mu}\big{(}((1-\tau)s_{u}+\tau
s_{0})\star u\big{)}$ $\displaystyle\leqslant E_{\mu}(t_{u}\star
u)=E_{\mu}(u),$

which implies $\sigma(a,\mu)\leqslant m_{r}^{-}(a,\mu)$.

For every $\gamma\in\Gamma$, we have $\gamma(0)\in(0,\mathcal{P}_{a,\mu}^{+})$
and $\gamma(1)\in E^{2m^{+}(a,\mu)}$. Then, by Lemma 4.3, we know
$t_{\theta(0)}>0>t_{\theta(1)}$ and since $t_{\alpha(\tau)\star\theta(\tau)}$
is continuous for $\tau$, there exists $\tau_{\gamma}\in[0,1]$ such that
$t_{\alpha(\tau_{\gamma})\star\theta(\tau_{\gamma})}=0$. This implies

$\max_{(\alpha,\theta)\in\gamma([0,1])}\tilde{E}_{\mu}(\alpha,\theta)\geqslant\tilde{E}_{\mu}\big{(}\alpha(\tau_{\gamma}),\theta(\tau_{\gamma})\big{)}=E_{\mu}\big{(}\alpha(\tau_{\gamma})\star\theta(\tau_{\gamma})\big{)}\geqslant
m_{r}^{-}(a,\mu).$

Therefore, $\sigma(a,\mu)\geqslant m_{r}(a,\mu)$. ∎

Proof of Theorem 1.2: Let

$X=\mathbb{R}\times S_{a,r},\quad\mathcal{F}=\Gamma,\quad and\quad
B=(0,A_{k})\cup(0,E^{2m^{+}(a,\mu)}).$

Then, using the terminology in [12, definition 5.1], $\Gamma$ is a homotopy
stable family of compact subset of $\mathbb{R}\times S_{a,r}$ with extend
closed boundary $(0,A_{k})\cup(0,E^{2m^{+}(a,\mu)})$. Let

$\varphi=\tilde{E}_{\mu}(s,u),\quad c=\sigma(a,\mu),\quad and\quad
F=\Big{\\{}(s,u)\in\mathbb{R}\times S_{a,r}:\tilde{E}_{\mu}(s,u)\geqslant
c\Big{\\}},$

we can check that $F$ satisfies assumptions (F’1) and (F’2) in [12, theorem
5.2].

Taking a minimizing sequence
$\\{\gamma_{n}=(\alpha_{n},\theta_{n})\\}\subset\Gamma$ for $\sigma(a,\mu)$
with properties that $\alpha_{n}\equiv 0$ and $\theta_{n}\geqslant 0$ for
every $\tau\in[0,1]$(if this is not the case, we just have to notice that
$\\{(0,\alpha_{n}\star\theta_{n})\\}$ is also a minimizing sequence). Then, by
[12, theorem 5.2], there exists a PS sequence $\\{(s_{n}\star
w_{n})\\}\subset\mathbb{R}\times S_{a,r}$ for
$\tilde{E}_{\mu}|_{\mathbb{R}\times S_{a,r}}$ at level $\sigma(a,\mu)$, that
is

$\partial_{s}\tilde{E}_{\mu}(s_{n},w_{n})\rightarrow 0,\quad
and\quad\lVert\partial_{u}\tilde{E}_{\mu}(s_{n},w_{n})\rVert_{(T_{w_{n}}S_{a,r})^{*}}\rightarrow
0\quad as\quad n\rightarrow\infty.$ (4.24)

Moreover,

$\lvert
s_{n}\rvert+dist_{W^{1,p}}\big{(}w_{n},\theta_{n}([0,1])\big{)}\rightarrow
0\quad as\quad n\rightarrow\infty.$ (4.25)

Thus, we have

$E_{\mu}(s_{n}\star
w_{n})=\tilde{E}_{\mu}(s_{n},w_{n})\rightarrow\sigma(a,\mu),\quad as\quad
n\rightarrow\infty,$

and

$\displaystyle dE_{\mu}(s_{n}\star w_{n})(s_{n}\star\varphi)$
$\displaystyle=\partial_{u}\tilde{E}_{\mu}(0,s_{n}\star
w_{n})(s_{n}\star\varphi)$
$\displaystyle=\partial_{u}\tilde{E}_{\mu}(s_{n},w_{n})\varphi$
$\displaystyle=o(1)\lVert\varphi\rVert=o(1)\lVert s_{n}\star\varphi\rVert$

for every $\varphi\in T_{w_{n}}S_{a,r}$, which implies
$\\{u_{n}\\}:=\\{s_{n}\star w_{n}\\}$ is a PS sequence for
$E_{\mu}|_{S_{a,r}}$ at level $\sigma(a,\mu)$. Since $E_{\mu}$ is invariant
under rotations, by [20, theorem 2.2], $\\{u_{n}\\}$ is also a PS sequence for
$E_{\mu}|_{S_{a}}$ at level $\sigma(a,\mu)$.

From (4.24), we have

$P_{\mu}(u_{n})=P_{\mu}(s_{n}\star
w_{n})=\partial_{s}\tilde{E}_{\mu}(s_{n},w_{n})\rightarrow 0$

as $n\rightarrow\infty$. Thus, by proposition 3.1, Lemma 4.5 and 4.6, one of
the cases in proposition 3.1 holds. If case (i) occurs, we have
$u_{n}\rightharpoonup u$ in $W^{1,p}(\mathbb{R}^{N})$ and

$E_{\mu}(u)\leqslant m^{-}(a,\mu)-\frac{1}{N}S^{\frac{N}{p}}.$ (4.26)

Since $u$ solves (1.1) for some $\lambda<0$, by Theorem 1.1 and Lemma 4.8,

$E_{\mu}(u)\geqslant m^{+}\big{(}\lVert u\rVert_{p},\mu\big{)}\geqslant
m^{+}(a,\mu).$

therefore,

$m^{+}(a,\mu)\leqslant m^{-}(a,\mu)-\frac{1}{N}S^{\frac{N}{p}},$

which contradicts with Lemma 4.5 and 4.6. This implies that case (ii) in
proposition 3.1 holds, that is $u_{n}\rightarrow u\in S_{a,r}$ in
$W^{1,p}(\mathbb{R}^{N})$, and $u$ solves (1.1) for some $\lambda<0$.
Moreover, noticing that $\theta_{n}(\tau)\geqslant 0$ for every
$\tau\in[0,1]$, then (4.25) implies $u$ is non-negative and hence positive by
strong maximum principle.$\hfill\qed$

##  5 Existence result to the case $q=p+\frac{p^{2}}{N}$

In this section, we prove Theorem 1.3 for $q=p+\frac{p^{2}}{N}$. Firstly, we
analyzed the properties of $E_{\mu}$ and $\mathcal{P}_{a,\mu}$, and then we
construct a mini-max structure.

######  Lemma 5.1.

$\mathcal{P}_{a,\mu}^{0}=\emptyset$, and $\mathcal{P}_{a,\mu}$ is a smooth
manifold of co-dimension $2$ in $W^{1,p}(\mathbb{R}^{N})$.

###### Proof.

If $u\in\mathcal{P}_{a,\mu}^{0}$, we have

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}},\quad and\quad p\lVert\nabla u\rVert_{p}^{p}=\mu
q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}+p^{*}\lVert u\rVert_{p^{*}}^{p^{*}},$

which implies $\lVert u\rVert_{p^{*}}=0$(since $q\gamma_{q}=p$). But we know
this impossible since $u\in S_{a}$. The rest of the proof is similar to one of
the Lemma 4.1, and hence is omitted. ∎

######  Lemma 5.2.

For every $u\in S_{a}$, the function $\Psi_{u}^{\mu}$ has a unique critical
point $t_{u}$ which is a strict maximum point at positive level. Moreover:

(i) $\mathcal{P}_{a,\mu}=\mathcal{P}_{a,\mu}^{-}$, and $s\star
u\in\mathcal{P}_{a,\mu}$ if and only if $s=t_{u}$.

(ii) $t_{u}<0$ is and only if $P_{\mu}(u)<0$.

(iii) The map $u\in S_{a}\longmapsto t_{u}\in\mathbb{R}$ is of class $C^{1}$.

###### Proof.

Here we just prove $\mathcal{P}_{a,\mu}=\mathcal{P}_{a,\mu}^{-}$, the rest of
the proof is similar to Lemma 4.3. For every $u\in\mathcal{P}_{a,\mu}$, we
have $t_{u}=0$ and hence $0$ is a strict maximum point of $\Psi_{u}^{\mu}$.
Now, $\big{(}\Psi_{u}^{\mu}\big{)}^{\prime\prime}(0)\leqslant 0$ implies
$u\in\mathcal{P}_{a,\mu}^{0}\cup\mathcal{P}_{a,\mu}^{-}$. By lemma 5.1, we
obtain $u\in\mathcal{P}_{a,\mu}^{-}$. ∎

######  Lemma 5.3.

We have $m(a,\mu)=m^{-}(a,\mu)>0$.

###### Proof.

By lemma 5.2, we know $m(a,\mu)=m^{-}(a,\mu)$. If $u\in\mathcal{P}_{a,\mu}$,
then

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}.$

Using the Gagliardo-Nirenberg inequality and Sobolev inequality, we have

$\lVert\nabla
u\rVert_{p}^{p}\leqslant\frac{\mu}{q}C_{N,q}^{q}a^{\frac{p^{2}}{N}}\lVert\nabla
u\rVert_{p}^{p}+S^{-\frac{p^{*}}{p}}\lVert\nabla u\rVert_{p}^{p^{*}}.$

Combining (1.11), we derive that

$\inf_{u\in\mathcal{P}_{a,\mu}}\lVert\nabla u\rVert_{p}>0.$

For every $u\in\mathcal{P}_{a,\mu}$, we have

$E_{\mu}(u)=\frac{1}{N}\lVert\nabla u\rVert_{p}^{p}-\frac{p\mu}{Nq}\lVert
u\rVert_{q}^{q}\geqslant\frac{1}{N}\Big{(}1-\frac{p}{q}C_{N,q}^{q}\mu
a^{\frac{p^{2}}{N}}\Big{)}\lVert\nabla u\rVert_{p}^{p},$

which implies $m(a,\mu)>0$. ∎

######  Lemma 5.4.

There exists $k>0$ sufficiently small such that

$0<\sup_{u\in A_{k}}E_{\mu}(u)<m(a,\mu),$

and

$E_{\mu}(u),P_{\mu}(u)>0\quad\forall u\in A_{k},$

where $A_{k}=\Big{\\{}u\in S_{a}:\lVert\nabla u\rVert_{p}\leqslant
k\Big{\\}}$.

###### Proof.

By the Gagliardo-Nirenberg inequality and Sobolev inequality, we have

$P_{\mu}(u)\geqslant\Big{(}1-\frac{p}{q}C_{N,q}^{q}\mu
a^{\frac{p^{2}}{N}}\Big{)}\lVert\nabla
u\rVert_{p}^{p}-S^{-\frac{p^{*}}{p}}\lVert\nabla u\rVert_{p}^{p^{*}},$

and

$\frac{1}{p}\lVert\nabla u\rVert_{p}^{p}\geqslant
E_{\mu}(u)\geqslant\Big{(}\frac{1}{p}-\frac{1}{q}C_{N,q}^{q}\mu
a^{\frac{p^{2}}{N}}\Big{)}\lVert\nabla
u\rVert_{p}^{p}-\frac{1}{pS^{p^{*}/p}}\lVert\nabla u\rVert_{p}^{p^{*}}.$

Thus, we can choose suitable $k>0$ such that the conclusion holds. ∎

By Lemma 5.4, we can construct a mini-max structure. Let

$E^{c}:=\Big{\\{}u\in S_{a}:E_{\mu}(u)\leqslant c\Big{\\}}.$

We introduce the mini-max class

$\Gamma:=\Big{\\{}\gamma=(\alpha,\theta)\in C([0,1],\mathbb{R}\times
S_{a,r}):\gamma(0)\in(0,A_{k}),\gamma(1)\in(0,E^{0})\Big{\\}},$

with associated mini-max level

$\sigma(a,\mu):=\inf_{\gamma\in\Gamma}\max_{(\alpha,\theta)\in\gamma([0,1])}\tilde{E}_{\mu}(\alpha,\theta),$

where

$\tilde{E}_{\mu}(s,u):=E_{\mu}(s*u).$

In order to use Proposition 3.1, we need the following lemmas.

######  Lemma 5.5.

We have $m(a,\mu)=m_{r}(a,\mu)=\sigma(a,\mu)$.

###### Proof.

The proof of $m(a,\mu)=m_{r}(a,\mu)$ is similar to Lemma 4.9, and hence we
omit it. Next, we prove $m_{r}(a,\mu)=\sigma(a,\mu)$.

For every $u\in\mathcal{P}_{a,\mu}\cap S_{a,r}$. By Lemma 5.2, $t_{u}=0$.
Choosing $s_{0}<0<s_{1}$ such that $\lVert\nabla(s_{0}\star
u)\rVert_{p}\leqslant k$ and $E_{\mu}(s_{1}\star u)\leqslant 0$, and Defining

$\gamma_{u}:\tau\in[0,1]\longmapsto\big{(}0,((1-\tau)s_{0}+\tau s_{1})\star
u\big{)}\in\mathbb{R}\times S_{a,r},$

then $\gamma_{u}\in\Gamma$. Thus

$\displaystyle\sigma(a,\mu)$
$\displaystyle\leqslant\max_{\tau\in[0,1]}\tilde{E}_{\mu}\big{(}0,((1-\tau)s_{0}+\tau
s_{1})\star u\big{)}$
$\displaystyle=\max_{\tau\in[0,1]}E_{\mu}\big{(}((1-\tau)s_{0}+\tau
s_{1})\star u\big{)}$ $\displaystyle\leqslant E_{\mu}(t_{u}\star
u)=E_{\mu}(u),$

which implies $\sigma(a,\mu)\leqslant m_{r}(a,\mu)$.

For every $\gamma\in\Gamma$, since $\gamma(0)\in(0,A_{k})$, by Lemma 5.4, we
have $P_{\mu}\big{(}\theta(0)\big{)}>0$. Now we claim that
$P_{\mu}\big{(}\theta(1)\big{)}<0$. Indeed, since $\gamma(1)\in(0,E^{0})$, we
have $E_{\mu}\big{(}\theta(1)\big{)}\leqslant 0$, that is
$\Psi_{\theta(1)}^{\mu}(0)\leqslant 0$. Then, by Lemma 5.2, $t_{\theta(1)}<0$
and hence $P_{\mu}\big{(}\theta(1)\big{)}<0$. We know
$\tau\longmapsto\alpha(\tau)\star\theta(\tau)$ is continuous for $[0,1]$ to
$W^{1,p}(\mathbb{R}^{N})$, so there exists $\tau_{\gamma}\in(0,1)$ such that
$P_{\mu}\big{(}\alpha(\tau_{\gamma})\star\theta(\tau_{\gamma})\big{)}=0$. This
implies

$\max_{(\alpha,\theta)\in\gamma([0,1])}\tilde{E}_{\mu}(\alpha,\theta)\geqslant\tilde{E}_{\mu}\big{(}\alpha(\tau_{\gamma}),\theta(\tau_{\gamma})\big{)}=E_{\mu}\big{(}\alpha(\tau_{\gamma})\star\theta(\tau_{\gamma})\big{)}\geqslant
m_{r}(a,\mu).$

Therefore, $\sigma(a,\mu)\geqslant m_{r}(a,\mu)$. ∎

######  Lemma 5.6.

We have $m(a,\mu)<\frac{1}{N}S^{\frac{N}{p}}$.

###### Proof.

Let

$W_{\varepsilon}(x)=\big{(}a^{-1}\lVert
u_{\varepsilon,\tau}\rVert_{p}\big{)}^{\frac{N-p}{p}}u_{\varepsilon}\big{(}a^{-1}\lVert
u_{\varepsilon}\rVert_{p}x\big{)},$ (5.1)

Then, we have

$\lVert W_{\varepsilon}\rVert_{p}^{p}=a^{p},\quad\lVert\nabla
W_{\varepsilon}\rVert_{p}^{p}=\lVert\nabla
u_{\varepsilon}\rVert_{p}^{p},\quad\lVert
W_{\varepsilon}\rVert_{p^{*}}^{p^{*}}=\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}},$

and

$\lVert W_{\varepsilon}\rVert_{q}^{q}=(a\lVert
u_{\varepsilon}\rVert_{p}^{-1})^{q(1-\gamma_{q})}\lVert
u_{\varepsilon}\rVert_{q}^{q}.$

Thus, there exists unique $\tau_{\varepsilon}\in\mathbb{R}$ such that
$\tau_{\varepsilon}\star W_{\varepsilon}\in\mathcal{P}_{a,\mu}$. By the
definition of $m(a,\mu)$, we have

$\displaystyle m(a,\mu)$ $\displaystyle\leqslant
E_{\mu}(\tau_{\varepsilon}\star W_{\varepsilon})$
$\displaystyle=\frac{1}{p}e^{p\tau_{\varepsilon}}\lVert\nabla
u_{\varepsilon}\rVert_{p}^{p}-\frac{\mu}{q}e^{q\gamma_{q}\tau_{\varepsilon}}\big{(}a\lVert
u_{\varepsilon}\rVert_{p}^{-1}\big{)}^{q(1-\gamma_{q})}\lVert
u_{\varepsilon}\rVert_{q}^{q}-\frac{1}{p^{*}}e^{p^{*}\tau_{\varepsilon}}\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}}$ (5.2)

If $\liminf_{\varepsilon\rightarrow 0}=-\infty$ or
$\limsup_{\varepsilon\rightarrow 0}=+\infty$, then

$m(a,\mu)\leqslant\liminf_{\varepsilon\rightarrow
0}E_{\mu}(\tau_{\varepsilon}\star W_{\varepsilon})\leqslant 0,$

which contradicts with Lemma 5.4. Therefore, there exists $t_{2}>t_{1}$ such
that $\tau_{\varepsilon}\in[t_{1},t_{2}]$. Now, (5) implies

$\displaystyle m(a,\mu)$ $\displaystyle\leqslant
S^{\frac{N}{p}}\Big{(}\frac{1}{p}e^{p\tau_{\varepsilon}}-\frac{1}{p^{*}}e^{p^{*}\tau_{\varepsilon}}\Big{)}+O(\varepsilon^{\frac{N-p}{p-1}})-C\lVert
u_{\varepsilon}\rVert_{p}^{q(\gamma_{q}-1)}\lVert
u_{\varepsilon}\rVert_{q}^{q}$
$\displaystyle\leqslant\frac{1}{N}S^{\frac{N}{p}}+O(\varepsilon^{\frac{N-p}{p-1}})-\left\\{\begin{array}[]{ll}C&N>p^{2}\\\
C\lvert\log\varepsilon\rvert^{-\frac{p}{N}}&N=p^{2}\\\
C\varepsilon^{\frac{p(p^{2}-N)}{N(p-1)}}&p^{\frac{3}{2}}<N<p^{2}\\\
C\varepsilon^{\frac{p^{3/2}-p}{p-1}}\lvert\log\varepsilon\rvert&N=p^{\frac{3}{2}}\end{array}\right.$
$\displaystyle<\frac{1}{N}S^{\frac{N}{p}},$

by taking $\varepsilon$ sufficiently small. ∎

Now, we give the proof of Theorem 1.3 in the case $q=p+\frac{p^{2}}{N}$.

Proof of Theorem 1.3: Let

$X=\mathbb{R}\times S_{a,r},\quad\mathcal{F}=\Gamma,\quad and\quad
B=(0,A_{k})\cup(0,E^{0}).$

Then, using the terminology in [12, definition 5.1], $\Gamma$ is a homotopy
stable family of compact subset of $\mathbb{R}\times S_{a,r}$ with extend
closed boundary $(0,A_{k})\cup(0,E^{0})$. Let

$\varphi=\tilde{E}_{\mu}(s,u),\quad c=\sigma(a,\mu),\quad and\quad
F=\Big{\\{}(s,u)\in\mathbb{R}\times S_{a,r}:\tilde{E}_{\mu}(s,u)\geqslant
c\Big{\\}},$

we can check that $F$ satisfies assumptions (F’1) and (F’2) in [12, theorem
5.2].

Similar to the proof of Theorem 1.2, there exists a PS sequence
$\\{(s_{n}\star w_{n})\\}\subset\mathbb{R}\times S_{a,r}$ for
$\tilde{E}_{\mu}|_{\mathbb{R}\times S_{a,r}}$ at level $\sigma(a,\mu)$ and we
can check that $\\{u_{n}\\}:=\\{s_{n}\star w_{n}\\}$ is a PS sequence for
$E_{\mu}|_{S_{a,r}}$ at level $\sigma(a,\mu)$. Thus, $\\{u_{n}\\}$ is also a
PS sequence for $E_{\mu}|_{S_{a}}$ at level $\sigma(a,\mu)$.

Since

$P_{\mu}(u_{n})=P_{\mu}(s_{n}\star
w_{n})=\partial_{s}\tilde{E}_{\mu}(s_{n},w_{n})\rightarrow 0$

as $n\rightarrow\infty$, by proposition 3.1 and Lemma 5.6, one of the cases in
proposition 3.1 holds. If case (i) occurs, we have $u_{n}\rightharpoonup u$ in
$W^{1,p}(\mathbb{R}^{N})$ and

$E_{\mu}(u)\leqslant m(a,\mu)-\frac{1}{N}S^{\frac{N}{p}}<0.$ (5.3)

Since $u$ solves (1.1) for some $\lambda<0$, by the Pohozaev identity
$P_{\mu}(u)=0$, we can derive that

$E_{\mu}(u)=\frac{1}{N}\lVert u\rVert_{P^{*}}^{p^{*}}>0,$

a contradiction with (5.3). This implies that case (ii) in proposition 3.1
holds, that is $u_{n}\rightarrow u\in S_{a,r}$ in $W^{1,p}(\mathbb{R}^{N})$,
and $u$ solves (1.1) for some $\lambda<0$. Moreover, we can choose $u$ is non-
negative and hence positive by strong maximum principle.

It remains to show that $u$ is a ground state. This is a direct result by
Proposition 2.1 and Lemma 4.9.$\hfill\qed$

##  6 Existence result for the case $p+\frac{p^{2}}{N}<q<p^{*}$

In this section, we always assume that the condition in Theorem 1.3 holds and
$p+\frac{p^{2}}{N}<q<p^{*}$. We will omit some of the proofs, since it is very
similar to the proofs in Section 5.

######  Lemma 6.1.

$\mathcal{P}_{a,\mu}^{0}=\emptyset$, and $\mathcal{P}_{a,\mu}$ is a smooth
manifold of co-dimension $2$ in $W^{1,p}(\mathbb{R}^{N})$.

###### Proof.

If $u\in\mathcal{P}_{a,\mu}^{0}$, we have

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}},\quad and\quad p\lVert\nabla u\rVert_{p}^{p}=\mu
q\gamma_{q}^{2}\lVert u\rVert_{q}^{q}+p^{*}\lVert u\rVert_{p^{*}}^{p^{*}},$

which implies

$\mu\gamma_{q}(q\gamma_{q}-p)\lVert u\rVert_{q}^{q}+(p^{*}-p)\lVert
u\rVert_{p^{*}}^{p^{*}}=0.$

Since $q\gamma_{q}>p$, we know this impossible. ∎

######  Lemma 6.2.

For every $u\in S_{a}$, the function $\Psi_{u}^{\mu}$ has a unique critical
point $t_{u}$ which is a strict maximum point at positive level. Moreover:

(i) $\mathcal{P}_{a,\mu}=\mathcal{P}_{a,\mu}^{-}$, and $s\star
u\in\mathcal{P}_{a,\mu}$ if and only if $s=t_{u}$.

(ii) $t_{u}<0$ is and only if $P_{\mu}(u)<0$.

(iii) The map $u\in S_{a}\longmapsto t_{u}\in\mathbb{R}$ is of class $C^{1}$.

######  Lemma 6.3.

We have $m(a,\mu)=m^{-}(a,\mu)>0$.

###### Proof.

If $u\in\mathcal{P}_{a,\mu}$, then

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}.$

Using the Gagliardo-Nirenberg inequality and Sobolev inequality, we have

$\lVert\nabla
u\rVert_{p}^{p}\leqslant\frac{\mu}{q}C_{N,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}}+S^{-\frac{p^{*}}{p}}\lVert\nabla
u\rVert_{p}^{p^{*}}.$

Since $q\gamma_{q}>p$, we derive that

$\inf_{u\in\mathcal{P}_{a,\mu}}\lVert\nabla u\rVert_{p}^{p}>0,$

and hence by $P_{\mu}(u)=0$, we know

$\inf_{u\in\mathcal{P}_{a,\mu}}\big{(}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}\big{)}>0$

For every $u\in\mathcal{P}_{a,\mu}$, we have

$E_{\mu}(u)=\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u\rVert_{q}^{q}+\frac{1}{N}\lVert u\rVert_{p^{*}}^{p^{*}},$

which implies $m(a,\mu)>0$. ∎

######  Lemma 6.4.

There exists $k>0$ sufficiently small such that

$0<\sup_{u\in A_{k}}E_{\mu}(u)<m(a,\mu),$

and

$E_{\mu}(u),P_{\mu}(u)>0\quad\forall u\in A_{k},$

where $A_{k}=\Big{\\{}u\in S_{a}:\lVert\nabla u\rVert_{p}\leqslant
k\Big{\\}}$.

###### Proof.

By the Gagliardo-Nirenberg inequality and Sobolev inequality, we have

$P_{\mu}(u)\geqslant\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}C_{N,p,q}^{q}a^{q(1-\gamma_{q})}\lVert
u\rVert_{q}^{q}-S^{-\frac{p^{*}}{p}}\lVert\nabla u\rVert_{p}^{p^{*}},$

and

$\frac{1}{p}\lVert\nabla u\rVert_{p}^{p}\geqslant
E_{\mu}(u)\geqslant\lVert\nabla
u\rVert_{p}^{p}-\frac{\mu}{q}C_{N,p,q}^{q}a^{q(1-\gamma_{q})}\lVert
u\rVert_{q}^{q}-\frac{1}{pS^{p^{*}/p}}\lVert\nabla u\rVert_{p}^{p^{*}}.$

Thus, we can choose suitable $k>0$ such that the conclusion holds. ∎

$E^{c}:=\Big{\\{}u\in S_{a}:E_{\mu}(u)\leqslant c\Big{\\}}.$
$\Gamma:=\Big{\\{}\gamma=(\alpha,\theta)\in C([0,1],\mathbb{R}\times
S_{a,r}):\gamma(0)\in(0,A_{k}),\gamma(1)\in(0,E^{0})\Big{\\}},$
$\sigma(a,\mu):=\inf_{\gamma\in\Gamma}\max_{(\alpha,\theta)\in\gamma([0,1])}\tilde{E}_{\mu}(\alpha,\theta),$
$\tilde{E}_{\mu}(s,u):=E_{\mu}(s*u).$

######  Lemma 6.5.

We have $m(a,\mu)=m_{r}(a,\mu)=\sigma(a,\mu)$.

######  Lemma 6.6.

We have $m(a,\mu)<\frac{1}{N}S^{\frac{N}{p}}$.

###### Proof.

Similar to Lemma 5.6, we have

$\displaystyle m(a,\mu)$
$\displaystyle\leqslant\frac{1}{N}S^{\frac{N}{p}}+O(\varepsilon^{\frac{N-p}{p-1}})-C\lVert
u_{\varepsilon}\rVert_{p}^{q(\gamma_{q}-1)}\lVert
u_{\varepsilon}\rVert_{q}^{q}$
$\displaystyle\leqslant\frac{1}{N}S^{\frac{N}{p}}+O(\varepsilon^{\frac{N-p}{p-1}})-\left\\{\begin{array}[]{ll}C&N>p^{2}\\\
C\lvert\log\varepsilon\rvert^{\frac{q(\gamma_{q}-1)}{p}}&N=p^{2}\\\
C\varepsilon^{N-\frac{q(p-\gamma_{q})(N-p)}{p(p-1)}}&p^{\frac{3}{2}}\leqslant
N<p^{2}\end{array}\right.$ $\displaystyle<\frac{1}{N}S^{\frac{N}{p}},$

by taking $\varepsilon$ sufficiently small. ∎

Now, we give the proof of Theorem 1.3 in the case $p+\frac{p^{2}}{N}<q<p^{*}$.

Proof of Theorem 1.3: Similar to Section 5, we can obtain a PS sequence
$\\{u_{n}\\}$ for $E_{\mu}|_{S_{a}}$ at level $\sigma(a,\mu)$ with the
property $P_{\mu}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$. Therefore, by
Proposition 3.1 and Lemma 6.6, one of the cases in proposition 3.1 holds. If
case (i) occurs, we have $u_{n}\rightharpoonup u$ in $W^{1,p}(\mathbb{R}^{N})$
and

$E_{\mu}(u)\leqslant m(a,\mu)-\frac{1}{N}S^{\frac{N}{p}}<0.$ (6.1)

Since $u$ solves (1.1) for some $\lambda<0$, by the Pohozaev identity
$P_{\mu}(u)=0$, we can derive that

$E_{\mu}(u)=\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u\rVert_{q}^{q}+\frac{1}{N}\lVert u\rVert_{p^{*}}^{p^{*}}>0,$

a contradiction with (6.1). This implies that case (ii) in proposition 3.1
holds. The rest of the proof is same to Similar to Section 3.1.$\hfill\qed$

##  7 Asymptotic behavior of $u_{a,\mu}^{\pm}$

In this section, the dependence of parameter $a$ will not be considered, so we
write $u_{a,\mu}^{\pm}$, $\mathcal{P}_{a,\mu}$, $S_{a}$, $m(a,\mu)$,
$\lambda_{a,\mu}$, … as $u_{\mu}^{\pm}$, $\mathcal{P}_{\mu}$, $S$, $m(\mu)$,
$\lambda_{\mu}$,….

###  7.1 Asymptotic behavior of $u_{\mu}^{+}$ as $\mu\rightarrow 0$

In this subsection, we always assume that the assumptions of Theorem 1.4(1)
hold. In fact, we can prove that $u_{\mu}^{-}\rightarrow 0$ in
$D^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$. Therefore, we need more
accurate estimate of how fast $u_{\mu}^{-}$ approaches $0$.

######  Lemma 7.1.

We have

$-\lambda_{\mu}^{+}\sim\lVert\nabla
u_{\mu}^{+}\rVert_{p}^{p}\sim\mu^{\frac{p}{p-q\gamma_{q}}}.$

###### Proof.

Since $u_{\mu}^{+}\in\mathcal{P}_{\mu}^{+}$, we have

$\lVert\nabla u_{\mu}^{+}\rVert_{p}^{p}=\mu\gamma_{q}\lVert
u_{\mu}^{+}\rVert_{q}^{q}+\lVert u_{\mu}^{+}\rVert_{p^{*}}^{p^{*}},$

and

$p\lVert\nabla u_{\mu}^{+}\rVert_{p}^{p}>\mu q\gamma_{q}^{2}\lVert
u_{\mu}^{+}\rVert_{q}^{q}+p^{*}\lVert u_{\mu}^{+}\rVert_{p^{*}}^{p^{*}}.$

It follows from the Gagliardo-Nirenberg inequality that

$(p^{*}-p)\lVert\nabla
u_{\mu}^{+}\rVert_{p}^{p}\leqslant\mu\gamma_{q}(p^{*}-q\gamma_{q})\lVert
u_{\mu}^{+}\rVert_{q}^{q}\leqslant\mu\gamma_{q}(p^{*}-q\gamma_{q})C_{N,p,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u_{\mu}^{+}\rVert_{p}^{q\gamma_{q}},$ (7.1)

which together with $q\gamma_{q}<p$ for $p<q<p+\frac{p^{2}}{N}$, implies

$\lVert\nabla u_{\mu}^{+}\rVert_{p}^{p}\leqslant
C\mu^{\frac{p}{p-q\gamma_{q}}}.$

Using Gagliardo-Nirenberg inequality again, we know

$\lVert u_{\mu}^{+}\rVert_{q}^{q}\leqslant
C\mu^{\frac{q\gamma_{q}}{p-q\gamma_{q}}}.$ (7.2)

Let $u\in S$ be fixed. Then, there exists unique $s_{u}(\mu)\in\mathbb{R}$
such that $s_{u}(\mu)\star\mathcal{P}_{\mu}^{+}$, that is

$e^{ps_{u}(\mu)}\lVert\nabla
u\rVert_{p}^{p}=\mu\gamma_{q}e^{q\gamma_{q}s_{u}(\mu)}\lVert
u\rVert_{q}^{q}+e^{p^{*}s_{u}(\mu)}\lVert u\rVert_{p^{*}}^{p^{*}},$

and

$pe^{ps_{u}(\mu)}\lVert\nabla u\rVert_{p}^{p}>\mu
q\gamma_{q}^{2}e^{q\gamma_{q}s_{u}(\mu)}\lVert
u\rVert_{q}^{q}+p^{*}e^{p^{*}s_{u}(\mu)}\lVert u\rVert_{p^{*}}^{p^{*}}.$

It follows that

$\bigg{(}\frac{\mu\gamma_{q}\lVert u\rVert_{q}^{q}}{\lVert\nabla
u\rVert_{p}^{p}}\bigg{)}^{\frac{1}{p-q\gamma_{q}}}<e^{s_{u}(\mu)}<\bigg{(}\frac{\mu\gamma_{q}(p^{*}-q\gamma_{q})\lVert
u\rVert_{q}^{q}}{(p^{*}-p)\lVert\nabla
u\rVert_{p}^{p}}\bigg{)}^{\frac{1}{p-q\gamma_{q}}},$

which implies $e^{s_{u}(\mu)}\sim\mu^{\frac{1}{p-q\gamma_{q}}}$. Thus, by
$s_{u}(\mu)\star u\in\mathcal{P}_{\mu}^{+}$ and $q\gamma_{q}<p$ for
$p<q<p+\frac{p^{2}}{N}$, we have

$E_{\mu}(s_{u}(\mu)\star
u)=\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}e^{ps_{u}(\mu)}\lVert\nabla
u\rVert_{p}^{p}+\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}e^{p^{*}s_{u}(\mu)}\lVert
u\rVert_{q}^{q}\sim-\mu^{\frac{p}{p-q\gamma_{q}}}.$

Therefore, by $E_{\mu}(s_{u}\star u)\geqslant m^{+}(\mu)$ and

$m^{+}(\mu)=\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u_{\mu}^{+}\rVert_{q}^{q}+\frac{1}{N}\lVert
u_{\mu}^{+}\rVert_{p^{*}}^{p^{*}}>\mu\gamma_{q}\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\lVert
u_{\mu}^{+}\rVert_{q}^{q},$

we obtain

$\lVert u_{\mu}^{+}\rVert_{q}^{q}\geqslant
C\mu^{\frac{q\gamma_{q}}{p-q\gamma_{q}}}.$ (7.3)

Now, (7.2) and (7.3) implies

$\lVert u_{\mu}^{+}\rVert_{q}^{q}\sim\mu^{\frac{q\gamma_{q}}{p-q\gamma_{q}}}.$

By the Pohozaev identity, we know

$\lambda_{\mu}^{+}a^{p}=\mu(\gamma_{q}-1)\lVert u_{\mu}^{+}\rVert_{q}^{q}.$

Therefore, by (7.1),

$-\lambda_{\mu}^{+}\sim\lVert\nabla
u_{\mu}^{+}\rVert_{p}^{p}\sim\mu^{\frac{p}{p-q\gamma_{q}}}.$

∎

######  Remark 7.1.

It is clear that $u_{\mu}^{+}\rightarrow 0$ in $D^{1,p}(\mathbb{R}^{N})$ and
$m^{+}(\mu)\rightarrow 0$ as $\mu\rightarrow 0$.

Proof of Theorem 1.4(1) Let

$w_{\mu}=\mu^{-\frac{N}{p(p-q\gamma_{q})}}u_{\mu}^{+}\Big{(}\mu^{-\frac{1}{p-q\gamma_{q}}}\cdot\Big{)}\in
S.$

By Lemma 7.1,

$\lVert\nabla w_{\mu}\rVert_{p}^{p}=\mu^{-\frac{p}{p-q\gamma_{q}}}\lVert\nabla
u_{\mu}^{+}\rVert_{p}^{p}\leqslant C,$

which implies $\\{w_{\mu}\\}$ is bounded in $W^{1,p}(\mathbb{R}^{N})$. Thus,
there exists $w\in W^{1,p}(\mathbb{R}^{N})$ such that $w_{\mu}\rightharpoonup
w$ in $W^{1,p}(\mathbb{R}^{N})$, $w_{\mu}\rightarrow w$ in
$L^{q}(\mathbb{R}^{N})$, $w_{\mu}\rightarrow w$ a.e. in $\mathbb{R}^{N}$.

We know $u_{\mu}^{+}$ solves (1.1) for some $\lambda_{\mu}^{+}$. Direct
calculations show that

$-\Delta_{p}w_{\mu}=\lambda_{\mu}^{+}\mu^{-\frac{p}{p-q\gamma_{q}}}w_{\mu}^{p-1}+w_{\mu}^{q-1}+\mu^{\frac{p^{2}}{(N-p)(p-q\gamma_{q})}}w_{\mu}^{p^{*}-1}.$

By Lemma 7.1, we know $\\{\lambda_{\mu}^{+}\mu^{-\frac{p}{p-q\gamma_{q}}}\\}$
is bounded, hence there exists $\sigma_{0}>0$ such that
$\lambda_{\mu}^{+}\mu^{-\frac{p}{p-q\gamma_{q}}}\rightarrow-\sigma_{0}$ as
$\mu\rightarrow 0$. Considering the mapping
$T:W^{1,p}(\mathbb{R}^{N})\rightarrow\Big{(}W^{1,p}(\mathbb{R}^{N})\Big{)}^{*}$
which is given by

$<Tu,v>=\int_{\mathbb{R}^{N}}\big{(}\lvert\nabla u\rvert^{p-2}\nabla
u\cdot\nabla v+\sigma_{0}\lvert u\rvert^{p-2}uv\big{)}dx.$

Then, similar to [14, Lemma 3.6], we can prove that $w_{\mu}\rightarrow w$ in
$W^{1,p}(\mathbb{R}^{N})$. Thus, $w$ satisfies the equation

$-\Delta_{p}w+\sigma_{0}w^{p-1}=w^{q-1}.$

Let
$\tilde{w}=\sigma_{0}^{\frac{1}{p-q}}w\big{(}\sigma_{0}^{-\frac{1}{p}}\cdot\big{)}$.
It is not difficult to shows that

$\sigma_{0}^{\frac{1}{p-q}}\mu^{-\frac{N}{p(p-q\gamma_{q})}}u_{\mu}^{+}\big{(}\sigma_{0}^{-\frac{1}{p}}\mu^{-\frac{1}{p-q\gamma_{q}}}\cdot\big{)}\rightarrow\tilde{w}$

in $W^{1,p}(\mathbb{R}^{N})$, and $\tilde{w}$ satisfies (1.12). By the
regularity and properties of $u_{\mu}^{+}$, we can derive that $\tilde{w}$ is
the ”ground state” of (1.12) and hence $\tilde{w}=\phi_{0}$. Now, using
$u_{\mu}^{+}\in S_{a}$, we can obtain that

$\sigma_{0}=\bigg{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\bigg{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

$\hfill\qed$

###  7.2 Asymptotic behavior of $u_{\mu}^{-}$ as $\mu\rightarrow 0$

In this subsection, we always assume that the assumptions of Theorem 1.4(2) or
(3) hold. Unlike $u_{\mu}^{+}$, we can prove that $\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p},\lVert u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\rightarrow
S^{\frac{N}{p}}$ as $\mu\rightarrow 0$.

######  Lemma 7.2.

Let $\mu>0$ satisfies (1.10) for $p<q<p+\frac{p^{2}}{N}$ and (1.11) for
$q=p+\frac{p^{2}}{N}$. Then,

$m^{-}(\mu)=\inf_{u\in S}\max_{s\in\mathbb{R}}E_{\mu}(s\star u).$

###### Proof.

For every $v\in\mathcal{P}_{\mu}^{-}$, by Lemma 4.3, 5.2 and 6.2, we know

$E_{\mu}(v)=\max_{s\in\mathbb{R}}E_{\mu}(s\star v)\geqslant\inf_{u\in
S}\max_{s\in\mathbb{R}}E_{\mu}(s\star u),$

and hence

$m^{-}(\mu)\geqslant\inf_{u\in S}\max_{s\in\mathbb{R}}E_{\mu}(s\star u).$

For every $v\in S_{a}$, by Lemma 4.3, 5.2 and 6.2, we know

$\max_{s\in\mathbb{R}}E_{\mu}(s\star v)=E_{\mu}(t_{v}\star
v)\geqslant\inf_{u\in\mathcal{P}_{\mu}^{-}}E_{\mu}(u),$

and hence

$\inf_{u\in S}\max_{s\in\mathbb{R}}E_{\mu}(s\star u)\geqslant m^{-}(\mu).$

∎

######  Lemma 7.3.

Let $\tilde{\mu}>0$ satisfies (1.10) for $p<q<p+\frac{p^{2}}{N}$ and (1.11)
for $q=p+\frac{p^{2}}{N}$. Then, the function $\mu\in(0,\tilde{\mu}]\mapsto
m^{-}(\mu)\in\mathbb{R}$ is non-increasing.

###### Proof.

Let $0<\mu_{1}<\mu_{2}\leqslant\tilde{\mu}$. By Lemma 7.2, we have

$\displaystyle m^{-}(\mu_{2})$ $\displaystyle=\inf_{u\in
S}\max_{s\in\mathbb{R}}E_{\mu_{2}}(s\star u)=\inf_{u\in
S}\max_{s\in\mathbb{R}}\Big{(}E_{\mu_{1}}(s\star
u)-\frac{\mu_{2}-\mu_{1}}{q}\lVert u\rVert_{q}^{q}\Big{)}$
$\displaystyle\leqslant\inf_{u\in S}\max_{s\in\mathbb{R}}E_{\mu_{1}}(s\star
u)=m^{-}(\mu_{1}).$

∎

######  Lemma 7.4.

We have $\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p},\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\rightarrow S^{\frac{N}{p}}$ and
$m^{-}(\mu)\rightarrow S^{\frac{N}{p}}/N$ as $\mu\rightarrow 0$.

###### Proof.

Using the fact $m^{-}(\mu)<S^{\frac{N}{p}}/N$ and slightly modifying the proof
of Lemma 3.1, we know $\\{u_{\mu}^{-}\\}$ is bounded in
$W^{1,p}(\mathbb{R}^{N})$. Thus, we can assume that $\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\rightarrow l$ as $\mu\rightarrow 0$.

We claim that $l\neq 0$. Suppose by contradiction that $l=0$, then
$E_{\mu}(u_{\mu}^{-})\rightarrow 0$ as $\mu\rightarrow 0$. However, by Lemma
4.4, 5.4, 6.4 and 7.3,, we know $E_{\mu}(u_{\mu}^{-})\geqslant
m^{-}(\tilde{\mu})>0$ for every $0<\mu\leqslant\tilde{\mu}$, a contradiction.

Now, by $P_{\mu}(u_{\mu}^{-})=0$, we deduce that

$\lVert u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}=\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-\mu\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}\rightarrow l$

as $\mu\rightarrow 0$. Therefore, by the Sobolev inequality we have
$l\geqslant Sl^{\frac{p}{p^{*}}}$ which implies $l\geqslant S^{\frac{N}{p}}$.
On the other hand, since

$\frac{l}{N}=\lim_{\mu\rightarrow 0}\bigg{(}\frac{1}{N}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\bigg{)}=\lim_{\mu\rightarrow
0}E_{\mu}(u_{\mu}^{-})\leqslant\frac{1}{N}S^{\frac{N}{p}},$

we obtain that $l=S^{\frac{N}{p}}$. We complete the proof. ∎

Proof of Theorem 1.4(2) Lemma 7.4 implies $\\{u_{\mu}^{-}\\}$ is a minimizing
sequence of the following minimizing problem:

$S=\inf_{u\in D^{1,p}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\lVert\nabla
u\rVert_{p}^{p}}{\lVert u\rVert_{p^{*}}^{p^{*}}}.$

Since $u_{\mu}^{-}$ is radially symmetric, by [28, Theorem 4.9], there exists
$\sigma_{\mu}>0$ such that

$w_{\mu}=\sigma_{\mu}^{\frac{N-p}{p}}u_{\mu}^{-}(\sigma_{\mu}\cdot)\rightarrow
U_{\varepsilon_{0}}$

in $D^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$ for some
$\varepsilon_{0}>0$. We know $U_{\varepsilon_{0}}\notin S$ for $N\leqslant
p^{2}$, and $\lVert w_{\mu}\rVert_{p}^{p}=a^{p}/\sigma_{\mu}^{p}$, by the
Fatou lemma, $\sigma_{\mu}\rightarrow 0$ as $\mu\rightarrow 0$.$\hfill\qed$

$U_{\varepsilon_{0}}\notin S$ for $N\leqslant p^{2}$ implies $w_{\mu}$ will
not converge to $U_{\varepsilon_{0}}$ in $W^{1,p}(\mathbb{R})$ as
$\mu\rightarrow 0$. However, since $U_{\varepsilon_{0}}\in S$ for $N>p^{2}$,
in the next, we will prove $u_{\mu}^{-}\rightarrow U_{\varepsilon_{0}}$ in
$W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$ for $N>p^{2}$.

Proof of Theorem 1.4(3) Since $\lVert
U_{\varepsilon}\rVert_{p}^{p}$=$\varepsilon^{p}\lVert U_{1}\rVert_{p}^{p}$, we
can choose $\varepsilon_{0}>0$ satisfies $\lVert
U_{\varepsilon_{0}}\rVert_{p}^{p}=a^{p}$. Hence, there exists unique
$t(\mu)\in\mathbb{R}$ such that $t(\mu)\star
U_{\varepsilon_{0}}\in\mathcal{P}_{\mu}^{-}$, that is

$e^{pt(\mu)}S^{\frac{N}{p}}=\mu\gamma_{q}e^{q\gamma_{q}t(\mu)}\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}+e^{p^{*}t(\mu)}S^{\frac{N}{p}}.$

Clearly, $t(0)=0$. Now, using implicit function theorem, $t(\mu)$ is of class
$C^{1}$ in a neighborhood of $0$. By direct calculation, we have

$t^{\prime}(0)=-\frac{\gamma_{q}\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{(p^{*}-p)S^{N/p}},$

which implies

$t(\mu)=t(0)+t^{\prime}(0)\mu+o(\mu)=-\frac{\gamma_{q}\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{(p^{*}-p)S^{N/p}}\mu+o(\mu).$

Consequently,

$\displaystyle E_{\mu}\big{(}t(\mu)\star U_{\varepsilon_{0}}\big{)}$
$\displaystyle=\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}e^{pt(\mu)}\lVert
U_{\varepsilon_{0}}\rVert_{p}^{p}+\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}e^{p^{*}t(\mu)}\lVert
U_{\varepsilon_{0}}\rVert_{p^{*}}^{p^{*}}$
$\displaystyle=\Big{(}\frac{1}{p}-\frac{1}{q\gamma_{q}}\Big{)}\bigg{(}1-\frac{p\gamma_{q}\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{(p^{*}-p)S^{N/p}}\mu+o(\mu)\bigg{)}S^{\frac{N}{p}}+$
$\displaystyle\qquad\qquad\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\bigg{(}1-\frac{p^{*}\gamma_{q}\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{(p^{*}-p)S^{N/p}}\mu+o(\mu)\bigg{)}S^{\frac{N}{p}}$
$\displaystyle=\frac{1}{N}S^{\frac{N}{p}}-\frac{\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{q}\mu+o(\mu).$

By the definition of $m^{-}(\mu)$, we have

$m^{-}(\mu)=\frac{1}{N}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u_{\mu}^{-}\rVert_{q}^{q}\leqslant\frac{1}{N}S^{\frac{N}{p}}-\frac{\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}}{q}\mu+o(\mu).$ (7.4)

By the Sobolev inequality, we have

$\displaystyle\lVert u_{\mu}^{-}\rVert_{p}^{p}$ $\displaystyle\geqslant
S\lVert u_{\mu}^{-}\rVert_{p^{*}}^{p}=S\Big{(}\lVert\nabla
u_{\mu}\rVert_{p}^{p}-\mu\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}\Big{)}^{\frac{p}{p^{*}}}$
$\displaystyle=S\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{\frac{p^{2}}{p^{*}}}\bigg{(}1-\frac{\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}}{S^{N/p}}\mu+o(\mu)\bigg{)}^{\frac{p}{p^{*}}}$
$\displaystyle=S\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{\frac{p^{2}}{p^{*}}}\bigg{(}1-\frac{p\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}}{p^{*}S^{N/p}}\mu+o(\mu)\bigg{)}.$

Thus,

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}\geqslant
S^{\frac{N}{p}}\bigg{(}1-\frac{p\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}}{p^{*}S^{N/p}}+o(\mu)\bigg{)}^{\frac{N}{p}}=S^{\frac{N}{p}}-\frac{(N-p)\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}}{p}\mu+o(\mu),$

which together with (7.4), implies

$\lVert u_{\mu}^{-}\rVert_{q}^{q}\geqslant\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}+o(1).$ (7.5)

Since $\\{u_{\mu}^{-}\\}$ is bounded in $W^{1,p}(\mathbb{R}^{N})$, there
exists $u\in W^{1,p}(\mathbb{R}^{N})$ such that $u_{\mu}^{-}\rightharpoonup u$
in $W^{1,p}(\mathbb{R}^{N})$, $u_{\mu}^{-}\rightarrow u$ in
$L^{q}(\mathbb{R}^{N})$ and $u_{\mu}\rightarrow u$ a.e. in $\mathbb{R}^{N}$ as
$\mu\rightarrow 0$. By (7.5), we know $u\neq 0$.

By the Pohozaev identity, we know

$\lambda_{\mu}^{-}a^{p}=\mu(\gamma_{q}-1)\lVert
u_{\mu}^{-}\rVert_{q}^{q}\rightarrow 0$

as $\mu\rightarrow 0$. Thus, by the weak convergence, $u$ is the solution of
the equation

$-\Delta_{p}U=U^{p^{*}-1},$

which implies $u=U_{\varepsilon,y}$ for some
$(\varepsilon,y)\in(\mathbb{R}^{+},\mathbb{R}^{N})$. Since $u_{\mu}^{-}$ is
radially symmetric, we have $y=0$ and hence $u=U_{\varepsilon}$. Now, by the
Fatou lemma and (7.5), we obtain

$\lVert U_{\varepsilon}\rVert_{p}^{p}=\lVert u\rVert_{p}^{p}\leqslant
a^{p}=\lVert U_{{\varepsilon_{0}}}\rVert_{p}^{p},\quad\lVert
U_{\varepsilon}\rVert_{q}^{q}=\lVert u\rVert_{q}^{q}=\lVert
U_{\varepsilon_{0}}\rVert_{q}^{q}.$

Therefore, $\varepsilon=\varepsilon_{0}$ and $u=U_{\varepsilon_{0}}$. Finally,
since

$\lVert U_{\varepsilon_{0}}\rVert_{p}^{p}=\lim_{\mu\rightarrow 0}\lVert
u_{\mu}^{-}\rVert_{p}^{p}=a^{p},\quad and\quad\lVert\nabla
U_{\varepsilon_{0}}\rVert_{p}^{p}=\lim_{\mu\rightarrow 0}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}=S^{\frac{N}{p}},$

the Brézis-Lieb lemma [5] implies $u_{\mu}\rightarrow U_{\varepsilon_{0}}$ in
$W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow 0$.$\hfill\qed$

###  7.3 Asymptotic behavior of $u_{\mu}^{-}$ as $\mu$ goes to its upper bound

In this subsection, we always assume that the assumptions of Theorem 1.5 hold.
Firstly, we assume that $q=p+\frac{p^{2}}{N}$ and we prove $\bar{\alpha}$ is
the upper bound of $\mu$ when $q=p+\frac{p^{2}}{N}$.

######  Lemma 7.5.

We have

$\sup_{u\in S}\frac{\lVert\nabla u\rVert_{p}^{p}}{\lVert
u\rVert_{q}^{q}}=+\infty.$

###### Proof.

By the Sobolev inequality, we just have to prove that

$\sup_{u\in S}\frac{\lVert u\rVert_{p^{*}}^{p}}{\lVert
u\rVert_{q}^{q}}=+\infty.$

Let

$u_{k}(x)=\frac{A_{k}\varphi_{k}(x)}{(1+\lvert x\rvert^{2})^{a_{k}}}\in S,$

where $A_{k}>0$ is a constant dependent on $k$,

$a_{k}=\frac{N-p}{2p}-\frac{1}{\log\log(k+2)},$

and $\varphi_{k}\in C_{c}^{\infty}(\mathbb{R}^{N})$ is a radial cut-off
function satisfies

$0\leqslant\varphi_{k}\leqslant 1,\quad\varphi_{k}=1\ in\ B_{k},\quad
and\quad\varphi_{k}=0\ in\ B_{k+1}^{c}.$

Since $u_{k}\in S$ and

$\displaystyle\lVert u_{k}\rVert_{p}^{p}$
$\displaystyle=A_{k}^{p}\int_{\mathbb{R}^{N}}\frac{\varphi_{k}^{p}(x)}{(1+\lvert
x\rvert^{2})^{pa_{k}}}dx\sim
A_{k}^{p}\int_{0}^{+\infty}\frac{\varphi_{k}^{p}(r)r^{N-1}}{(1+r^{2})^{pa_{k}}}dr$
$\displaystyle\sim
A_{k}^{p}\int_{0}^{k}\frac{r^{N-1}}{(1+r^{2})^{pa_{k}}}dr\sim\frac{A_{k}^{p}k^{N-2pa_{k}}}{N-2pa_{k}}\sim
A_{k}^{p}k^{N-2pa_{k}}$

as $k\rightarrow\infty$, we have $A_{k}\sim k^{2a_{k}-N/p}$ as
$k\rightarrow\infty$. Therefore,

$\lVert
u_{k}\rVert_{q}^{q}=A_{k}^{q}\int_{\mathbb{R}^{N}}\frac{\varphi_{k}^{q}(x)}{(1+\lvert
x\rvert^{2})^{qa_{k}}}dx\sim
k^{2qa_{k}-N-p}\int_{0}^{k}\frac{r^{N-1}}{(1+r^{2})^{qa_{k}}}dr\sim\frac{1}{k^{p}},$

and

$\lVert
u_{k}\rVert_{p^{*}}^{p^{*}}=A_{k}^{p^{*}}\int_{\mathbb{R}^{N}}\frac{A_{k}^{p^{*}}}{(1+\lvert
x\rvert^{2})^{p^{*}a_{k}}}dx\sim
k^{2p^{*}a_{k}-\frac{Np^{*}}{p}}\int_{0}^{k}\frac{r^{N-1}}{(1+r^{2})^{p^{*}a_{k}}}dr\sim\frac{1}{(N-p^{*}a_{k})k^{p^{*}}},$

as $k\rightarrow\infty$, which implies

$\frac{\lVert u_{k}\rVert_{p^{*}}^{p}}{\lVert
u_{k}\rVert_{q}^{q}}\sim\frac{1}{(N-p^{*}a_{k})^{p/p^{*}}}\rightarrow+\infty$

as $k\rightarrow\infty$. ∎

######  Lemma 7.6.

For $\mu\geqslant\bar{\alpha}$, we have
$\mathcal{P}_{\mu}=\mathcal{P}_{\mu}^{-}\neq\emptyset$.

###### Proof.

For every $\mu\geq\bar{\alpha}$, by Lemma 7.5, there exists $u\in S$ such that
$\lVert\nabla u\rVert_{p}^{p}>\mu\gamma_{q}\lVert u\rVert_{q}^{q}$. Then,

$\Psi_{u}^{\mu}(s)=\frac{1}{p}e^{ps}\big{(}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\lVert
u\rVert_{q}^{q}\big{)}-\frac{1}{p^{*}}e^{p^{*}s}\lVert
u\rVert_{p^{*}}^{p^{*}}$

has a critical point $t_{u}\in\mathbb{R}$. By proposition 2.1, we know
$t_{u}\star u\in\mathcal{P}_{\mu}$ which implies
$\mathcal{P}_{\mu}\neq\emptyset$.

If there exists $v\in\mathcal{P}_{\mu}^{0}\cup\mathcal{P}_{\mu}^{+}$, we have

$\lVert\nabla v\rVert_{p}^{p}=\mu\gamma_{q}\lVert v\rVert_{q}^{q}+\lVert
v\rVert_{p^{*}}^{p^{*}},\quad and\quad p\lVert\nabla
v\rVert_{p}^{p}\geqslant\mu q\gamma_{q}^{2}\lVert v\rVert_{q}^{q}+p^{*}\lVert
v\rVert_{p^{*}}^{p^{*}},$

which implies $\lVert v\rVert_{p^{*}}^{p^{*}}\leqslant 0$(since
$q\gamma_{q}=p$), a contradiction since $v\in S$. ∎

######  Lemma 7.7.

For $\mu\geqslant\bar{\alpha}$, there is $m^{-}(\mu)=0$, and $m^{-}(\mu)$ can
not be attained by any $u\in S$.

###### Proof.

For every $\mu\geqslant\bar{\alpha}$, by Lemma 7.5, there exists
$\\{u_{n}\\}\subset S$ such that

$\frac{\lVert\nabla u_{n}\rVert_{p}^{p}}{\lVert
u_{n}\rVert_{q}^{q}}>\mu\gamma_{q},\quad and\quad\frac{\lVert\nabla
u_{n}\rVert_{p}^{p}}{\lVert u_{n}\rVert_{q}^{q}}\rightarrow\mu\gamma_{q}$

as $n\rightarrow\infty$. Without loss of generality, by scaling
$\frac{as_{n}^{N/p}}{\lVert u_{n}\rVert_{p}^{p}}u_{n}(sx)$ if necessary, we
may assume that $\lVert u_{n}\rVert_{q}^{q}=1$. Then, we have $\lVert\nabla
u_{n}\rVert_{p}^{p}>\mu\gamma_{q}$ and $\lVert\nabla
u_{n}\rVert_{p}^{p}\rightarrow\mu\gamma_{q}$ as $n\rightarrow\infty$.

Now, we can obtain the function

$\Psi_{u_{n}}^{\mu}(s)=\frac{1}{p}e^{ps}\big{(}\lVert\nabla
u_{n}\rVert_{p}^{p}-\mu\gamma_{q}\big{)}-\frac{1}{p^{*}}e^{p^{*}s}\lVert
u_{n}\rVert_{p^{*}}^{p^{*}}$

has a critical point $t_{n}\in\mathbb{R}$. Hence, $t_{n}\star
u_{n}\in\mathcal{P}_{\mu}^{-}$ and we have

$\lVert u_{n}\rVert_{p}^{p}-\mu\gamma_{q}=e^{(p^{*}-p)t_{n}}\lVert
u_{n}\rVert_{p^{*}}^{p^{*}}.$ (7.6)

By the Sobolev inequality $S\lVert u_{n}\rVert_{p}^{p}\leqslant\lVert\nabla
u_{n}\rVert_{p}^{p}$ and Hölder inequality $\lVert
u_{n}\rVert_{q}^{q}\leqslant\lVert u_{n}\rVert_{p^{*}}^{q\gamma_{q}}\lVert
u_{n}\rVert_{p}^{q(1-\gamma_{q})}$, we obtain $\lVert
u_{n}\rVert_{p^{*}}^{p^{*}}\sim 1$. Thus, (7.6) implies
$t_{n}\rightarrow-\infty$ as $n\rightarrow\infty$.

By the definition of $m^{-}(\mu)$, we have $m^{-}(\mu)\leqslant
E_{\mu}(t_{n}\star u_{n})$, that is

$m^{-}(\mu)\leqslant\frac{1}{N}e^{pt_{n}}\big{(}\lVert\nabla
u_{n}\rVert_{p}^{p}-\mu\gamma_{q}\big{)}\rightarrow 0$

as $n\rightarrow\infty$, which implies $m^{-}(\mu)\leqslant 0$. For every
$u\in\mathcal{P}_{\mu}^{-}$, we have

$E_{\mu}(u)=\frac{1}{N}\lVert u\rVert_{p^{*}}^{p^{*}},$

which implies $m^{-}(\mu)\geqslant 0$. Therefore, $m^{-}(\mu)=0$.

If there exists $u\in\mathcal{P}_{\mu}^{-}$ such that $E_{\mu}(u)=0$. Then, it
must have $u\equiv 0$ which contradicts with $u\in S$. ∎

Theorem 1.3 and Lemma 7.7 implies $\bar{\alpha}$ is the upper bound of $\mu$.
Therefore, we can study the asymptotic behavior of $u_{a,\mu}^{-}$ as
$\mu\rightarrow\bar{\alpha}$. We give the asymptotic behavior of
$\lambda_{\mu}^{-}$ as follows.

######  Lemma 7.8.

For $\mu<\bar{\alpha}$, we have

$-\lambda_{\mu}^{-}\sim\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\sim(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

as $\mu\rightarrow\bar{\alpha}$.

###### Proof.

By the Gagliardo-Nirenberg inequality and Sobolev inequality,

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}=\mu\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}+\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\leqslant\frac{\mu}{\bar{\alpha}}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}+S^{\frac{p^{*}}{p}}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p^{*}},$

which implies

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}\geqslant
C(\alpha-\mu)^{\frac{N-p}{p}},$ (7.7)

as $\mu\rightarrow\bar{\alpha}$.

Let $\varphi=\frac{a}{\lVert\psi_{0}\rVert_{p}}\psi_{0}\in S$, where
$\psi_{0}$ is a minimizer of Gagliardo-Nirenberg inequality. Then, direct
calculations shows that $t_{\mu}\star\varphi\in\mathcal{P}_{\mu}^{-}$, where

$e^{t_{\mu}}=\frac{\lVert\psi_{0}\rVert_{p}\lVert\nabla\psi_{0}\rVert_{p}^{p/(p^{*}-p)}}{a\lVert\psi_{0}\rVert_{p^{*}}^{p^{*}/(p^{*}-p)}}\Big{(}1-\frac{\mu}{\bar{\alpha}}\Big{)}^{\frac{1}{p^{*}-p}}.$

Therefore,

$m^{-}(\mu)\leqslant
E_{\mu}(t_{\mu}\star\varphi)=\frac{a^{p}\lVert\nabla\psi_{0}\rVert_{p}^{p}}{N\lVert\psi_{0}\rVert_{p}^{p}}\Big{(}1-\frac{\mu}{\bar{\alpha}}\Big{)}e^{pt_{\mu}}=\frac{1}{N}\Big{(}1-\frac{\mu}{\bar{\alpha}}\Big{)}^{\frac{N}{p}}\frac{\lVert\nabla\psi_{0}\rVert_{p}^{N}}{\lVert\psi_{0}\rVert_{p^{*}}^{N}}.$

Since

$E_{\mu}(u_{\mu}^{-})=\frac{1}{N}\lVert u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}},$

we have

$\lVert u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\leqslant
C(\bar{\alpha}-\mu)^{\frac{N}{p}}.$

Now, by the Hölder inequality, we obtain

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}=\mu\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}+\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\leqslant\frac{\mu}{\bar{\alpha}}\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p}+\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\leqslant
C(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

as $\mu\rightarrow\bar{\alpha}$, which together with (7.7), implies

$\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\sim(\bar{\alpha}-\mu)^{\frac{N-p}{p}}.$

Using the Gagliardo-Nirenberg inequality and Sobolev inequality again, we have

$\lVert u_{\mu}^{-}\rVert_{q}^{q}\leqslant C\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\leqslant C(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

and

$\mu\gamma_{q}\lVert u_{\mu}^{-}\rVert_{q}^{q}=\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-\lVert
u_{\mu}^{-}\rVert_{p}^{p^{*}}\geqslant\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-S^{\frac{p^{*}}{p}}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p^{*}}\geqslant C(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

as $\mu\rightarrow\bar{\alpha}$. Therefore,

$\lVert u_{\mu}^{-}\rVert_{q}^{q}\sim(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

as $\mu\rightarrow\bar{\alpha}$. By the Pohozaev identity
$\lambda_{\mu}^{-}a^{p}=\mu(\gamma_{q}-1)\lVert u_{\mu}^{-}\rVert_{q}^{q}$, we
know

$-\lambda_{\mu}^{-}\sim(\bar{\alpha}-\mu)^{\frac{N-p}{p}},$

as $\mu\rightarrow\bar{\alpha}$. ∎

######  Remark 7.2.

We have $u_{\mu}^{-}\rightarrow 0$ in $D^{1,p}(\mathbb{R}^{N})$ and
$m^{-}(\mu)\rightarrow 0$ as $\mu\rightarrow\bar{\alpha}$.

Proof of Theorem 1.5(1) Let

$w_{\mu}=s_{\mu}^{\frac{N}{p}}u_{\mu}^{-}(s_{\mu}\cdot)\in S,$

where $s_{\mu}=(\bar{\alpha}-\mu)^{-(N-p)/p^{2}}$. By Lemma 7.8,

$\lVert\nabla w_{\mu}\rVert_{p}^{p}=s_{\mu}^{p}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\leqslant C,$

which implies $\\{w_{\mu}\\}$ is bounded in $W^{1,p}(\mathbb{R}^{N})$. Thus,
there exists $w\in W^{1,p}(\mathbb{R}^{N})$ such that $w_{\mu}\rightharpoonup
w$ in $W^{1,p}(\mathbb{R}^{N})$, $w_{\mu}\rightarrow w$ in
$L^{q}(\mathbb{R}^{N})$, $w_{\mu}\rightarrow w$ a.e. in $\mathbb{R}^{N}$.

Direct calculations shows that

$-\Delta_{p}w_{\mu}=\lambda_{\mu}^{-}s_{\mu}^{p}w_{\mu}^{p-1}+\mu
w_{\mu}^{q-1}+s_{\mu}^{-\frac{p^{2}}{N-p}}w_{\mu}^{p^{*}-1}.$

By Lemma 7.8, we know $\\{\lambda_{\mu}^{+}s_{\mu}^{p}\\}$ is bounded, hence
there exists $\sigma_{0}>0$ such that
$\lambda_{\mu}^{+}s_{\mu}^{p}\rightarrow-\sigma_{0}$ as $\mu\rightarrow 0$.
Similar to the proof of Theorem 1.4(1), we can prove that $w_{\mu}\rightarrow
w$ in $W^{1,p}(\mathbb{R}^{N})$. Thus, $w$ satisfies the equation

$-\Delta_{p}w+\sigma_{0}w^{p-1}=\bar{\alpha}w^{q-1}.$

Let
$\tilde{w}=(\bar{\alpha}\sigma_{0})^{\frac{1}{p-q}}w\big{(}\sigma_{0}^{-\frac{1}{p}}\cdot\big{)}$.
It is not difficult to show that

$(\bar{\alpha}\sigma_{0})^{\frac{1}{p-q}}s_{\mu}^{\frac{N}{p}}u_{\mu}^{-}\big{(}\sigma_{0}^{-\frac{1}{p}}s_{\mu}\cdot\big{)}\rightarrow\tilde{w}$

in $W^{1,p}(\mathbb{R}^{N})$, and $\tilde{w}$ satisfies (1.12). By the
regularity and properties of $u_{\mu}^{-}$, we can derive that $\tilde{w}$ is
the ”ground state” of (1.12) and hence $\tilde{w}=\phi_{0}$. Now, using $w\in
S_{a}$, we can obtain that

$\sigma_{0}=\bar{\alpha}^{\frac{p^{2}}{N(q-p)-p^{2}}}\bigg{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\bigg{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

$\hfill\qed$

Now, we assume that $p+\frac{p^{2}}{N}<q<p^{*}$. Obviously, the upper bound of
$\mu$ is $+\infty$.

######  Lemma 7.9.

We have

$-\lambda_{\mu}^{-}\sim\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}\sim\mu^{-\frac{p}{q\gamma_{q}-p}},$

as $\mu\rightarrow+\infty$.

###### Proof.

By the Gagliardo-Nirenberg inequality and Sobolev inequality,

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}=\mu\gamma_{q}\lVert
u_{\mu}^{-}\rVert_{q}^{q}+\lVert
u_{\mu}^{-}\rVert_{p^{*}}^{p^{*}}\leqslant\mu\gamma_{q}a^{q(1-\gamma_{q})}C_{N,p,q}^{q}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{q\gamma_{q}}+S^{\frac{p^{*}}{p}}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p^{*}},$

which implies

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}\geqslant
C\mu^{-\frac{p}{q\gamma_{q}-p}},$ (7.8)

as $\mu\rightarrow+\infty$.

Let $u\in S$ be fixed. Then, there exists $t_{\mu}\in\mathbb{R}$ such that
$t_{\mu}\star u\in\mathcal{P}_{\mu}^{-}$, that is

$e^{pt_{\mu}}\lVert\nabla
u\rVert_{p}^{p}=\mu\gamma_{q}e^{q\gamma_{q}t_{\mu}}\lVert
u\rVert_{q}^{q}+e^{p^{*}t_{\mu}}\lVert u\rVert_{p^{*}}^{p^{*}},$

which implies

$e^{t_{\mu}}\leqslant C\mu^{-\frac{1}{q\gamma_{q}-p}},$

as $\mu\rightarrow+\infty$. Therefore,

$E_{\mu}(t_{\mu}\star u)=\frac{1}{N}e^{pt_{\mu}}\lVert\nabla
u\rVert_{p}^{p}+\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}e^{p^{*}t_{\mu}}\lVert
u\rVert_{q}^{q}\leqslant C\mu^{-\frac{p}{q\gamma_{q}-p}},$

as $\mu\rightarrow+\infty$. Since,

$E_{\mu}(u_{\mu}^{-})=\frac{1}{N}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}+\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u_{\mu}^{-}\rVert_{q}^{q}\geqslant\frac{1}{N}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p},$

we have

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}\leqslant E_{\mu}(t_{\mu}\star
u)\leqslant C\mu^{-\frac{p}{q\gamma_{q}-p}},$

as $\mu\rightarrow+\infty$, which together with (7.8), implies

$\lVert\nabla u_{\mu}^{-}\rVert_{p}^{p}\sim\mu^{-\frac{p}{q\gamma_{q}-p}}$

as $\mu\rightarrow+\infty$.

Using the Gagliardo-Nirenberg inequality and Sobolev inequality again, we have

$\lVert u_{\mu}^{-}\rVert_{q}^{q}\leqslant C\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{q\gamma_{q}}\leqslant
C\mu^{-\frac{q\gamma_{q}}{q\gamma_{q}-p}},$

and

$\mu\gamma_{q}\lVert u_{\mu}^{-}\rVert_{q}^{q}=\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-\lVert
u_{\mu}^{-}\rVert_{p}^{p^{*}}\geqslant\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p}-S^{\frac{p^{*}}{p}}\lVert\nabla
u_{\mu}^{-}\rVert_{p}^{p^{*}}\geqslant C\mu^{-\frac{p}{q\gamma_{q}-p}}$

as $\mu\rightarrow+\infty$. Therefore,

$\lVert u_{\mu}^{-}\rVert_{q}^{q}\sim\mu^{-\frac{q\gamma_{q}}{q\gamma_{q}-p}}$

as $\mu\rightarrow+\infty$. By the Pohozaev identity
$\lambda_{\mu}^{-}a^{p}=\mu(\gamma_{q}-1)\lVert u_{\mu}^{-}\rVert_{q}^{q}$, we
know

$-\lambda_{\mu}^{-}\sim\mu^{-\frac{p}{q\gamma_{q}-p}}$

as $\mu\rightarrow+\infty$. ∎

######  Remark 7.3.

We have $u_{\mu}^{-}\rightarrow 0$ in $D^{1,p}(\mathbb{R}^{N})$ and
$m^{-}(\mu)\rightarrow 0$ as $\mu\rightarrow+\infty$.

Proof of Theorem 1.5(2) Let

$w_{\mu}=\mu^{\frac{N}{p(q\gamma_{q}-p)}}u_{\mu}^{-}\Big{(}\mu^{\frac{1}{q\gamma_{q}-p}}\cdot\Big{)}\in
S.$

Similar to the proof of Theorem 1.4(1), we can prove that there exists $w\in
W^{1,p}(\mathbb{R}^{N})$ such that $w_{\mu}\rightarrow w$ in
$W^{1,p}(\mathbb{R}^{N})$ and $w$ satisfies

$-\Delta_{p}w+\sigma_{0}w^{p-1}=w^{q-1}$

for some $\sigma_{0}>0$.

Let
$\tilde{w}=\sigma_{0}^{\frac{1}{p-q}}w\big{(}\sigma_{0}^{-\frac{1}{p}}\cdot\big{)}$.
Then

$\sigma_{0}^{\frac{1}{p-q}}\mu^{\frac{N}{q\gamma_{q}-p}}u_{\mu}^{-}\Big{(}\sigma_{0}^{-\frac{1}{p}}\mu^{\frac{1}{q\gamma_{q}-p}}\cdot\Big{)}\rightarrow\tilde{w}$

in $W^{1,p}(\mathbb{R}^{N})$ as $\mu\rightarrow+\infty$ and we can prove that
$\tilde{w}=\phi_{0}$. Finally, using $w\in S_{a}$, we have

$\sigma_{0}=\bigg{(}\frac{a^{p}}{\lVert\phi_{0}\rVert_{p}^{p}}\bigg{)}^{\frac{p(q-p)}{p^{2}-N(q-p)}}.$

$\hfill\qed$

##  8 Nonexistence result

In this section, we prove the nonexistence result for $\mu<0$. The proof is
not complicated and is a direct application of the result in [3].

Proof of Theorem 1.6 Let $u$ be a critical point of $E_{\mu}|_{S_{a}}$. Then,
$u$ solves (1.1) for some $\lambda\in\mathbb{R}$. By the Pohozaev identity, we
have

$\lambda a^{p}=\mu(\gamma_{q}-1)\lVert u\rVert_{q}^{q},$

which implies $\lambda<0$, since $\mu<0$, $\gamma_{q}<1$ and $u\in S_{a}$.

Using the Sobolev inequality and the fact that $P_{\mu}(u)=0$, we deduce that

$\lVert\nabla u\rVert_{p}^{p}=\mu\gamma_{q}\lVert u\rVert_{q}^{q}+\lVert
u\rVert_{p^{*}}^{p^{*}}<\lVert u\rVert_{p^{*}}^{p^{*}}\leqslant
S^{-\frac{p^{*}}{p}}\lVert\nabla u\rVert_{p}^{p^{*}},$

which implies $\lVert\nabla u\rVert_{p}^{p}>S^{\frac{N}{p}}$. Therefore,

$E_{\mu}(u)=\frac{1}{N}\lVert\nabla
u\rVert_{p}^{p}-\mu\gamma_{q}\Big{(}\frac{1}{q\gamma_{q}}-\frac{1}{p^{*}}\Big{)}\lVert
u\rVert_{q}^{q}>\frac{1}{N}S^{\frac{N}{p}}.$

We complete the proof of (1).

In order to prove (2), we use corollary 4.2 in [3]. Let $Q=\Delta_{p}$,
$\gamma=0$ and $g(u)=\lambda u+\mu u^{q-1}+u^{p^{*}-1}$. We know

$\alpha^{*}=\frac{N-p}{N-1}>0,$

thus,

$\sigma^{*}=p-1+\frac{p-\gamma}{\alpha^{*}}=\frac{N(p-1)}{N-p}.$

By (1), since $\lambda<0$, we have

$\liminf_{s\rightarrow 0^{+}}s^{-\sigma^{*}}g(s)=+\infty.$

Now, by [3, corollary 4.2], (1.1) has no positive solution for any
$\mu<0$.$\hfill\qed$

##  9 Multiplicity result

In this section, we will prove the multiplicity result. Thus, we always assume
that the assumptions of Theorem 1.7 holds.

Firstly, we introduce the concept of genus. Let $X$ be a Banach space and $A$
be a subset of $X$. The set $A$ is said to be symmetric if $u\in A$ implies
$-u\in A$. Denote by $\Sigma$ the family of closed symmetric subsets $A$ of
$X$ such that $0\notin A$, that is

$\Sigma=\\{A\subset X\backslash\\{0\\}:\mbox{A is closed and symmetric with
respect to the origin}\\}.$

For $A\in\Sigma$, define the genus $\gamma(A)$ by

$\gamma(A)=\min\\{k\in\mathbb{N}:\exists\phi\in
C(A,\mathbb{R}^{k}\backslash\\{0\\})\ \mbox{and}\ \phi(x)=-\phi(-x),\forall
x\in X\\}.$

If such odd map $\phi$ does not exist, we define $\gamma(A)=+\infty$. For all
$k\in\mathbb{N}_{+}$, let

$\Sigma_{k}=\\{A:A\in\Sigma\ and\ \gamma(A)\geqslant k\\}.$

For every $\delta>0$ and $A\in\Sigma$, let

$A_{\delta}=\\{x\in X:\inf\nolimits_{y\in A}\lVert
x-y\rVert_{X}\leqslant\delta\\}.$

We have following lemma with respect to genus.

######  Lemma 9.1.

[23, section 7] Let $A,B\in\Sigma$. Then the following statements hold.

(i) If $\gamma(A)\geqslant 2$, then $A$ contains infinitely many distinct
points.

(ii) If there exists an odd mapping $f\in C(A,B)$, then
$\gamma(A)\leqslant\gamma(B)$. In particular, if $f$ is a homeomorphism
between $A$ and $B$, then $\gamma(A)=\gamma(B)$.

(iii) Let $\mathbb{S}^{N-1}$ is the sphere in $\mathbb{R}^{N}$, then
$\gamma(\mathbb{S}^{N-1})=N$.

(iv) If $\gamma(B)<+\infty$, then
$\gamma(\overline{A-B})\geqslant\gamma(A)-\gamma(B)$.

(v) If $A$ is compact, then $\gamma(A)<\infty$ and there exists $\delta>0$
such that $\gamma(A)=\gamma(A_{\delta})$.

Let $\varphi\in C^{1}(X,\mathbb{R})$ be an even functional and

$V=\\{v\in X:\psi(v)=1\\},$

where $\psi\in C^{2}(X,\mathbb{R})$ and $\psi^{\prime}(v)\neq 0$ for all $v\in
V$. We define the set of critical points of $\varphi|_{V}$ at level $c$ as

$K^{c}=\\{u\in V:\varphi(u)=c,\varphi|_{V}^{\prime}(u)=0\\}.$

The following conclusion is the key to proving the result of multiplicity.

######  Proposition 9.1.

Assume that $\varphi|_{V}$ is bounded from below and satisfies the $(PS)_{c}$
condition for all $c<0$. Moreover, we also assume that
$\Sigma_{k}\neq\emptyset$ for $k\in\mathbb{N}_{+}$. Define a sequence of mini-
max values $-\infty<c_{1}\leqslant c_{2}\leqslant...\leqslant
c_{n}\leqslant...<+\infty$ as follows

$c_{k}:=\inf_{A\in\Sigma_{k}}\sup_{u\in A}\varphi(u)\quad\forall k=1,2,..,n.$

We have following statements hold.

(i) If $c_{k}<0$, then $c_{k}$ is a critical value of $\varphi|_{V}$.

(ii) If there exists $c<0$ such that $c_{k}=c_{k+1}=...=c_{k+l}=c,$ then
$\gamma(K^{c})\geqslant l+1$. In particular, $\varphi|_{V}$ has infinitely
many critical points at level $c$ if $l\geqslant 2$.

###### Proof.

The proof is very similar to [18, Theorem 2.1], if we replace [18, Lemma 2.3]
with following quantitative deformation lemma. ∎

For every $c,d\in\mathbb{R}$ with $c<d$, define

$\varphi|_{V}^{c}:=\\{u\in V:\varphi(u)\leqslant c\\},\quad
and\quad\varphi^{-1}([c,d]):=\\{u\in X:c\leqslant\varphi(u)\leqslant d\\}.$

Then, we have following quantitative deformation lemma.

######  Lemma 9.2.

[35, Lemma 5.15] Let $\varphi\in C^{1}(X,\mathbb{R})$, $W\subset V$,
$c\in\mathbb{R}$, and $\varepsilon,\delta>0$ such that

$\lVert\varphi|_{V}^{\prime}(u)\rVert\geqslant\frac{8\varepsilon}{\delta}\quad\forall
u\in\varphi^{-1}([c-2\varepsilon,c+2\varepsilon]\cap W_{2\delta}).$

Then there exists $\eta\in C([0,1]\times V,V)$ such that

(i) $\eta(t,u)=u$ if $t=0$ or if
$u\notin\varphi^{-1}([c-2\varepsilon,c+2\varepsilon])\cap W_{2\delta}$.

(ii) $\eta(1,\varphi^{c+\varepsilon}\cap W)\in\varphi^{c-\varepsilon}$.

(iii) $\varphi(\eta(\cdot,u))$ is non-increasing in $t\in[0,1]$ for all $v\in
V$.

(iv) $\eta(t,u)$ is odd in $V$ for all $t\in[0,1]$ if $\varphi$ is even in
$V$.

Let $\tau\in C^{\infty}(\mathbb{R}^{+},[0,1])$ be a non-increasing function
satisfies

$\tau(t)=1\ \mbox{for}\ t\in[0,R_{0}],\quad\mbox{and}\quad\tau(t)=0\
\mbox{for}\ t\in[R_{1},+\infty),$

where $R_{0}$ and $R_{1}$ are obtained by Lemma 4.2. Define the truncated
functional as follows

$E_{\tau}(u)=\frac{1}{p}\lVert\nabla u\rVert_{p}^{p}-\frac{\mu}{q}\lVert
u\rVert_{q}^{q}-\frac{1}{p^{*}}\tau(\lVert\nabla u\rVert_{p})\lVert
u\rVert_{P^{*}}^{p^{*}}.$

For $u\in S_{a}$, by the Gagliardo-Nirenberg inequality and Sobolev
inequality, there is

$E_{\tau}(u)\geqslant\frac{1}{p}\lVert\nabla
u\rVert_{p}^{p}-\frac{\mu}{q}C_{N,p,q}^{q}a^{q(1-\gamma_{q})}\lVert\nabla
u\rVert_{p}^{q\gamma_{q}}-\frac{1}{p^{*}S^{p^{*}/p}}\tau(\lVert\nabla
u\rVert_{p})\lVert\nabla u\rVert_{p}^{p^{*}}=\tilde{h}(\lVert\nabla
u\rVert_{p}),$

where

$\tilde{h}(t)=\frac{1}{p}t^{p}-\frac{\mu}{q}C_{N,p,q}^{q}a^{q(1-\gamma_{q})}t^{q\gamma_{q}}-\frac{\tau(t)}{p^{*}S^{p^{*}/p}}t^{p^{*}}.$

By Lemma 4.2, we know that $\tilde{h}(t)<0$ for $t\in(0,R_{0})$ and
$\tilde{h}(t)>0$ for $t\in(R_{0},+\infty)$.

######  Lemma 9.3.

We have (i) $E_{\tau}\in C^{1}(W_{rad}^{1,p}(\mathbb{R}^{N}),\mathbb{R})$.

(ii) $E_{\tau}|_{S_{a,r}}$ is coercive and bounded from below. Moreover, if
$E_{\tau}(u)\leqslant 0$ on $S_{a,r}$, then $\lVert\nabla u\rVert_{p}\leqslant
R_{0}$ and $E_{\tau}(u)=E_{\mu}(u)$.