# Normalized solutions for Schrödinger equations with critical Sobolev
exponent and mixed nonlinearities

Juncheng Wei  Department of Mathematics, University of British Columbia,
Vancouver, B.C., Canada, V6T 1Z2<EMAIL_ADDRESS>and  Yuanze Wu  School of
Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R.
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the following nonlinear Schrödinger equations with
mixed nonlinearities:

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+\mu|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\\ &u\in
H^{1}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}u^{2}=a^{2},\end{aligned}\right.$

where $N\geq 3$, $\mu>0$, $\lambda\in\mathbb{R}$ and $2<q<2^{*}$. We prove in
this paper

  1. $(1)$

Existence of solutions of mountain-pass type for $N=3$ and
$2<q<2+\frac{4}{N}$.

  2. $(2)$

Existence and nonexistence of ground states for $2+\frac{4}{N}\leq q<2^{*}$
with $\mu>0$ large.

  3. $(3)$

Precisely asymptotic behaviors of ground states and mountain-pass solutions as
$\mu\to 0$ and $\mu$ goes to its upper bound.

Our studies answer some questions proposed by Soave in [49, Remarks 1.1, 1.2
and 8.1].

Keywords: Normalized solution; Ground state; Variational method; Critical
nonlinearity; Mixed nonlinearities.

AMS Subject Classification 2010: 35B09; 35B33; 35B40; 35J20.

##  1\. Introduction

In this paper, we consider the following nonlinear scalar field equation:

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+\mu|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\\ &u\in
H^{1}(\mathbb{R}^{N}),\end{aligned}\right.$ (1.1)

where $N\geq 3$, $\mu>0$, $\lambda\in\mathbb{R}$ and
$2<q<2^{*}=\frac{2N}{N-2}$.

(1.1) is a special case of the following model,

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda u+f(u)\quad\text{in
}\mathbb{R}^{N},\\\ &u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.$ (1.2)

which is related to finding the stationary waves of nonlinear Schrödinger
equations:

$\displaystyle i\psi_{t}+\Delta\psi+g(|\psi|^{2})\psi=0\quad\text{in
}\mathbb{R}^{N}.$ (1.3)

Indeed, a stationary wave of (1.3) is of the form $\psi(t,x)=e^{i\lambda
t}u(x)$ where $\lambda\in\mathbb{R}$ is a constant and $u(x)$ is a time-
independent function, then it is well-known that $\psi$ is a solution of (1.3)
if and only if $u$ is a solution of (1.2) with $f(u)=g(|u|^{2})u$. As pointed
out in [49, 50], in general, the function $u$ is complex valued and thus,
(1.2) can be regarded as a complex valued system coupled by the nonlinearities
$f(u)=g(|u|^{2})u$.

As pointed out in [29, 36], the studies on (1.2) can be traced back to the
semi-classical papers [12, 13, 42, 43, 52]. In these studies, there are two
different methods to study (1.2). The first one is to fix the number
$\lambda<0$ in (1.2) and restrict the unknown $u$ to be real valued. In this
case, (1.2) is a single equation and under some mild assumptions on $f(u)$,
the solutions of (1.2) are critical points of the functional

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

in the usual Sobolev space $H^{1}(\mathbb{R}^{N})$, where
$F(u)=\int_{0}^{u}f(t)dt$. In this case, particular attention is devoted to
least action solutions, namely solutions minimizing $\mathcal{J}(u)$ among all
non-trivial solutions. We also refer the readers to [2, 3, 4, 5] and the
references therein for the recent results on the special case (1.1) in this
direction. Another one is to fix the $L^{2}$ norm of the unknown $u$, that is,
to find solutions of (1.2) with prescribed mass. In this case, (1.2) is always
rewritten as follows

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda u+f(u)\quad\text{in
}\mathbb{R}^{N},\\\ &u\in
H^{1}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}u^{2}=a^{2},\end{aligned}\right.$
(1.4)

where $\int_{\mathbb{R}^{N}}u^{2}=a^{2}$ is the prescribed mass, and in this
case, $\lambda\in\mathbb{R}$ is a part of the unknown which appears in (1.4)
as a Lagrange multiplier. In particular, (1.1) is rewritten as

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda
u+\mu|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\\ &u\in
H^{1}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}u^{2}=a^{2}.\end{aligned}\right.$
(1.5)

Similar to the first case, under some mild assumptions on $f(u)$, the
solutions of (1.4) are critical points of the functional

$\displaystyle\mathcal{F}_{\mu}(u)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\big{(}\frac{1}{2}|\nabla
u|^{2}-F(u)\big{)}dx$

on the smooth manifold

$\displaystyle\mathcal{S}_{a}=\\{u\in
H^{1}(\mathbb{R}^{N})\mid\|u\|_{2}^{2}=a^{2}\\}.$

In particular, the solutions of (1.5) are critical points of the
$C^{2}$-functional

$\displaystyle\mathcal{E}_{\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{\mu}{q}\|u\|_{q}^{q}-\frac{1}{2^{*}}\|u\|_{2^{*}}^{2^{*}}$

on $\mathcal{S}_{a}$, where we denote the usual norm in
$L^{p}(\mathbb{R}^{N})$ by $\|\cdot\|_{p}$. In this case, solutions of (1.4)
are always called normalized solutions which are particularly relevant for the
nonlinear Schrödinger equation (1.3) since the mass is preserved along the
time evolution in (1.3). Thus, normalized solutions of (1.4) seems to be
particularly meaningful from the physical viewpoint, moreover, these solutions
often offer a good insight of the dynamical properties of the stationary
solutions for the nonlinear Schrödinger equation (1.3), such as stability or
instability (cf. [11, 18]). In this case, particular attention is also devoted
to least action solutions which are also called ground states for normalized
solutions, namely solutions minimizing $\mathcal{F}_{\mu}(u)$ among all non-
trivial solutions. The studies on normalized solutions of (1.4) is a hot topic
in the community of nonlinear PDEs nowadays, thus, it is impossible for us to
provide a complete references. We just refer the readers to [1, 9, 8, 10, 15,
17, 23, 33, 35, 36, 37, 46, 47, 40, 48, 49, 50, 51] and the references
therein. In these references, we would like to highlight [49, 50, 36] to the
readers for their detail introductions and references on normalized solutions
of (1.4) and new directions on the study of normalized solutions of autonomous
problems. We also would like to point out [26, 27, 28, 29, 30, 31, 32] and the
references therein for the studies on normalized solutions of problems with
trapping potentials.

It is well-known that the number $p=2+\frac{4}{N}$ plays an important role in
studying normalized solutions which is called the $L^{2}$ critical exponent or
mass critical exponent in the literature. Since
$2^{*}=\frac{2N}{N-2}>2+\frac{4}{N}$, the nonlinearity of (1.5) grows faster
than $|u|^{p-2}u$ at infinity and thus, it is well-known that
$\mathcal{E}_{\mu}(u)$ is unbounded from below on $\mathcal{S}_{a}$, which
makes one to find new constraints to prove the existence of ground states of
$\mathcal{E}_{\mu}(u)$ on $\mathcal{S}_{a}$. The new constraint, which is
introduced by Bartsch and Soave in [9] for the general problem (1.4) and is
widely used nowadays in studying normalized solutions, is the following
$L^{2}$-Pohozaev manifold:

$\displaystyle\mathcal{P}_{a,\mu}=\\{u\in\mathcal{S}_{a}\mid\|\nabla
u\|_{2}^{2}=\mu\gamma_{q}\|u\|_{q}^{q}+\|u\|_{2^{*}}^{2^{*}}\\},$

where

$\displaystyle\gamma_{q}=\frac{N(q-2)}{2q}.$ (1.6)

By the Pohozaev identity of (1.5), $\mathcal{P}_{a,\mu}$ contains all
nontrivial solutions of (1.5), thus, we have the following definition of
ground states of (1.5).

######  Definition 1.1.

We say $(u_{0},\lambda_{0})$ is a ground state of (1.5) if $u_{0}$ is a
critical point of $\mathcal{E}_{\mu}|_{\mathcal{S}_{a}}(u)$ with
$\mathcal{E}_{\mu}|_{\mathcal{S}_{a}}(u_{0})=\inf_{u\in\mathcal{P}_{a,\mu}}\mathcal{E}_{\mu}(u)$.

The $L^{2}$-Pohozaev manifold $\mathcal{P}_{a,\mu}$ is quite related to the
fibering maps

$\displaystyle\Psi(u,s)=\frac{e^{2s}}{2}\|\nabla u\|_{2}^{2}-\frac{\mu
e^{q\gamma_{q}s}}{q}\|u\|_{q}^{q}-\frac{e^{2^{*}s}}{2^{*}}\|u\|_{2^{*}}^{2^{*}},$

which is introduced by Jeanjean in [33] for the general problem (1.4) and is
well studied by Soave in [49]. According to the fibering maps $\Psi(u,s)$,
$\mathcal{P}_{a,\mu}$ can be naturally divided into the following three parts:

$\displaystyle\mathcal{P}_{+}^{a,\mu}=\\{u\in\mathcal{S}_{a}\mid 2\|\nabla
u\|_{2}^{2}>\mu q\gamma^{2}_{q}\|u\|_{q}^{q}+2^{*}\|u\|_{2^{*}}^{2^{*}}\\},$
$\displaystyle\mathcal{P}_{0}^{a,\mu}=\\{u\in\mathcal{S}_{a}\mid 2\|\nabla
u\|_{2}^{2}=\mu q\gamma^{2}_{q}\|u\|_{q}^{q}+2^{*}\|u\|_{2^{*}}^{2^{*}}\\},$
$\displaystyle\mathcal{P}_{-}^{a,\mu}=\\{u\in\mathcal{S}_{a}\mid 2\|\nabla
u\|_{2}^{2}<\mu q\gamma^{2}_{q}\|u\|_{q}^{q}+2^{*}\|u\|_{2^{*}}^{2^{*}}\\}.$

Let

$\displaystyle
m_{a,\mu}^{\pm}=\inf_{u\in\mathcal{P}_{\pm}^{a,\mu}}\mathcal{E}_{\mu}(u),$
(1.7)

then Soave proved the following results in [49, Theorems 1.1 and 1.4]:

  1. $(1)$

For $2<q<2+\frac{4}{N}$, there exists $\alpha_{N,q}>0$ such that if $\mu
a^{q-q\gamma_{q}}<\alpha_{N,q}$ then
$m_{a,\mu}^{+}=\inf_{u\in\mathcal{P}_{+}^{a,\mu}}\mathcal{E}_{\mu}(u)=\inf_{u\in\mathcal{P}_{a,\mu}}\mathcal{E}_{\mu}(u)<0$
and it can be attained by some $u_{a,\mu,+}$ which is real valued, positive,
radially symmetric and radially decreasing. Moreover, (1.5) has a ground state
$(u_{a,\mu,+},\lambda_{a,\mu,+})$ with $\lambda_{a,\mu,+}<0$, and
$m_{a,\mu}^{+}\to 0$ and $\|\nabla u_{a,\mu,+}\|_{2}\to 0$ as $\mu\to 0$.

  2. $(2)$

For $2+\frac{4}{N}\leq q<2^{*}$, there exists $\alpha_{N,q}>0$ such that if
$\mu a^{q-q\gamma_{q}}<\alpha_{N,q}$ then
$m_{a,\mu}^{-}=\inf_{u\in\mathcal{P}_{-}^{a,\mu}}\mathcal{E}_{\mu}(u)=\inf_{u\in\mathcal{P}_{a,\mu}}\mathcal{E}_{\mu}(u)\in(0,\frac{1}{N}S^{\frac{N}{2}})$
and it can be attained by some $u_{a,\mu,-}$ which is real valued, positive,
radially symmetric and radially decreasing, where $S$ is the optimal constant
in the Sobolev embedding, that is,

$\displaystyle\|u\|_{2^{*}}^{2}\leq S^{-1}\|\nabla u\|_{2}^{2}\quad\text{for
all }u\in D^{1,2}(\mathbb{R}^{N}).$ (1.8)

Moreover, (1.5) has a ground state $(u_{a,\mu,-},\lambda_{a,\mu,-})$ with
$\lambda_{a,\mu,-}<0$, and $m_{a,\mu}^{-}\to\frac{1}{N}S^{\frac{N}{2}}$ and
$\|\nabla u_{a,\mu,-}\|_{2}\to S^{\frac{N}{2}}$ as $\mu\to 0$.

In the $L^{2}$-subcritical case $2<q<2+\frac{4}{N}$, since
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ is unbounded from below, it could be
naturally to expect that $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ has a
second critical point of mountain-pass type, which is also positive, real
valued and radially symmetric. This natural expectation has been pointed out
by Soave in [49, Remark 1.1] which can be summarized to be the following
question:

  1. $(Q_{1})$

Does $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ has a critical point of
mountain-pass type in the $L^{2}$-subcritical case $2<q<2+\frac{4}{N}$?

######  Remark 1.1.

In preparing this paper, we notice that in the very recent work [34], the
question $(Q_{1})$ has been solved for $N\geq 4$. Thus, it only need to
consider the case $N=3$ for the question $(Q_{1})$.

Besides, since Soave only considered the case that $\mu a^{q-q\gamma_{q}}>0$
small in [49, Theorem 1.1], it is also natural to ask what will happen if
$\mu>0$ and $\mu a^{q-q\gamma_{q}}>0$ is large. This natural question has been
proposed by Soave in [49] as an open problem, which can be summarized to be
the following one:

  1. $(Q_{2})$

Does $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ have a ground state if $\mu>0$
and $\mu a^{q-q\gamma_{q}}>0$ large?

In [49], Soave conjectures that the answer of $(Q_{2})$ is negative in
general.

Finally, in these results, the asymptotic behavior is only for $\|\nabla
u_{a,\mu,-}\|_{2}$ in the cases of $2+\frac{4}{N}\leq q<2^{*}$. Thus, it is
also natural to ask if it is possible to characterize the asymptotic behavior
of $u_{a,\mu,-}$, and not only of $\|\nabla u_{a,\mu,-}\|_{2}$. In [49, Remark
8.1], Soave pointed out that in dimensions $N=3,4$, it could be proved that
$\|\nabla u_{a,\mu,-}\|_{2}\to S^{\frac{N}{2}}$, but
$u_{a,\mu,-}\rightharpoonup 0$ in $H^{1}$ while, in dimensions $H\geq 5$, both
$u_{a,\mu,-}\rightharpoonup 0$ and
$u_{a,\mu,-}\rightharpoonup\widetilde{u}\not=0$ could happen. He also
conjectures that the weak limit of $\\{u_{a,\mu,-}\\}$ will be the Aubin-
Talanti babbles in the higher dimensions $N\geq 5$. Soave’s conjecture can be
slightly generalized to the following question:

  1. $(Q_{3})$

Can we capture the precisely asymptotic behavior of $u_{a,\mu,-}$ as $\mu\to
0$?

In this paper, we are interested in these questions and we shall give some
answers to them, which will give more information on the ground states of
(1.5). Our first result, which is devoted to the existence and nonexistence of
ground states, can be stated as follows.

######  Theorem 1.1.

Let $N\geq 3$, $2<q<2^{*}$ and $a,\mu>0$.

  1. $(1)$

If $N=3$ and $2<q<2+\frac{4}{N}$, then for $\mu
a^{q-q\gamma_{q}}<\alpha_{N,q}$, $m_{a,\mu}^{-}$ can be attained by some
$u_{a,\mu,-}$ which is real valued, positive, radially symmetric and radially
decreasing, and thus, (1.5) has a second solution $u_{a,\mu,-}$ with some
$\lambda_{a,\mu,-}<0$.

  2. $(2)$

If $q=2+\frac{4}{N}$, then $m_{a,\mu}^{-}$ can not be attained for $\mu
a^{q-q\gamma_{q}}\geq\alpha_{N,q}$ and thus, (1.5) has no ground states for
$\mu a^{q-q\gamma_{q}}\geq\alpha_{N,q}$.

  3. $(3)$

If $2+\frac{4}{N}<q<2^{*}$, then for all $\mu>0$, and $m_{a,\mu}^{-}$ can be
attained by some $u_{a,\mu,-}$ which is real valued, positive, radially
symmetric and radially decreasing, and thus, (1.5) has a ground state
$u_{a,\mu,-}$ with some $\lambda_{a,\mu,-}<0$ for all $\mu>0$.

######  Remark 1.2.

  1. $(a)$

$(1)$ of Theorem 1.1, which together the results of [34], gives a completely
positive answer to the question $(Q_{1})$.

  2. $(b)$

As pointed out in the very recent work [34], the crucial point in studying
$(Q_{1})$ is to obtain a good energy estimate of $m_{a,\mu}^{-}$ for
$2<q<2+\frac{4}{N}$ such that the compactness of minimizing sequence or $(PS)$
sequence at the energy level $m_{a,\mu}^{-}$ still holds. As for other
concave-convex problems (cf. [6]) and observed in [34], the threshold of such
compactness should be $m_{a,\mu}^{+}+\frac{1}{N}S^{\frac{N}{2}}$. Since
$m_{a,\mu}^{-}$ is a mountain-pass level, the classical idea, which can be
traced back to [16], is to use the ground state $u_{a,\mu,+}$ and the Aubin-
Talanti babbles to construct a good path, whose energy can be well controlled
from above to make sure that it is smaller than the threshold
$m_{a,\mu}^{+}+\frac{1}{N}S^{\frac{N}{2}}$. This strategy is already used in
[34] to prove the existence of critical points of
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ of mountain-pass type for $N\geq 4$
and $2<q<2+\frac{4}{N}$. Unlike [34] in which nonradial test function
composing of $u_{a,\mu,+}$ and a bubble at $\infty$ is used, here we directly
use the radial superposition of $u_{a,\mu,+}$ and the Aubin-Talenti bubble.
This test function seems to be more natural and it works for all dimensions.

  3. $(c)$

$(2)$ and $(3)$ of Theorem 1.1 give partial answers to $(Q_{2})$ and they are
proved by observing the non-increasing of $m_{a,\mu}^{-}$ and suitable choices
of test functions. These two conclusions imply that the $L^{2}$-critical and
supercritical perturbations have quite different influence on (1.5). Moreover,
it seems that the critical mass of ground states also exists for (1.5) in the
$L^{2}$-critical case.

Our next result will be devoted to the precisely asymptotic behaviors of the
solutions found in [49, Theorem 1.1], [34, Theorem 1.6] and Theorem 1.1 as
$\mu\to 0$. To state this result, let us first introduce some necessary
notations. By [55, Theorem B], the Gagliardo-Nirenberg inequality,

$\displaystyle\|u\|_{q}\leq C_{N,q}\|u\|_{2}^{1-\gamma_{q}}\|\nabla
u\|_{2}^{\gamma_{q}}\quad\text{for all }u\in H^{1}(\mathbb{R}^{N}),$ (1.9)

has a minimizer $\phi_{0}$, which satisfies

$\displaystyle\left\\{\begin{aligned}
&-\Delta\phi_{0}+\nu_{0}\phi_{0}=\sigma_{0}\phi_{0}^{q-1}\quad\text{in
}\mathbb{R}^{N},\\\ &\phi_{0}(0)=\max_{x\in\mathbb{R}^{N}}\phi_{0}(x),\\\
&\phi_{0}(x)>0\quad\text{in }\mathbb{R}^{N},\\\ &\phi_{0}(x)\to 0\quad\text{as
}|x|\to+\infty,\end{aligned}\right.$ (1.10)

where $\nu_{0}=\frac{4}{N(q-2)}(1-\frac{(q-2)(N-2)}{4})$,
$\sigma_{0}=\frac{4}{N(q-2)}$ and $C_{N,q}$ is the best constant in the
Gagliardo-Nirenberg inequality. On the other hand, the Aubin-Talanti babbles,

$\displaystyle
U_{\varepsilon}(x)=[N(N-2)]^{\frac{N-2}{4}}\bigg{(}\frac{\varepsilon}{\varepsilon^{2}+|x|^{2}}\bigg{)}^{\frac{N-2}{2}},$
(1.11)

is the only solutions to the following equation:

$\displaystyle\left\\{\begin{aligned} &-\Delta u=u^{2^{*}-1}\quad\text{in
}\mathbb{R}^{N},\\\ &u(0)=\max_{x\in\mathbb{R}^{N}}u(x),\\\
&u(x)>0\quad\text{in }\mathbb{R}^{N},\\\ &u(x)\to 0\quad\text{as
}|x|\to+\infty.\end{aligned}\right.$

Now, our second result can be stated as follows.

######  Theorem 1.2.

Let $N\geq 3$, $2<q<2^{*}$ and $a,\mu>0$ such that $\mu>0$ is sufficiently
small. Let $\widetilde{u}_{\mu}$ be the minimizer of
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ in $\mathcal{P}_{+}^{a,\mu}$ and
$\widehat{u}_{\mu}$ be the minimizer of
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ in $\mathcal{P}_{-}^{a,\mu}$. Then

  1. $(1)$

For $2<q<2+\frac{4}{N}$,
$\widetilde{w}_{a,\mu}(x)=s_{\mu}^{\frac{N}{2}}\widetilde{u}_{\mu}(s_{\mu}x)\to\nu_{a}^{\frac{1}{q-2}}\phi_{0}(\sqrt{\nu_{a}}x)$
strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$, where $\phi_{0}$ is the
unique solution of (1.10),

$\displaystyle\nu_{a}=\bigg{(}\frac{a^{2}}{\|\phi_{0}\|_{2}^{2}}\bigg{)}^{\frac{2(q-2)}{4-N(q-2)}}.$
(1.12)

and $s_{\mu}\sim\mu^{\frac{1}{2-q\gamma_{q}}}$ is the unique solution of the
following system:

$\displaystyle\left\\{\begin{aligned}
&s_{\mu}^{2}\|\nabla\psi_{\nu_{a},1}\|_{2}^{2}-\mu\gamma_{q}\|\psi_{\nu_{a},1}\|_{q}^{q}s_{\mu}^{q\gamma_{q}}-\|\psi_{\nu_{a},1}\|_{2^{*}}^{2^{*}}s_{\mu}^{2^{*}}=0,\\\
&2s_{\mu}^{2}\|\nabla\psi_{\nu_{a},1}\|_{2}^{2}-\mu
q\gamma_{q}^{2}\|\psi_{\nu_{a},1}\|_{q}^{q}s_{\mu}^{q\gamma_{q}}-2^{*}\|\psi_{\nu_{a},1}\|_{2^{*}}^{2^{*}}s_{\mu}^{2^{*}}>0,\end{aligned}\right.$
(1.13)

where $\psi_{\nu_{a},1}(x)=\nu_{a}^{\frac{1}{q-2}}\phi_{0}(\sqrt{\nu_{a}}x)$.
Moreover, up to translations and rotations, $\widetilde{u}_{\mu}$ is the
unique ground state of (1.5) for $\mu>0$ sufficiently small.

  2. $(2)$

For $N\geq 5$, $\widehat{u}_{\mu}\to U_{\varepsilon_{0}}$ strongly in
$H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$, where $U_{\varepsilon_{0}}$ is the
Aubin-Talanti babble satisfying $\|U_{\varepsilon_{0}}\|_{2}^{2}=a^{2}$.
Moreover, up to translations and rotations, $\widehat{u}_{\mu}$ is the unique
minimizer of $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ in
$\mathcal{P}_{-}^{a,\mu}$ for $\mu>0$ sufficiently small.

  3. $(3)$

For $N=3,4$,
$\widehat{w}_{a,\mu}(x)=\varepsilon_{\mu}^{\frac{N-2}{2}}\widehat{u}_{\mu}(\varepsilon_{\mu}x)\to
U_{\varepsilon_{0}}$ strongly in $D^{1,2}(\mathbb{R}^{N})$ for some
$\varepsilon_{0}>0$ as $\mu\to 0$ up to a subsequence, where
$\varepsilon_{\mu}$ satisfies

$\displaystyle\mu\sim\left\\{\begin{aligned}
&\varepsilon_{\mu}^{6-q}e^{-2\varepsilon_{\mu}^{-2}},\quad N=4,2<q<4,\\\
&\varepsilon_{\mu}^{\frac{q}{2}-1},\quad N=3,3<q<6,\\\
&\frac{\varepsilon_{\mu}^{\frac{1}{2}}}{\ln(\frac{1}{\varepsilon_{\mu}})},\quad
N=3,q=3,\\\ &\varepsilon_{\mu}^{5-\frac{3q}{2}},\quad
N=3,2<q<3.\end{aligned}\right.$

######  Remark 1.3.

  1. $(1)$

The precise asymptotic behaviors of $\widetilde{u}_{\mu}$ and
$\widehat{u}_{\mu}$ stated in $(1)$ and $(2)$ of Theorem 1.2 are captured by
comparing the energy values and norms by full using the variational formulas
of $\widetilde{u}_{\mu}$ and $\widehat{u}_{\mu}$, and minimizers of the
Gagliardo-Nirenberg inequality and the Aubin-Talanti bubbles. In this
argument, the unique determination of minimizers of the Gagliardo–Nirenberg
inequality (1.9) for $2<q<2+\frac{4}{N}$ and Aubin-Talanti bubbles for $N\geq
5$ in $\mathcal{S}_{a}$, respectively, is crucial. Moreover, $(2)$ of Theorem
1.2 also gives a positive answer to Soave’s conjecture on $(Q_{3})$.

  2. $(2)$

For the local uniqueness, the standard strategy is to assume the contrary and
obtain a contradiction by full using the non-degeneracy of minimizers of the
Gagliardo–Nirenberg inequality and Aubin-Talanti bubbles in passing to the
limit (cf. [27, 21]), which is powerful in studying problems with potentials.
Since (1.5) is autonomous, we can use a different method, based on the
precisely asymptotic behaviors of $\widetilde{u}_{\mu}$ and the implicit
function theorem, to prove the local uniqueness of $\widetilde{u}_{\mu}$ in a
more direct way. It is worth pointing out that our method is also based on the
non-degenerate of minimizers of the Gagliardo–Nirenberg inequality. For
$\widehat{u}_{\mu}$, we remark that since the linear operator of the limit
equation is different from that of (1.5), our direct methods, based on
implicit function theorem, is invalid. Thus, we will still use the standard
method, that is to assume the contrary and obtain a contradiction by full
using the non-degeneracy of Aubin-Talanti bubbles.

  3. $(3)$

Since we loss the $L^{2}$-integrability of the Aubin-Talanti babbles
$\\{U_{\varepsilon}\\}$ for $N=3,4$, the asymptotic behavior of
$\widehat{u}_{\mu}$ as $\mu\to 0$ for $N=3,4$ is much weaker than that of
$N\geq 5$ in the sense that, the convergence is only for subsequences, which
also leads us to loss the local uniqueness of $\widehat{u}_{\mu}$ for $\mu>0$
sufficiently small in these two cases. We also remark that since we loss the
$L^{2}$-integrability of the Aubin-Talanti babbles $\\{U_{\varepsilon}\\}$ for
$N=3,4$, the asymptotic behavior of $\widehat{u}_{\mu}$ can not be obtained by
merely using variational arguments to compare the energy values and norms as
that for $(2)$ of Theorem 1.2. Thus, to capture the precisely asymptotic
behavior of $\widehat{u}_{\mu}$, we drive some uniformly pointwise estimates
of $\widehat{u}_{\mu}$ by the maximum principle (cf. [20]) and some ODE
technique used in [7] (see also [24, 38]). With these additional estimates, we
obtain the precisely asymptotic behavior of $\widehat{u}_{\mu}$ for $N=3,4$.
It is worth pointing out that, in the case $N=3$ and $2<q<3$, since the
nonlinearity decays too slow at infinity, we need to further employ the
bootstrapping argument to drive the desired estimates.

Our final result is devoted to the asymptotic behavior of the minimizers of
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ in $\mathcal{P}_{-}^{a,\mu}$ as
$\mu$ close to its upper-bound in the cases of $2+\frac{4}{N}\leq q<2^{*}$. It
can be stated as follows.

######  Theorem 1.3.

Assume $N\geq 3$, $2+\frac{4}{N}\leq q<2^{*}$ and $\mu,a>0$. Let
$\widehat{u}_{\mu}$ be the minimizer of
$\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ in $\mathcal{P}_{-}^{a,\mu}$, found
in [49, Theorem 1.1] for $q=2+\frac{4}{N}$ with $0<\mu
a^{q-q\gamma_{q}}<\alpha_{N,q}$ and found in Theorem 1.1 for
$2+\frac{4}{N}<q<2^{*}$ with all $\mu>0$. Then

  1. $(1)$

For $q=2+\frac{4}{N}$,
$\widehat{v}_{\mu}=(\frac{a}{\|\phi_{0}\|_{2}})^{\frac{N-2}{2}}s_{\mu}^{\frac{N}{2}}\widehat{u}_{\mu}(\frac{a}{\|\phi_{0}\|_{2}}s_{\mu}x)\to(\nu_{a}^{\prime})^{\frac{1}{q-2}}\phi_{0}(\sqrt{\nu_{a}^{\prime}}x)$
strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to\alpha_{N,q,a}$ up to a
subsequence, where $\alpha_{N,q,a}=a^{q\gamma_{q}-q}\alpha_{N,q}$ for some
$\nu_{a}^{\prime}>0$ and
$s_{\mu}=(1-\frac{\mu}{\alpha_{N,q,a}})^{-\frac{N-2}{4}}$.

  2. $(2)$

For $2+\frac{4}{N}<q<2^{*}$,
$\widehat{v}_{\mu}=s_{\mu}^{\frac{N}{2}}\widehat{u}_{\mu}(s_{\mu}x)\to\nu_{a}^{\frac{1}{q-2}}\phi_{0}(\sqrt{\nu_{a}}x)$
strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to+\infty$, where
$s_{\mu}=\mu^{\frac{1}{q\gamma_{q}-2}}$. Moreover, up to translations and
rotations, $\widehat{u}_{\mu}$ is also the unique ground state of (1.5) for
$\mu>0$ sufficiently large.

######  Remark 1.4.

  1. (1)

The ideas in proving Theorem 1.3 are similar to that of Theorem 1.2. However,
in the $L^{2}$-critical case $q=2+\frac{4}{N}$, the convergence of
$\widehat{u}_{\mu}$ is much weaker than that in the $L^{2}$-supcritical case
$2+\frac{4}{N}<q<2^{*}$ in the sense that, it only holds for subsequences. The
main reason is that in the $L^{2}$-critical case $q=2+\frac{4}{N}$, we have
$\|\varphi\|_{2}^{2}=const.$ for all $\varphi$ being a minimizer of the
Gagliardo–Nirenberg inequality (1.9). Thus, the precise mass
$\|\widehat{u}_{\mu}\|_{2}^{2}=a^{2}$ is invalid in determining a unique
minimizer of the Gagliardo–Nirenberg inequality (1.9) in the case
$q=2+\frac{4}{N}$. Moreover, unlike the studies for problems with homogeneous
nonlinearities (cf. [26, 27]), combining nonlinearities ($L^{2}$-critical and
$L^{2}$-supercritical) of (1.5) makes the asymptotic behavior of
$\widehat{u}_{\mu}$ to be more complicated, which also make us loss the local
uniqueness of $\widehat{u}_{\mu}$ for $\mu>0$ close to its upper bound in this
case. Indeed, as $\mu$ goes to its upper bound in the $L^{2}$-critical case,
comparing with the studies for problems with homogeneous nonlinearities, the
Sobolev critical term of (1.5) is an additionally inhomogenous perturbation in
passing to the limit, which makes the oscillations occurring.

Notations. Throughout this paper, $C$ and $C^{\prime}$ are indiscriminately
used to denote various absolutely positive constants. $a\sim b$ means that
$C^{\prime}b\leq a\leq Cb$ and $a\lesssim b$ means that $a\leq Cb$.

##  2\. Asymptotic behavior of $u_{a,\mu,+}$

By [49, Theorem 1.1], $m_{a,\mu}^{+}$ can always be attained by some
$u_{a,\mu,+}$ for $2<q<2+\frac{4}{N}$ and $\mu
a^{q-q\gamma_{q}}<\alpha_{N,q}$, where $m_{a,\mu}^{+}$ is given by (1.7) and
$u_{a,\mu,+}$ is real valued, positive, radially symmetric and radially
decreasing. Our goal in this section is to give an asymptotic behavior of
$u_{a,\mu,+}$ as $\mu\to 0$, which is more precisely than that in [49, Theorem
1.4], and capture the precisely decaying rate of $u_{a,\mu,+}$ as $\mu\to 0$.
We recall that by [49, Theorem 1.1], $u_{a,\mu,+}$ is a solution of (1.5) for
some $\lambda_{a,\mu,+}<0$. To simplify the notation, we shall denote
$u_{\mu,+}=u_{a,\mu,+}$ and $\lambda_{\mu,+}=\lambda_{a,\mu,+}$, since we will
fix $a>0$ in what follows. Let us begin with

######  Lemma 2.1.

Let $2<q<2+\frac{4}{N}$. Then $-\lambda_{\mu,+}\sim\|\nabla
u_{\mu,+}\|_{2}^{2}\sim\mu^{\frac{2}{2-q\gamma_{q}}}$ as $\mu\to 0$.

###### Proof.

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

$\displaystyle\|\nabla
u_{\mu,+}\|_{2}^{2}=\mu\gamma_{q}\|u_{\mu,+}\|_{q}^{q}+\|u_{\mu,+}\|_{2^{*}}^{2^{*}}$
(2.1)

and

$\displaystyle 2\|\nabla u_{\mu,+}\|_{2}^{2}>\mu
q\gamma_{q}^{2}\|u_{\mu,+}\|_{q}^{q}+2^{*}\|u_{\mu,+}\|_{2^{*}}^{2^{*}}.$

It follows from the Gagliardo–Nirenberg inequality that

$\displaystyle\|\nabla
u_{\mu,+}\|_{2}^{2}\lesssim\mu\|u_{\mu,+}\|_{q}^{q}\lesssim\mu\|\nabla
u_{\mu,+}\|_{2}^{q\gamma_{q}},$

which together with $q\gamma_{q}<2$ for $2<q<2+\frac{4}{N}$, implies

$\displaystyle\|\nabla
u_{\mu,+}\|_{2}^{2}\lesssim\mu^{\frac{2}{2-q\gamma_{q}}}.$ (2.2)

Thus, by (2.1) and (2.2), we also have

$\displaystyle\mu\|u_{\mu,+}\|_{q}^{q}\lesssim\mu^{\frac{2}{2-q\gamma_{q}}}.$
(2.3)

Let us define

$\displaystyle
V_{\varepsilon}(x)=U_{\varepsilon}(x)\varphi(R_{\varepsilon}^{-1}x)$ (2.4)

where $U_{\varepsilon}(x)$ is the Aubin-Talanti babbles given by (1.11) and
$\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$ is a radial cut-off function with
$\varphi\equiv 1$ in $B_{1}$, $\varphi\equiv 0$ in $B_{2}^{c}$, and
$R_{\varepsilon}$ is chosen such that $V_{\varepsilon}\in\mathcal{S}_{a}$.
More precisely, for $N\geq 5$, we choose $\varepsilon=\varepsilon_{0}$ and
$R_{\varepsilon_{0}}=+\infty$ such that
$V_{\varepsilon_{0}}=U_{\varepsilon_{0}}\in\mathcal{S}_{a}$ while for $N=3,4$,
we choose $\varepsilon>0$ sufficiently small and then in the later two cases,
we have

$\displaystyle
a^{2}=\int_{\mathbb{R}^{N}}(U_{\varepsilon}(x)\varphi(R_{\varepsilon}^{-1}x))^{2}\sim\varepsilon^{2}\int_{1}^{R_{\varepsilon}\varepsilon^{-1}}r^{3-N}\sim\left\\{\begin{aligned}
\varepsilon^{2}\ln(R_{\varepsilon}\varepsilon^{-1}),\text{ for }N=4,\\\
\varepsilon R_{\varepsilon},\text{ for }N=3,\end{aligned}\right.$ (2.5)

which implies $R_{\varepsilon}\varepsilon^{-1}\to+\infty$ as $\varepsilon\to
0$. Then it is well-known (cf. [44, (4.2)–(4.5)] or [53, Chapter III]) that

$\displaystyle\|\nabla
V_{\varepsilon}\|_{2}^{2}=S^{\frac{N}{2}}+O((R_{\varepsilon}\varepsilon^{-1})^{2-N}),\quad\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}=S^{\frac{N}{2}}+O((R_{\varepsilon}\varepsilon^{-1})^{-N})$
(2.6)

for $\varepsilon>0$ sufficiently small, which implies

$\displaystyle\|\nabla V_{\varepsilon}\|_{2}^{2}\sim
S^{\frac{N}{2}}\sim\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}$ (2.7)

for $\varepsilon>0$ sufficiently small. Now, we fix
$\varepsilon=\varepsilon_{0}$ and $R_{\varepsilon_{0}}=+\infty$ for $N=5$, and
fix $\varepsilon>0$ sufficiently small and choose $R_{\varepsilon}$ as that in
(2.5) for $N=3,4$ such that (2.7) holds for all $N\geq 3$. By [49, Lemma 4.2],
there exists $t(\mu)>0$ such that
$(V_{\varepsilon})_{t(\mu)}\in\mathcal{P}_{+}^{a,\mu}$ for $\mu>0$
sufficiently small, where

$\displaystyle(V_{\varepsilon})_{t(\mu)}=[t(\mu)]^{\frac{N}{2}}V_{\varepsilon}(t(\mu)x).$

Then

$\displaystyle[t(\mu)]^{2}\|\nabla
V_{\varepsilon}\|_{2}^{2}=\mu\gamma_{q}\|V_{\varepsilon}\|_{q}^{q}[t(\mu)]^{q\gamma_{q}}+\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}[t(\mu)]^{2^{*}}$

and

$\displaystyle 2[t(\mu)]^{2}\|\nabla V_{\varepsilon}\|_{2}^{2}>\mu
q\gamma_{q}^{2}\|V_{\varepsilon}\|_{q}^{q}[t(\mu)]^{q\gamma_{q}}+2^{*}\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}[t(\mu)]^{2^{*}}.$

Since $q\gamma_{q}<2$ for $2<q<2+\frac{4}{N}$, by

$\displaystyle(2^{*}-2)[t(\mu)]^{2}\|\nabla
V_{\varepsilon}\|_{2}^{2}<\mu(2^{*}-q\gamma_{q})\gamma_{q}\|V_{\varepsilon}\|_{q}^{q}[t(\mu)]^{q\gamma_{q}},$

it is easy to see that $t(\mu)\to 0$ as $\mu\to 0$ for all $N\geq 3$. It
follows that

$\displaystyle[t(\mu)]^{2}\sim\mu[t(\mu)]^{q\gamma_{q}}\quad\text{as }\mu\to
0,$

which implies $t(\mu)\sim\mu^{\frac{1}{2-q\gamma_{q}}}$ as $\mu\to 0$. Thus,
by $q\gamma_{q}<2$ for $2<q<2+\frac{4}{N}$ once more,

$\displaystyle\mathcal{E}_{\mu}((V_{\varepsilon})_{t(\mu)})=(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla
V_{\varepsilon}\|_{2}^{2}[t(\mu)]^{2}+(\frac{1}{q\gamma_{q}}-\frac{1}{2^{*}})\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}[t(\mu)]^{2^{*}}\sim-\mu^{\frac{2}{2-q\gamma_{q}}}.$

Therefore, by $\mathcal{E}_{\mu}((V_{\varepsilon})_{t(\mu)})\geq
m_{a,\mu}^{+}$ and $m_{a,\mu}^{+}\gtrsim-\mu\|u_{\mu,+}\|_{q}^{q}$, we have

$\displaystyle\mu\|u_{\mu,+}\|_{q}^{q}\gtrsim\mu^{\frac{2}{2-q\gamma_{q}}},$

which together with (2.3), implies

$\displaystyle\mu\|u_{\mu,+}\|_{q}^{q}\sim\mu^{\frac{2}{2-q\gamma_{q}}}.$

By the regularity of $u_{\mu,+}$ and the Pohozaev identity,
$\lambda_{\mu,+}\sim-\mu\|u_{\mu,+}\|_{q}^{q}$, and by (2.2) and
$u_{\mu,+}\in\mathcal{P}_{+}^{a,\mu}$, $\|\nabla
u_{\mu,+}\|_{2}^{2}\sim\mu\|u_{\mu,+}\|_{q}^{q}$. Therefore,

$\displaystyle-\lambda_{\mu,+}\sim\|\nabla
u_{\mu,+}\|_{2}^{2}\sim\mu^{\frac{2}{2-q\gamma_{q}}}$

as $\mu\to 0$. It completes the proof. ∎

By the well-known uniqueness result (cf. [39]) and the scaling invariance of
(1.10),

$\displaystyle\phi_{0}(x)=\bigg{(}\frac{\nu_{0}}{\sigma_{0}}\bigg{)}^{\frac{1}{q-2}}w(\sqrt{\nu_{0}}x),$

where $w$ is the unique solution of the following equation:

$\displaystyle\left\\{\begin{aligned} &-\Delta u+u=u^{q-1}\quad\text{in
}\mathbb{R}^{N},\\\ &u(0)=\max_{x\in\mathbb{R}^{N}}u(x),\\\
&u(x)>0\quad\text{in }\mathbb{R}^{N},\\\ &u(x)\to 0\quad\text{as
}|x|\to+\infty,\end{aligned}\right.$ (2.8)

A direct calculation also shows that

$\displaystyle\psi_{\nu,\sigma}(x)=(\frac{\nu}{\sigma})^{\frac{1}{q-2}}\phi_{0}(\sqrt{\frac{\nu}{\sigma}}x)$
(2.9)

for $\nu,\sigma>0$ are all minimizers of the Gagliardo-Nirenberg inequality
(1.9). Let $\nu_{a}$ be given by (1.12), then for $q\not=2+\frac{4}{N}$,
$\psi_{\nu_{a},1}\in\mathcal{S}_{a}$ and $\psi_{\nu_{a},1}$ is a minimizer of
the Gagliardo–Nirenberg inequality, that is,

$\displaystyle\|\psi_{\nu_{a},1}\|_{q}^{q}=C_{N,q}^{q}a^{q-q\gamma_{q}}\|\nabla\psi_{\nu_{a},1}\|_{2}^{q\gamma_{q}}.$
(2.10)

For the sake of simplicity, we re-denote $\psi_{a}=\psi_{\nu_{a},1}$.

######  Proposition 2.1.

Let $2<q<2+\frac{4}{N}$. Then $w_{\mu,+}\to\psi_{a}$ strongly in
$H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$, where
$w_{\mu,+}=s_{\mu}^{-\frac{N}{2}}u_{\mu,+}(s_{\mu}^{-1}x)$ with $s_{\mu}$
being the unique solution of (1.13). Moreover, up to translations and
rotations, $u_{\mu,+}$ is the unique ground state of (1.5) for $\mu>0$
sufficiently small.

###### Proof.

Since $\psi_{a}\in\mathcal{S}_{a}$, by [49, Lemma 4.2], there exists a unique
$s_{\mu}>0$ such that $(\psi_{a})_{s_{\mu}}\in\mathcal{P}_{+}^{a,\mu}$ for
$\mu>0$ sufficiently small where
$(\psi_{a})_{s_{\mu}}=s_{\mu}^{\frac{N}{2}}\psi_{a}(s_{\mu}x)$. That is,

$\displaystyle
s_{\mu}^{2}\|\nabla\psi_{a}\|_{2}^{2}=\mu\gamma_{q}\|\psi_{a}\|_{q}^{q}s_{\mu}^{q\gamma_{q}}+\|\psi_{a}\|_{2^{*}}^{2^{*}}s_{\mu}^{2^{*}}$
(2.11)

and

$\displaystyle 2s_{\mu}^{2}\|\nabla\psi_{a}\|_{2}^{2}>\mu
q\gamma_{q}^{2}\|\psi_{a}\|_{q}^{q}s_{\mu}^{q\gamma_{q}}+2^{*}\|\psi_{a}\|_{2^{*}}^{2^{*}}s_{\mu}^{2^{*}}.$
(2.12)

As that in the proof of Lemma 2.1, we have

$\displaystyle\|\nabla(\psi_{a})_{s_{\mu}}\|_{2}^{2-q\gamma_{q}}<C_{N,q}^{q}\gamma_{q}\mu
a^{q-q\gamma_{q}}\frac{2^{*}-q\gamma_{q}}{2^{*}-2}.$ (2.13)

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

$\displaystyle\|\nabla
u_{\mu,+}\|_{2}^{2-q\gamma_{q}}<C_{N,q}^{q}\gamma_{q}\mu
a^{q-q\gamma_{q}}\frac{2^{*}-q\gamma_{q}}{2^{*}-2}.$ (2.14)

Now, using $(\psi_{a})_{s_{\mu}}$ as a test function of $m_{a,\mu}^{+}$ and by
(2.10),

$\displaystyle
m_{a,\mu}^{+}\leq\mathcal{E}_{\mu}((\psi_{a})_{s_{\mu}})=\frac{1}{N}\|\nabla(\psi_{a})_{s_{\mu}}\|_{2}^{2}-\frac{\mu
a^{q-q\gamma_{q}}C_{N,q}^{q}}{q}(1-\frac{q\gamma_{q}}{2^{*}})\|\nabla(\psi_{a})_{s_{\mu}}\|_{2}^{q\gamma_{q}}.$

By the Gagliardo–Nirenberg inequality (1.9),

$\displaystyle
m_{a,\mu}^{+}=\mathcal{E}_{\mu}(u_{\mu,+})\geq\frac{1}{N}\|\nabla
u_{\mu,+}\|_{2}^{2}-\frac{\mu
a^{q-q\gamma_{q}}C_{N,q}^{q}}{q}(1-\frac{q\gamma_{q}}{2^{*}})\|\nabla
u_{\mu,+}\|_{2}^{q\gamma_{q}}.$

Let us consider the function

$\displaystyle f(t)=\frac{1}{N}t^{2}-\frac{\mu
a^{q-q\gamma_{q}}C_{N,q}^{q}}{q}(1-\frac{q\gamma_{q}}{2^{*}})t^{q\gamma_{q}}.$

A direct calculation shows that $f(t)$ is strictly decreasing in $(0,t_{0})$,
where

$\displaystyle t_{0}=\bigg{(}C_{N,q}^{q}\gamma_{q}\mu
a^{q-q\gamma_{q}}\frac{2^{*}-q\gamma_{q}}{2^{*}-2}\bigg{)}^{\frac{1}{2-q\gamma_{q}}}.$

Thus, by (2.13) and (2.14),

$\displaystyle\|\nabla
u_{\mu,+}\|_{2}^{2}\geq\|\nabla(\psi_{a})_{s_{\mu}}\|_{2}^{2}.$ (2.15)

By (2.11) and (2.12), we can use similar arguments as that used in the proof
of Lemma 2.1 to show that $s_{\mu}\sim\mu^{\frac{1}{2-q\gamma_{q}}}$ as
$\mu\to 0$. It then follows from (2.9) and (2.11) that

$\displaystyle
s_{\mu}=(1+o_{\mu}(1))\bigg{(}\frac{\mu\gamma_{q}\|\psi_{a}\|_{q}^{q}}{\|\nabla\psi_{a}\|_{2}^{2}}\bigg{)}^{\frac{1}{2-q\gamma_{q}}}=(1+o_{\mu}(1))\bigg{(}\frac{\mu\gamma_{q}\|\phi_{0}\|_{q}^{q}}{\|\nabla\phi_{0}\|_{2}^{2}}\bigg{)}^{\frac{1}{2-q\gamma_{q}}}.$

Since by the Pohozaev identity satisfied by $\phi$, we have
$\frac{1}{N}\|\nabla\phi_{0}\|_{2}^{2}=\frac{(q-2)\sigma_{0}}{2q}\|\phi_{0}\|_{q}^{q}$.
By (1.6),
$s_{\mu}=[(\frac{1}{\sigma_{0}}+o_{\mu}(1))\mu]^{\frac{1}{2-q\gamma_{q}}}$.
Let

$\displaystyle w_{\mu,+}=s_{\mu}^{-\frac{N}{2}}u_{\mu,+}(s_{\mu}^{-1}x).$

Since $u_{\mu,+}$ satisfies (1.5), $w_{\mu,+}$ satisfies the following
equation:

$\displaystyle-\Delta w_{\mu,+}=\lambda_{\mu,+}s_{\mu}^{-2}w_{\mu,+}+\mu
s_{\mu}^{-2+\frac{N}{2}(q-2)}w_{\mu,+}^{q-1}+s_{\mu}^{-2+\frac{N}{2}(2^{*}-2)}w_{\mu,+}^{2^{*}-1}.$
(2.16)

By Lemma 2.1 and

$\displaystyle\int_{\mathbb{R}^{N}}w_{\mu,+}^{2}=\int_{\mathbb{R}^{N}}u_{\mu,+}^{2}\equiv
a^{2},$

we have

$\displaystyle\|\nabla
w_{\mu,+}\|_{2}^{2}+\|w_{\mu,+}\|_{2}^{2}=s_{\mu}^{-2}\|\nabla
u_{\mu,+}\|_{2}^{2}+a^{2}\sim 1.$

Therefore, $\\{w_{\mu,+}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$. It follows
that $w_{\mu,+}\rightharpoonup w_{*}$ weakly in $H^{1}(\mathbb{R}^{N})$ as
$\mu\to 0$ up to a subsequence. Note that $w_{\mu,+}$ is radial, by Struss’s
radial lemma (cf. [12, Lemma A.IV, Theorem A.I’] or [44, Lemma 3.1]) and the
Sobolev embedding theorem, $w_{\mu,+}\to w_{*}$ strongly in
$L^{q}(\mathbb{R}^{N})$ as $\mu\to 0$ up to a subsequence. By Lemma 2.1 once
more, $\\{\lambda_{\mu,+}\mu^{\frac{-2}{2-q\gamma_{q}}}\\}$ is bounded. Thus,
$\lambda_{\mu,+}\mu^{\frac{-2}{2-q\gamma_{q}}}\to\alpha_{*}$ as $\mu\to 0$ up
to a subsequence. On the other hand, by $q\gamma_{q}<2$ for
$2<q<2+\frac{4}{N}$,

$\displaystyle
s_{\mu}^{-2+\frac{N}{2}(2^{*}-2)}\sim\mu^{\frac{2^{*}-2}{2-q\gamma_{q}}}\to 0$

as $\mu\to 0$. Now, using (2.15) and (2.16), it is standard to show that
$w_{\mu,+}\to w_{*}$ strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$ up to a
subsequence, where $w_{*}$ is the unique solution of the following equation:

$\displaystyle\left\\{\begin{aligned} &-\Delta
u+\alpha_{*}u=\sigma_{0}u^{q-1}\quad\text{in }\mathbb{R}^{N},\\\
&u(0)=\max_{x\in\mathbb{R}^{N}}u(x),\\\ &u(x)>0\quad\text{in
}\mathbb{R}^{N},\\\ &u(x)\to 0\quad\text{as
}|x|\to+\infty,\end{aligned}\right.$ (2.17)

by the well-known uniqueness result (cf. [39]) and the scaling invariance of
(1.10),
$w_{*}(x)=(\frac{\alpha_{*}}{\sigma_{0}})^{\frac{1}{q-2}}w(\sqrt{\alpha_{*}}x)$,
where $w$ is the unique solution of (2.8). It follows from
$\|w_{\mu,+}\|_{2}^{2}=a^{2}$ and the strong convergence of $\\{w_{\mu,+}\\}$
in $H^{1}(\mathbb{R}^{N})$ that $\|w_{*}\|_{2}^{2}=a^{2}$, which implies
$\alpha_{*}=\nu_{a}\nu_{0}$ where $\nu_{a}$ is given by (1.12). Thus,
$w_{*}=\psi_{a}$. Since $\psi_{a}$ is unique, $w_{\mu,+}\to\psi_{a}$ strongly
in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$. The system (1.13) directly comes
from (2.11) and (2.12). It remains to prove the local uniqueness of
$u_{\mu,+}$ for $\mu>0$ sufficiently small. Let us consider the following
system:

$\displaystyle\left\\{\begin{aligned}
&\mathcal{F}(w,\alpha,\beta,\gamma)=\Delta w-\alpha\nu_{0}w+\beta
w^{q-1}+\gamma w^{2^{*}-1},\\\
&\mathcal{G}(w,\alpha,\beta,\gamma)=\|w\|_{2}^{2}-a^{2},\end{aligned}\right.$
(2.18)

where $\alpha,\beta,\gamma>0$ are parameters. It is easy to see that
$\mathcal{F}(\psi_{a},\nu_{a},\sigma_{0},0)=0$ and
$\mathcal{G}(\psi_{a},\nu_{a},\sigma_{0},0)=0$. Let

$\displaystyle\mathcal{L}(\psi_{a},\nu_{a},\sigma_{0},0)=\left(\begin{aligned}
\partial_{w}\mathcal{F}(\psi_{a},\nu_{a},\sigma_{0},0)\quad\partial_{\alpha}\mathcal{F}(\psi_{a},\nu_{a},\sigma_{0},0)\\\
\partial_{w}\mathcal{G}(\psi_{a},\nu_{a},\sigma_{0},0)\quad\partial_{\alpha}\mathcal{G}(\psi_{a},\nu_{a},\sigma_{0},0)\end{aligned}\right)$

be the linearization of the system (2.18) at $(\psi_{a},\nu_{a},\sigma_{0},0)$
in $H^{1}(\mathbb{R}^{N})\times\mathbb{R}$, that is,

$\displaystyle\partial_{w}\mathcal{F}(\psi_{a},\nu_{a},\sigma_{0},0)=\Delta-\nu_{a}\nu_{0}+(q-1)\sigma_{0}\psi_{a}^{q-2},\quad\partial_{\alpha}\mathcal{F}(\psi_{a},\nu_{a},\sigma_{0},0)=-\nu_{0}\psi_{a}$

and

$\displaystyle\partial_{w}\mathcal{G}(\psi_{a},\nu_{a},\sigma_{0},0)=2\psi_{a},\quad\partial_{\alpha}\mathcal{G}(\psi_{a},\nu_{a},\sigma_{0},0)=0.$

Then $\mathcal{L}(\psi_{a},\nu_{a},\sigma_{0},0)[(\phi,\tau)]=0$ if and only
if

$\displaystyle\left\\{\begin{aligned}
&\Delta\phi-\nu_{a}\nu_{0}\phi+(q-1)\sigma_{0}\psi_{a}^{q-2}\phi-\tau\nu_{0}\psi_{a}=0,\\\
&\int_{\mathbb{R}^{N}}\psi_{a}\phi=0.\end{aligned}\right.$

Let us consider the following system:

$\displaystyle\left\\{\begin{aligned}
&\Delta\phi-\nu_{a}\nu_{0}\phi+(q-1)\sigma_{0}\psi_{a}^{q-2}\phi-\tau\nu_{0}\psi_{a}=g,\\\
&\int_{\mathbb{R}^{N}}\psi_{a}\phi=b,\end{aligned}\right.$ (2.19)

where $(g,b)\in H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$ with

$\displaystyle H_{rad}^{1}(\mathbb{R}^{N})=\\{u\in
H^{1}(\mathbb{R}^{N})\mid\text{$u$ is radial}\\}.$

Then $\phi=\phi_{g}+\tau\nu_{0}\phi_{a}$, where $\phi_{g}$ and $\phi_{a}$
satisfies

$\displaystyle\Delta\phi_{g}-\nu_{a}\nu_{0}\phi_{g}+(q-1)\sigma_{0}\psi_{a}^{q-2}\phi_{g}=g$
(2.20)

and

$\displaystyle\Delta\phi_{a}-\nu_{a}\nu_{0}\phi_{a}+(q-1)\sigma_{0}\psi_{a}^{q-2}\phi_{a}=\psi_{a},$
(2.21)

respectively. By [54, (5.2) and (5.3)],
$\phi_{a}=\frac{1}{q-2}\psi_{a}+\frac{1}{2}(x\cdot\nabla\psi_{a})$ and

$\displaystyle\int_{\mathbb{R}^{N}}\phi_{a}\psi_{a}=(\frac{1}{q-2}-\frac{4}{N})\|\psi_{a}\|_{2}^{2}\not=0$

since $q\not=2+\frac{4}{N}$. Thus, the unique solution of (2.19) is given by
$(\phi_{g}+\tau_{b,g}\nu_{0}\phi_{a},\tau_{b,g})$ where

$\displaystyle\tau_{b,g}=\frac{b-\int_{\mathbb{R}^{N}}\phi_{g}\psi_{a}}{\nu_{0}\int_{\mathbb{R}^{N}}\phi_{a}\psi_{a}}.$

Since $q<2^{*}$, it is well-known that $\psi_{a}$ is nondegenerate (cf. [41,
Theorem 2.12] and [45, Lemma 4.2]). Thus, by $q<2^{*}$, (2.20) only has zero
solution in $H_{rad}^{1}(\mathbb{R}^{N})$ for $g=0$, which implies the linear
operator
$\mathcal{L}(\psi_{a},\nu_{a},\sigma_{0},0):H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}\to
H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$ is bijective. Moreover, it is
standard to show that

$\displaystyle|\tau_{b,g}|+\|\phi_{g}+\tau_{b,g}\phi_{a}\|_{H^{1}}\lesssim|b|+\|g\|_{H^{1}}.$

Now, by the implicit function theorem, there exists a unique $C^{1}$-curve
$(w^{\beta,\gamma},\alpha^{\beta,\gamma})$ in
$H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}^{3}$ for $|\beta-\sigma_{0}|<<1$
and $|\gamma|<<1$ such that
$(w^{\sigma_{0},0},\alpha^{\sigma_{0},0})=(\psi_{a},\nu_{a})$, and

$\displaystyle\mathcal{F}(w^{\beta,\gamma},\alpha^{\beta,\gamma},\beta,\gamma)\equiv
0,\quad\mathcal{G}(w^{\beta,\gamma},\alpha^{\beta,\gamma},\beta,\gamma)\equiv
0.$

We recall that $w_{\mu,+}$ is radial and satisfies (2.16), and
$w_{\mu,+}\to\psi_{a}$ strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0$ with
$\|w_{\sigma,+}\|_{2}^{2}=a^{2}$, thus, by the uniqueness of $s_{\mu}$
determined by (1.13), we must have $w_{\mu,+}=w^{\beta(\mu),\gamma(\mu)}$ for
$\beta(\mu)=\mu s_{\mu}^{-2+\frac{N}{2}(q-2)}$ and
$\gamma(\mu)=s_{\mu}^{-2+\frac{N}{2}(2^{*}-2)}$ with $\mu>0$ sufficiently
small. On the other hand, if $\widetilde{u}_{\mu,+}$ is another ground state
of (1.5) with some $\widetilde{\lambda}_{\mu,+}\in\mathbb{R}$ for $\mu>0$
sufficiently small, then by [49, Theorem 1.3],
$\widetilde{u}_{\mu,+}=e^{i\theta}\widehat{u}_{\mu,+}$ where $\theta$ is a
constant and $\widehat{u}_{\mu,+}$ is real valued and positive. Since by the
Pohozaev identity, we always have $\widetilde{\lambda}_{\mu,+}<0$. By applying
the well-known Gidas-Ni-Nirenberg theorem (cf. [25]), $\widetilde{u}_{\mu,+}$
must be radially symmetric. Now, by running the arguments as used above once
more, we know that
$\widehat{w}_{\mu,+}=s_{\mu}^{-\frac{N}{2}}\widehat{u}_{\mu,+}(s_{\mu}^{-1}x)\to\psi_{a}$
strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$ with
$\|w_{\sigma,+}\|_{2}^{2}=a^{2}$. It follows from the uniqueness of
$w^{\beta(\mu),\gamma(\mu)}$ that $\widehat{u}_{\mu,+}=u_{\mu,+}$ for $\mu>0$
sufficiently small. Thus, $u_{\mu,+}$ is the unique ground state of (1.5) for
$\mu>0$ sufficiently small up to translations and rotations. ∎

##  3\. Existence and nonexistence of $u_{a,\mu,-}$

In this section, we shall mainly study the question $(Q_{1})$. Since in the
very recent work [34], the question $(Q_{1})$ has been solved for $N\geq 4$.
we only consider the case $N=3$ and prove that $m_{a,\mu}^{-}$ can also be
attained by some $u_{a,\mu,-}$ for $2<q<2+\frac{4}{N}$ in the case $N=3$ under
some additional assumptions, where $m_{a,\mu}^{-}$ is also given by (1.7) and
$u_{a,\mu,-}$ is also real valued, positive, radially symmetric and radially
decreasing. The crucial point in this study is the following energy estimates.

######  Lemma 3.1.

Let $N=3$, $2<q<2+\frac{4}{N}$ and $\mu,a>0$. Then for $\mu
a^{q-q\gamma_{q}}<\alpha_{N,q}$,

$\displaystyle
m_{a,\mu}^{-}=\inf_{u\in\mathcal{P}_{-}^{a,\mu}}\mathcal{E}_{\mu}(u)<m_{a,\mu}^{+}+\frac{1}{3}S^{\frac{3}{2}}.$
(3.1)

###### Proof.

Since $N=3$, we have
$U_{\varepsilon}=3^{\frac{1}{4}}(\frac{\varepsilon}{\varepsilon^{2}+|x|^{2}})^{\frac{1}{2}}$.
Let $W_{\varepsilon}=\chi(x)U_{\epsilon}$ where $\chi(x)$ is a cut-off
function such that $\chi(x)=1$ for $|x|\leq 1$ and $\chi(x)=0$ for $|x|>2$. By
simple computations, we have that

$\displaystyle\|\nabla
W_{\varepsilon}\|_{2}^{2}=S^{\frac{3}{2}}+O(\varepsilon),\quad\|W_{\varepsilon}\|_{6}^{6}=S^{\frac{3}{2}}+O(\varepsilon^{3})$
(3.2)

and

$\displaystyle\|W_{\varepsilon}\|_{p}^{p}\sim\left\\{\begin{aligned}
&\varepsilon^{3-\frac{p}{2}},\quad 3<p<6;\\\
&\varepsilon^{\frac{3}{2}}\ln\frac{1}{\varepsilon},\quad p=3;\\\
&\varepsilon^{\frac{p}{2}},\quad 2\leq p<3.\end{aligned}\right.$ (3.3)

Now, we define $\widehat{W}_{\varepsilon,t}=u_{\mu,+}+tW_{\varepsilon}$ and
$\overline{W}_{\varepsilon,t}=s^{\frac{1}{2}}\widehat{W}_{\varepsilon,t}(sx)$.
Then it is well-known that

$\displaystyle\|\nabla\overline{W}_{\varepsilon,t}\|_{2}^{2}=\|\nabla\widehat{W}_{\varepsilon,t}\|_{2}^{2},\quad\|\overline{W}_{\varepsilon,t}\|_{6}^{6}=\|\widehat{W}_{\varepsilon,t}\|_{6}^{6},$
(3.4)

and

$\displaystyle\|\overline{W}_{\varepsilon,t}\|_{2}^{2}=s^{-2}\|\widehat{W}_{\varepsilon,t}\|_{2}^{2},\quad\|\overline{W}_{\varepsilon,t}\|_{q}^{q}=s^{q\gamma_{q}-q}\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}.$
(3.5)

We choose $s=\frac{\|\widehat{W}_{\varepsilon,t}\|_{2}}{a}$, then
$\overline{W}_{\varepsilon,t}\in\mathcal{S}_{a}$. By [49, Lemma 4.2], there
exist $\tau_{\varepsilon,t}>0$ such that
$(\overline{W}_{\varepsilon,t})_{\tau_{\varepsilon,t}}\in\mathcal{P}_{-}^{a,\mu}$,
where
$(\overline{W}_{\varepsilon,t})_{\tau_{\varepsilon,t}}=\tau_{\varepsilon,t}^{\frac{3}{2}}\overline{W}_{\varepsilon,t}(\tau_{\varepsilon,t}x)$.
Thus,

$\displaystyle\|\nabla\overline{W}_{\varepsilon,t}\|_{2}^{2}\tau_{\varepsilon,t}^{2-q\gamma_{q}}=\mu\gamma_{q}\|\overline{W}_{\varepsilon,t}\|_{q}^{q}+\|\overline{W}_{\varepsilon,t}\|_{2^{*}}^{2^{*}}\tau_{\varepsilon,t}^{2^{*}-q\gamma_{q}}.$
(3.6)

Since $u_{\mu,+}\in\mathcal{P}_{+}^{a,\mu}$, by [49, Lemma 4.2],
$\tau_{\varepsilon,0}>1$. By (3.2) and (3.6), we also know that
$\tau_{\varepsilon,t}\to 0$ as $t\to+\infty$ uniformly for $\varepsilon>0$
sufficiently small. Since $\tau_{\varepsilon,t}$ is unique by [49, Lemma 4.2],
it is standard to show that $\tau_{\varepsilon,t}$ is continuous for $t$,
which implies that there exists $t_{\varepsilon}>0$ such that
$\tau_{\varepsilon,t_{\varepsilon}}=1$. It follows that

$\displaystyle m_{\mu,a}^{-}\leq\sup_{t\geq
0}\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t}).$ (3.7)

Recall that $u_{\mu,+}\in\mathcal{S}_{a}$ and $W_{\varepsilon}$ are positive,
by (3.2), (3.4) and (3.5), there exists $t_{0}>0$ such that

$\displaystyle\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t})=(\frac{1}{2}\|\nabla\widehat{W}_{\varepsilon,t}\|_{2}^{2}-\frac{\mu}{q}s^{q\gamma_{q}-q}\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}-\frac{1}{6}\|\widehat{W}_{\varepsilon,t}\|_{6}^{6})<m_{\mu,a}^{+}+\frac{1}{3}S^{\frac{3}{2}}-\sigma^{\prime}$
(3.8)

for $t<\frac{1}{t_{0}}$ and $t>t_{0}$ with $\sigma^{\prime}>0$. Since
$u_{\mu,+}$ is radial solution of (1.5) and exponentially decays to zero as
$r\to+\infty$,

$\displaystyle\int_{\mathbb{R}^{3}}u_{\mu,+}W_{\varepsilon}\sim\varepsilon^{\frac{5}{2}}\int_{1}^{\frac{1}{\varepsilon}}(\frac{1}{1+r^{2}})^{\frac{1}{2}}r^{2}\sim\varepsilon^{\frac{1}{2}}$

and

$\displaystyle\int_{\mathbb{R}^{3}}u_{\mu,+}W_{\varepsilon}^{5}\sim\varepsilon^{\frac{1}{2}}\int_{1}^{\frac{1}{\varepsilon}}(\frac{1}{1+r^{2}})^{\frac{5}{2}}r^{2}\sim\varepsilon^{\frac{1}{2}}.$
(3.9)

Thus, by (3.3),

$\displaystyle
s^{2}=\frac{\|\widehat{W}_{\varepsilon,t}\|_{2}^{2}}{a^{2}}=1+\frac{2t}{a^{2}}\int_{\mathbb{R}^{3}}u_{\mu,+}W_{\varepsilon}+t^{2}\|W_{\varepsilon}\|_{2}^{2}=1+O(\varepsilon^{\frac{1}{2}})$

for $t_{0}^{-1}\leq t\leq t_{0}$. Since it is easy to see that
$f(t)=(1+t)^{q}-1-t^{q}-qt-qt^{q-1}\geq 0$ for all $t\geq 0$ in the case of
$q\geq 3$, by (3.4), (3.5) and the fact that $u_{\mu,+}$ is a solution of
(1.5) for some $\lambda_{\mu,+}<0$,

$\displaystyle\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\|\nabla\widehat{W}_{\varepsilon,t}\|_{2}^{2}-\frac{\mu}{q}s^{q\gamma_{q}-q}\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}-\frac{1}{6}\|\widehat{W}_{\varepsilon,t}\|_{6}^{6}$
$\displaystyle\leq$ $\displaystyle
m_{\mu,a}^{+}+\mathcal{E}_{\mu}(tW_{\varepsilon})-\int_{\mathbb{R}^{3}}(tW_{\varepsilon})^{5}u_{\mu,+}$
$\displaystyle+t(\lambda_{\mu,+}\int_{\mathbb{R}^{3}}u_{\mu,+}W_{\varepsilon}+\frac{\mu}{a^{2}}(\gamma_{q}-1)\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}\int_{\mathbb{R}^{3}}u_{\mu,+}W_{\varepsilon})+O(\varepsilon)$
$\displaystyle=$ $\displaystyle
m_{\mu,a}^{+}+\mathcal{E}_{\mu}(tW_{\varepsilon})-\int_{\mathbb{R}^{3}}(tW_{\varepsilon})^{5}u_{\mu,+}+O(\varepsilon)$

for $t_{0}^{-1}\leq t\leq t_{0}$, where we have used the fact that
$\lambda_{\mu,+}a^{2}=\lambda_{\mu,+}\|u_{\mu,+}\|_{2}^{2}=\mu(\gamma_{q}-1)\|u_{\mu,+}\|_{q}^{q}$
which comes from the Pohozaev identity satisfied by $u_{\mu,+}$. Now, for
$t_{0}^{-1}\leq t\leq t_{0}$, by (3.2), (3.3) and (3.9),

$\displaystyle\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t})\leq
m_{\mu,a}^{+}+\frac{1}{3}S^{\frac{3}{2}}+O(\varepsilon)-C\varepsilon^{\frac{1}{2}}<m_{\mu,a}^{+}+\frac{1}{3}S^{\frac{3}{2}}$

by taking $\varepsilon>0$ sufficiently small. It follows from (3.8) that

$\displaystyle\sup_{t\geq
0}\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t})<m_{\mu,a}^{+}+\frac{1}{3}S^{\frac{3}{2}}.$
(3.10)

The conclusion then follows from (3.7). ∎

######  Remark 3.1.

It is worth pointing our that the above argument also works for $N\geq 4$. In
these cases, we have

$\displaystyle\|\nabla
W_{\varepsilon}\|_{2}^{2}=S^{\frac{N}{2}}+O(\varepsilon^{N-2}),\quad\|W_{\varepsilon}\|_{6}^{6}=S^{\frac{N}{2}}+O(\varepsilon^{N})$

and

$\displaystyle\|W_{\varepsilon}\|_{q}^{q}\sim\varepsilon^{N-\frac{(N-2)q}{2}},\quad\|W_{\varepsilon}\|_{2}^{2}\sim\left\\{\begin{aligned}
&\varepsilon^{2}\ln\frac{1}{\varepsilon},\quad N=4,\\\ &\varepsilon^{2},\quad
N\geq 5.\end{aligned}\right.$

Moreover, similar to (3.9),

$\displaystyle\int_{\mathbb{R}^{N}}u_{\mu,+}^{p}W_{\varepsilon}\sim\varepsilon^{\frac{N-2}{2}}\quad\text{for
all }p\geq 1.$

It follows that

$\displaystyle\mathcal{E}_{\mu}(\overline{W}_{\varepsilon,t})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\|\nabla\widehat{W}_{\varepsilon,t}\|_{2}^{2}-\frac{\mu}{q}s^{q\gamma_{q}-q}\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}-\frac{1}{6}\|\widehat{W}_{\varepsilon,t}\|_{6}^{6}$
$\displaystyle\leq$ $\displaystyle
m_{\mu,a}^{+}+\mathcal{E}_{\mu}(tW_{\varepsilon})$
$\displaystyle+t(\lambda_{\mu,+}\int_{\mathbb{R}^{N}}u_{\mu,+}W_{\varepsilon}+\frac{\mu}{a^{2}}(\gamma_{q}-1)\|\widehat{W}_{\varepsilon,t}\|_{q}^{q}\int_{\mathbb{R}^{N}}u_{\mu,+}W_{\varepsilon})+O(\varepsilon^{2}\ln\frac{1}{\varepsilon})$
$\displaystyle=$ $\displaystyle
m_{\mu,a}^{+}+\mathcal{E}_{\mu}(tW_{\varepsilon})+O(\varepsilon^{N-2})$
$\displaystyle\leq$ $\displaystyle
m_{\mu,a}^{+}+\frac{1}{N}S^{\frac{N}{2}}-C\varepsilon^{N-\frac{(N-2)q}{2}}+O(\varepsilon^{2}\ln\frac{1}{\varepsilon})$
$\displaystyle<$ $\displaystyle m_{\mu,a}^{+}+\frac{1}{N}S^{\frac{N}{2}}$

for $t_{0}^{-1}\leq t\leq t_{0}$ by taking $\varepsilon>0$ sufficiently small
since $N\geq 4$ and $q>2$. Our proof is slightly simpler than that of [34]
since our test function is radial and we do not need other variational
formulas of $m_{\mu,a}^{-}$.

For every $c>0$ such that $\mu c^{q-q\gamma_{q}}<\alpha_{N,q}$, let
$u\in\mathcal{P}_{\pm}^{c,\mu}$, then $v_{b}=\frac{b}{c}u\in\mathcal{S}_{b}$
for all $b>0$. By [49, Lemma 4.2], there exists $\tau_{\pm}(b)>0$ such that

$\displaystyle(v_{b})_{\tau_{\pm}(b)}=(\tau_{\pm}(b))^{\frac{N}{2}}v_{b}(\tau_{\pm}(b)x)\in\mathcal{P}_{\pm}^{b,\mu},$

where $b>0$ such that $\mu b^{q-q\gamma_{q}}<\alpha_{N,q}$. Clearly,
$\tau_{\pm}(c)=1$.

######  Lemma 3.2.

Let $2<q<2+\frac{4}{N}$. For every $c>0$ such that $\mu
c^{q-q\gamma_{q}}<\alpha_{N,q}$, $\tau^{\prime}_{\pm}(c)$ exist and

$\displaystyle\tau^{\prime}_{\pm}(c)=\frac{\mu
q\gamma_{q}\|u\|_{q}^{q}+2^{*}\|u\|_{2^{*}}^{2^{*}}-2\|\nabla
u\|_{2}^{2}}{c(2\|\nabla u\|_{2}^{2}-\mu
q\gamma_{q}^{2}\|u\|_{q}^{q}-2^{*}\|u\|_{2^{*}}^{2^{*}})}.$ (3.11)

Moreover, $\mathcal{E}_{\mu}((v_{b})_{\tau_{\pm}(b)})<\mathcal{E}_{\mu}(u)$
for all $b>c$ such that $\mu b^{q-q\gamma_{q}}<\alpha_{N,q}$.

###### Proof.

The proof is mainly inspired by [19]. Since
$(v_{b})_{\tau_{\pm}(b)}\in\mathcal{P}_{\pm}^{b,\mu}$, we have

$\displaystyle(\frac{b}{c}\tau(b))^{2}\|\nabla
u\|_{2}^{2}=(\frac{b}{c})^{q}(\tau(b))^{q\gamma_{q}}\mu\gamma_{q}\|u\|_{q}^{q}+(\frac{b}{c}\tau(b))^{2^{*}}\|u\|_{2^{*}}^{2^{*}}.$

Now, if we define the function

$\displaystyle\Phi(b,\tau)=(\frac{b\tau}{c})^{2}\|\nabla
u\|_{2}^{2}-(\frac{b}{c})^{q}\tau^{q\gamma_{q}}\mu\gamma_{q}\|u\|_{q}^{q}-(\frac{b\tau}{c})^{2^{*}}\|u\|_{2^{*}}^{2^{*}},$

then $\Phi(b,\tau(b))\equiv 0$ for $b>0$ such that $\mu
b^{q-q\gamma_{q}}<\alpha_{N,q}$. Since $u\in\mathcal{P}_{\pm}^{c,\mu}$,

$\displaystyle\partial_{\tau}\Phi(c,1)=2\|\nabla u\|_{2}^{2}-\mu
q\gamma_{q}^{2}\|u\|_{q}^{q}-2^{*}\|u\|_{2^{*}}^{2^{*}}\not=0.$

It follows from the implicit function theorem that $\tau^{\prime}_{\pm}(c)$
exist and (3.11) holds. By (1.6) and $q<2^{*}$, $1-\gamma_{q}>0$. Thus, by
$u\in\mathcal{P}_{\pm}^{c,\mu}$,

$\displaystyle 1+c\tau^{\prime}(c)$ $\displaystyle=$ $\displaystyle
1+\frac{\mu q\gamma_{q}\|u\|_{q}^{q}+2^{*}\|u\|_{2^{*}}^{2^{*}}-2\|\nabla
u\|_{2}^{2}}{2\|\nabla u\|_{2}^{2}-\mu
q\gamma_{q}^{2}\|u\|_{q}^{q}-2^{*}\|u\|_{2^{*}}^{2^{*}}}$ $\displaystyle=$
$\displaystyle\frac{\mu q\gamma_{q}(1-\gamma_{q})\|u\|_{q}^{q}}{2\|\nabla
u\|_{2}^{2}-\mu q\gamma_{q}^{2}\|u\|_{q}^{q}-2^{*}\|u\|_{2^{*}}^{2^{*}}}.$

Since $(v_{b})_{\tau_{\pm}(b)}\in\mathcal{P}_{\pm}^{b,\mu}$ and
$u\in\mathcal{P}_{\pm}^{c,\mu}$,

$\displaystyle\mathcal{E}_{\mu}((v_{b})_{\tau_{\pm}(b)})$ $\displaystyle=$
$\displaystyle(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla(v_{b})_{\tau(b)}\|_{2}^{2}+(\frac{1}{q\gamma_{q}}-\frac{1}{2^{*}})\|(v_{b})_{\tau(b)}\|_{2^{*}}^{2^{*}}$
$\displaystyle=$
$\displaystyle(\frac{b}{c}\tau(b))^{2}(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla
u\|_{2}^{2}+(\frac{b}{c}\tau(b))^{2^{*}}(\frac{1}{q\gamma_{q}}-\frac{1}{2^{*}})\|u\|_{2^{*}}^{2^{*}}$
$\displaystyle=$ $\displaystyle(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla
u\|_{2}^{2}+(\frac{1}{q\gamma_{q}}-\frac{1}{2^{*}})\|u\|_{2^{*}}^{2^{*}}+o(b-c)$
$\displaystyle+\frac{1+c\tau^{\prime}(c)}{c}(2(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla
u\|_{2}^{2}+2^{*}(\frac{1}{q\gamma_{q}}-\frac{1}{2^{*}})\|u\|_{2^{*}}^{2^{*}})(b-c)$
$\displaystyle=$
$\displaystyle\mathcal{E}_{\mu}(u)-\frac{\mu(1-\gamma_{q})\|u\|_{q}^{q}}{c}(b-c)$
$\displaystyle+o(b-c).$

Therefore,

$\displaystyle\frac{d\mathcal{E}_{\mu}((v_{b})_{\tau_{\pm}(b)})}{db}|_{b=c}=-\frac{\mu(1-\gamma_{q})\|u\|_{q}^{q}}{c}<0.$

Since $c>0$, which satisfies $\mu c^{q-q\gamma_{q}}<\alpha_{N,q}$, is
arbitrary and $(v_{b})_{\tau_{\pm}(b)}\in\mathcal{P}_{\pm}^{b,\mu}$, we have
$\mathcal{E}_{\mu}((v_{b})_{\tau_{\pm}(b)})<\mathcal{E}_{\mu}(u)$ for all
$b>c$ such that $\mu b^{q-q\gamma_{q}}<\alpha_{N,q}$. ∎

With Lemma 3.2 in hands, we can obtain the following.

######  Proposition 3.1.

Let $2<q<2+\frac{4}{N}$ and $\mu a^{q-q\gamma_{q}}<\alpha_{N,q}$. If
$m_{a,\mu}^{-}<m_{a,\mu}^{+}+\frac{1}{N}S^{\frac{N}{2}}$ then

$\displaystyle
m_{a,\mu}^{-}=\inf_{u\in\mathcal{P}_{-}^{a,\mu}}\mathcal{E}_{\mu}(u)$

can be attained by some $u_{a,\mu,-}$ which is real valued, positive, radially
symmetric and decreasing in $r=|x|$. Moreover, (1.5) has a second solution
$u_{a,\mu,-}$ with some $\lambda_{a,\mu,-}<0$ which is real valued, positive,
radially symmetric and radially decreasing.

###### Proof.

Let $\\{u_{n}\\}\subset\mathcal{P}_{-}^{a,\mu}$ be a minimizing sequence. Then
by taking $|u_{n}|$ and adapting the Schwarz symmetrization to $|u_{n}|$ if
necessary, we can obtain a new minimizing sequence, say $\\{u_{n}\\}$ again,
such that $u_{n}$ are all real valued, nonnegative, radially symmetric and
decreasing in $r=|x|$. Since $\\{u_{n}\\}\subset\mathcal{P}_{-}^{a,\mu}$, we
have

$\displaystyle\mathcal{E}_{\mu}(u_{n})=\frac{\mu}{q}(\frac{q\gamma_{q}}{2}-1)\|u_{n}\|_{q}^{q}+\frac{1}{N}\|u_{n}\|_{2^{*}}^{2^{*}}.$
(3.12)

Thus, by the Hölder and Young inequalities and
$\\{u_{n}\\}\subset\mathcal{P}_{-}^{a,\mu}$ again, we known that $\\{u_{n}\\}$
is bounded in $H^{1}(\mathbb{R}^{N})$ and thus, $u_{n}\rightharpoonup u_{0}$
weakly in $H^{1}(\mathbb{R}^{N})$ as $n\to\infty$ up to a subsequence. Since
$u_{n}$ are all radial, by Struss’s radial lemma (cf. [12, Lemma A.IV, Theorem
A.I’] or [44, Lemma 3.1]) and the Sobolev embedding theorem, $u_{n}\to u_{0}$
strongly in $L^{q}(\mathbb{R}^{N})$ as $n\to\infty$ up to a subsequence.
Without loss of generality, we assume that $u_{n}\rightharpoonup u_{0}$ weakly
in $H^{1}(\mathbb{R}^{N})$ and $u_{n}\to u_{0}$ strongly in
$L^{q}(\mathbb{R}^{N})$ as $n\to\infty$. We claim that $u_{0}\not=0$. If not,
then $u_{n}\to 0$ strongly in $L^{q}(\mathbb{R}^{N})$ as $n\to\infty$. It
follows from $\\{u_{n}\\}\subset\mathcal{P}_{-}^{a,\mu}$ that

$\displaystyle\|\nabla u_{n}\|_{2}^{2}=\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1),$

which together with the Sobolev inequality (1.8), implies that either
$u_{n}\to 0$ strongly in $D^{1,2}(\mathbb{R}^{N})$ as $n\to\infty$ or
$\|\nabla u_{n}\|_{2}^{2}=\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1)\geq
S^{\frac{N}{2}}+o_{n}(1)$. Hence, by (3.12), either $m_{a,\mu}^{-}=0$ or
$m_{a,\mu}^{-}\geq\frac{1}{N}S^{\frac{N}{2}}$, which contradicts
$\mathcal{E}_{\mu}(u)\gtrsim 1$ for $u\in\mathcal{P}_{-}^{c,\mu}$ and Lemma
3.1. We remark that $\mathcal{E}_{\mu}(u)\gtrsim 1$ for
$u\in\mathcal{P}_{-}^{c,\mu}$ comes from similar arguments as used for [50,
Lemma 5.7]. Therefore, we must have $u_{0}\not=0$. Let $v_{n}=u_{n}-u_{0}$.
Then there are two cases:

  1. $(i)$

$v_{n}\to 0$ strongly in $H^{1}(\mathbb{R}^{N})$ as $n\to\infty$ up to a
subsequence.

  2. $(ii)$

$\|\nabla v_{n}\|_{2}^{2}+\|v_{n}\|_{2}^{2}\gtrsim 1$.

In the case $(i)$, $u_{0}\in\mathcal{P}_{-}^{a,\mu}$ and $m_{a,\mu}^{-}$ is
attained by $u_{0}$ which is real valued, radially symmetric, nonnegative and
decreasing in $r=|x|$. By [49, Proposition 1.5], $u_{0}$ is a solution of
(1.5) with some $\lambda_{0}\in\mathbb{R}$ which appears as a Lagrange
multiplier. By multiplying (1.5) with $u_{0}$ and integrating by parts, and
using $u_{0}\in\mathcal{P}_{-}^{a,\mu}$, we have

$\displaystyle\lambda_{0}a^{2}=\mu(\gamma_{q}-1)\|u_{0}\|_{q}^{q}<0,$

which implies $\lambda_{0}<0$. Now, by the maximum principle and classical
elliptic estimates, we know that $u_{0}$ is positive. It remains to consider
the case $(ii)$. Let $\|u_{0}\|_{2}^{2}=t_{0}^{2}$, then by the Fatou lemma,
$0<t_{0}\leq a$. There are two subcases:

  1. $(ii_{1})$

$\|v_{n}\|_{2^{*}}\to 0$ as $n\to\infty$ up to a subsequence.

  2. $(ii_{2})$

$\|v_{n}\|_{2^{*}}^{2^{*}}\gtrsim 1$.

In the subcase $(ii_{1})$, by [49, Lemma 4.2], there exists $s_{0}>0$ such
that $(u_{0})_{s_{0}}\in\mathcal{P}_{-}^{t_{0},\mu}$. By [49, Lemma 4.2] once
more, $\\{u_{n}\\}\subset\mathcal{P}_{-}^{a,\mu}$ and $u_{n}\to u_{0}$
strongly in $L^{2^{*}}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})$ as
$n\to\infty$ up to a subsequence,

$\displaystyle
m_{a,\mu}^{-}+o_{n}(1)=\mathcal{E}_{\mu}(u_{n})\geq\mathcal{E}_{\mu}((u_{n})_{s_{0}})\geq\mathcal{E}_{\mu}((u_{0})_{s_{0}})+o_{n}(1).$

By Lemma 3.2, we have $m_{t_{0},\mu}^{-}\geq m_{a,\mu}^{-}$. Thus,
$\mathcal{E}_{\mu}((u_{0})_{s_{0}})=m_{t_{0},\mu}^{-}$ and
$m_{t_{0},\mu}^{-}=m_{a,\mu}^{-}$. If $t_{0}<a$ then by taking
$(u_{0})_{s_{0}}$ as the test function in the proof of Lemma 3.2, we know that
$m_{t_{0},\mu}^{-}>m_{a,\mu}^{-}$, which is a contradiction. Thus, in the
subcase $(ii_{1})$, we must have $t_{0}=a$ and so that $m_{a,\mu}^{-}$ is
attained by $(u_{0})_{s_{0}}$ which is real valued, radially symmetric,
nonnegative and decreasing in $r=|x|$. As above, we can show that
$(u_{0})_{s_{0}}$ is positive and $(u_{0})_{s_{0}}$ is a solution of (1.5)
with some $\lambda_{0}^{\prime}<0$. It remains to consider the subcase
$(ii_{2})$. Let

$\displaystyle s_{n}=\bigg{(}\frac{\|\nabla
v_{n}\|_{2}^{2}}{\|v_{n}\|_{2^{*}}^{2^{*}}}\bigg{)}^{\frac{1}{2^{*}-2}}.$

Then in the subcase $(ii_{2})$, $s_{n}\lesssim 1$ and by the Sobolev
inequality (1.8),

$\displaystyle\|\nabla(v_{n})_{s_{n}}\|_{2}^{2}=\|(v_{n})_{s_{n}}\|_{2^{*}}^{2^{*}}\geq\frac{1}{N}S^{\frac{N}{2}}.$

Since $0<t_{0}\leq a$, by [49, Lemma 4.2], there exists $\tau_{0}>0$ such that
$(u_{0})_{\tau_{0}}\in\mathcal{P}_{-}^{t_{0},\mu}$. We claim that
$s_{n}\geq\tau_{0}$ up to a subsequence. Suppose the contrary that
$s_{n}<\tau_{0}$ for all $n$. Then by [49, Lemma 4.2] once more, the Brezis-
Lieb lemma (cf. [56, Lemma 1.32]), Lemma 3.2, the fact that $u_{n}\to u_{0}$
strongly in $L^{q}(\mathbb{R}^{N})$ as $n\to\infty$ and the boundedness of
$\\{s_{n}\\}$,

$\displaystyle m_{a,\mu}^{-}+o_{n}(1)$ $\displaystyle=$
$\displaystyle\mathcal{E}_{\mu}(u_{n})$ $\displaystyle\geq$
$\displaystyle\mathcal{E}_{\mu}((u_{n})_{s_{n}})$ $\displaystyle=$
$\displaystyle\mathcal{E}_{\mu}((u_{0})_{s_{n}})+\mathcal{E}_{0}((v_{n})_{s_{n}})+o_{n}(1)$
$\displaystyle\geq$ $\displaystyle
m_{t_{0},\mu}^{+}+\frac{1}{N}S^{\frac{N}{2}}+o_{n}(1)$ $\displaystyle\geq$
$\displaystyle m_{a,\mu}^{+}+\frac{1}{N}S^{\frac{N}{2}}+o_{n}(1),$

which is impossible. Thus, we must have $s_{n}\geq\tau_{0}$ up to a
subsequence. Without loss of generality, we may assume that
$s_{n}\geq\tau_{0}$ for all $n\in\mathbb{N}$. Again, by [49, Lemma 4.2], the
Brezis-Lieb lemma (cf. [56, Lemma 1.32]) and the fact that $u_{n}\to u_{0}$
strongly in $L^{q}(\mathbb{R}^{N})$ as $n\to\infty$,

$\displaystyle
m_{a,\mu}^{-}+o_{n}(1)=\mathcal{E}_{\mu}(u_{n})\geq\mathcal{E}_{\mu}((u_{n})_{\tau_{0}})=\mathcal{E}_{\mu}((u_{0})_{\tau_{0}})+\mathcal{E}_{0}((v_{n})_{\tau_{0}})+o_{n}(1).$

Since $s_{n}\geq\tau_{0}$, by [49, Proposition 2.2],
$\mathcal{E}_{0}((v_{n})_{\tau_{0}})\geq 0$, which, together with Lemma 3.2,
implies that $t_{0}=a$ and $m_{a,\mu}^{-}$ is attained by
$(u_{0})_{\tau_{0}}$. Clearly, $(u_{0})_{\tau_{0}}$ is real valued, radially
symmetric, nonnegative and decreasing in $r=|x|$. As above, we can show that
$(u_{0})_{\tau_{0}}$ is positive and $(u_{0})_{\tau_{0}}$ is a solution of
(1.5) with some $\lambda_{0}^{\prime\prime}<0$. Therefore, we have proved that
$m_{a,\mu}^{-}$ can always be attained by some $u_{a,\mu,-}$ which is real
valued, radially symmetric, positive and decreasing in $r=|x|$. By [49,
Proposition 1.5], (1.5) has a second solution $u_{a,\mu,-}$ which is real
valued, radially symmetric, positive and decreasing in $r=|x|$. ∎

Our next goal in this section is to prove the existence and nonexistence of
ground states for $\mu a^{q-q\gamma_{q}}\geq\alpha_{N,q}$ in the
$L^{2}$-critical and supercritical cases, which gives partial answers to the
question $(Q_{2})$. In these two cases, $2+\frac{4}{N}\leq q<2^{*}$, which
implies

$\displaystyle q\gamma_{q}\geq 2.$

We recall that the constant $\alpha_{N,q}$ is given by [49, Theorem 1.1]. For
$q=2+\frac{4}{N}$, by [49, (5,1)],

$\displaystyle\alpha_{N,q}=C_{N,q}^{-q}(1+\frac{2}{N})=\frac{1}{C_{N,q}^{q}\gamma_{q}},$
(3.13)

where $C_{N,q}$ is the optimal constant in the Gagliardo–Nirenberg inequality
(1.9).

######  Lemma 3.3.

Let $N\geq 3$ and $2+\frac{4}{N}\leq q<2^{*}$. Then $m_{a,\mu}^{-}$ is
strictly decreasing for $0<\mu<a^{q\gamma_{q}-q}\alpha_{N,q}$ and is
nonincreasing for $\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$, where
$m_{a,\mu}^{-}$ is given by (1.7). Moreover,
$0<m_{a,\mu}^{-}<\frac{1}{N}S^{\frac{N}{2}}$ for all $\mu>0$ in the case of
$2+\frac{4}{N}<q<2^{*}$ while, $m_{a,\mu}^{-}=0$ for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$ in the case of $q=2+\frac{4}{N}$.

###### Proof.

Modified the proof of [49, Lemma 8.2] in a trivial way (or by Lemma 3.2 and
[49, Theorem 1.1]), we can show that $m_{a,\mu}^{-}$ is strictly decreasing
for $0<\mu<a^{q\gamma_{q}-q}\alpha_{N,q}$. For $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$, let us consider the fibering map

$\displaystyle\Psi_{u}(t)=\frac{t^{2}}{2}\|\nabla u\|_{2}^{2}-\frac{\mu
t^{q\gamma_{q}}}{q}\|u\|_{q}^{q}-\frac{t^{2^{*}}}{2^{*}}\|u\|_{2^{*}}^{2^{*}},$

as that in [49]. For $2+\frac{4}{N}<q<2^{*}$, it has been proved in [49, Lemma
6.1] that for every $u\in\mathcal{S}_{a}$, there exists $t_{u}>0$ such that
$\Psi_{u}(t)$ is strictly increasing in $(0,t_{u})$, is strictly decreasing in
$(t_{u},+\infty)$ and

$\displaystyle(u)_{t_{u}}=t_{u}^{\frac{N}{2}}u(t_{u}x)\in\mathcal{P}_{-}^{a,\mu}.$

Moreover, by [49, Lemma 6.2], we have $m_{a,\mu}^{-}>0$ for all $\mu>0$ in the
$L^{2}$-supercritical case $2+\frac{4}{N}<q<2^{*}$. It follows that we can
always choose $v_{\varepsilon}\in\mathcal{P}_{-}^{a,\mu}$ such that
$\mathcal{E}_{\mu}(v_{\varepsilon})<m_{a,\mu}^{-}+\varepsilon$ in the
$L^{2}$-supercritical case $2+\frac{4}{N}<q<2^{*}$. Then by similar arguments
as used for [49, Lemma 8.2] (or by Lemma 3.2), we have

$\displaystyle m_{a,\mu^{\prime}}^{-}<m_{a,\mu}^{-}+\varepsilon\quad\text{for
all }\mu^{\prime}>\mu.$

Since $\varepsilon>0$ and $\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$ are
arbitrary, $m_{a,\mu}^{-}$ is nonincreasing for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$ in the $L^{2}$-supercritical case
$2+\frac{4}{N}<q<2^{*}$. It follows from [49, Lemma 6.4] that
$m_{a,\mu}^{-}<\frac{1}{N}S^{\frac{N}{2}}$ for all $\mu>0$.

In the $L^{2}$-critical case $q=2+\frac{4}{N}$, since

$\displaystyle\sup_{u\in\mathcal{S}_{a}}\frac{\|\nabla
u\|_{2}}{\|u\|_{q}}=+\infty.$

For all $\mu>0$, we can always choose $u\in\mathcal{S}_{a}$ such that
$\frac{\|\nabla u\|_{2}}{\|u\|_{q}}>\mu\gamma_{q}$. Indeed, if
$\sup_{u\in\mathcal{S}_{a}}\frac{\|\nabla u\|_{2}}{\|u\|_{q}}\lesssim 1$, then
by the Gagliardo-Nirenberg inequality,

$\displaystyle\|\nabla u\|_{2}\lesssim\|u\|_{q}\lesssim\|\nabla
u\|_{2}^{\gamma_{q}}\quad\text{for all }u\in\mathcal{S}_{a},$

which implies

$\displaystyle\sup_{u\in\mathcal{S}_{a}}\|\nabla u\|_{2}\lesssim 1.$

It is impossible since in any ball $B_{R}(0)$, the eigenvalue problem $-\Delta
u=\lambda u$, with Dirichlet boundary conditions, has a sequence of
eigenvalues $\lambda_{j}\to+\infty$ as $j\to\infty$. We note that
$q\gamma_{q}=2$ in the $L^{2}$-critical case $q=2+\frac{4}{N}$. Thus,

$\displaystyle\Psi_{u}^{\prime}(t)=(\|\nabla
u\|_{2}^{2}-\mu\gamma_{q}\|u\|_{q}^{q})t-t^{2^{*}-1}\|u\|_{2^{*}}^{2^{*}}=0$

has a unique solution $t_{u}>0$ for $u\in\mathcal{S}_{a}$ such that
$\frac{\|\nabla u\|_{2}}{\|u\|_{q}}>\mu\gamma_{q}$. Moreover, $\Psi_{u}(t)$ is
strictly increasing in $(0,t_{u})$, is strictly decreasing in
$(t_{u},+\infty)$ and

$\displaystyle(u)_{t_{u}}=t_{u}^{\frac{N}{2}}u(t_{u}x)\in\mathcal{P}_{a,\mu}=\mathcal{P}_{-}^{a,\mu}.$

Thus, $\mathcal{P}_{a,\mu}=\mathcal{P}_{-}^{a,\mu}\not=\emptyset$ and
$\Psi_{u}(t_{u})=\max_{t\geq 0}\Psi_{u}(t)$ for all $\mu>0$ and all
$u\in\mathcal{S}_{a}$ such that $\frac{\|\nabla
u\|_{2}}{\|u\|_{q}}>\mu\gamma_{q}$. Now, as in the $L^{2}$-supercritical case
$2+\frac{4}{N}<q<2^{*}$, by similar arguments as used for [49, Lemma 8.2], we
can show that $m_{a,\mu}^{-}$ is nonincreasing for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$ in the $L^{2}$-critical case $q=2+\frac{4}{N}$.
It remains to prove that $m_{a,\mu}^{-}=0$ for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$ in the case of $q=2+\frac{4}{N}$. Let
$\\{\varphi_{n}\\}$ be the minimizing sequence of the Gagliardo-Nirenberg
inequality (1.9). Then by scaling
$\frac{at_{n}^{\frac{N}{2}}}{\|\varphi_{n}\|_{2}}\varphi_{n}(t_{n}x)$ if
necessary, we may assume that $\|\varphi_{n}\|_{2}^{2}=a^{2}$,
$\|\varphi_{n}\|_{q}^{q}=1$ and
$\|\nabla\varphi_{n}\|_{2}^{2}=C_{N,q}^{-\frac{2}{\gamma_{q}}}a^{\frac{2(\gamma_{q}-1)}{\gamma_{q}}}+o_{n}(1)$.
Let us consider the following function:

$\displaystyle h_{\varphi_{n}}(\mu,t)$ $\displaystyle=$ $\displaystyle
t^{2}(\|\nabla\varphi_{n}\|_{2}^{2}-\mu\gamma_{q}\|\varphi_{n}\|_{q}^{q})-t^{2^{*}}\|\varphi_{n}\|_{2^{*}}^{2^{*}}$
$\displaystyle=$
$\displaystyle(C_{N,q}^{-\frac{2}{\gamma_{q}}}a^{\frac{2(\gamma_{q}-1)}{\gamma_{q}}}+o_{n}(1)-\mu\gamma_{q})t^{2}-t^{2^{*}}\|\psi\|_{2^{*}}^{2^{*}}$
$\displaystyle=$
$\displaystyle\gamma_{q}(\alpha_{N,q}a^{q\gamma_{q}-q}+o_{n}(1)-\mu)t^{2}-t^{2^{*}}\|\psi\|_{2^{*}}^{2^{*}},$

where we have used (3.13). By [49, Lemma 5.1], there exists a unique
$t_{n}(\mu)>0$ such that $h_{\varphi_{n}}(\mu,t_{n}(\mu))=0$ for
$0<\mu<a^{q\gamma_{q}-q}\alpha_{N,q}$. Thus,
$(\varphi_{n})_{t_{n}(\mu)}\in\mathcal{P}_{a,\mu}$ for
$0<\mu<a^{q\gamma_{q}-q}\alpha_{N,q}$, where
$(\varphi_{n})_{t_{n}(\mu)}=[t_{n}(\mu)]^{\frac{N}{2}}\varphi_{n}(t_{n}(\mu)x)$.
Moreover, since $\|\varphi_{n}\|_{q}^{q}=1$, by the Hölder inequality,
$\|\varphi_{n}\|_{2^{*}}\gtrsim 1$. It follows that $t(\mu)\to 0$ as $\mu\to
a^{q\gamma_{q}-q}\alpha_{N,q}$, which implies

$\displaystyle\mathcal{E}_{\mu}((\psi)_{t(\mu)})=\frac{1}{N}\|\varphi_{n}\|_{2^{*}}^{2^{*}}[t_{n}(\mu)]^{2^{*}}=o_{n}(1)$

as $\mu\to a^{q\gamma_{q}-q}\alpha_{N,q}$ in the $L^{2}$-critical case
$q=2+\frac{4}{N}$. Thus, we must have $m_{a,\mu}^{-}\leq 0$ for
$\mu=a^{q\gamma_{q}-q}\alpha_{N,q}$. By the monotone property of
$m_{a,\mu}^{-}$ stated in Lemma 3.2, $m_{a,\mu}^{-}\leq 0$ for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$. Recall that we always have

$\displaystyle\mathcal{E}_{\mu}(u)=\frac{1}{N}\|u\|_{2^{*}}^{2^{*}}\geq
0\quad\text{for all }u\in\mathcal{P}_{a,\mu},$ (3.14)

thus, we must have $m_{a,\mu}^{-}=0$ for $\mu\geq
a^{q\gamma_{q}-q}\alpha_{N,q}$. ∎

With Lemma 3.3 in hands, we can obtain the following.

######  Proposition 3.2.

Let $N\geq 3$ and $2+\frac{4}{N}\leq q<2^{*}$.

  1. $(1)$

If $2+\frac{4}{N}<q<2^{*}$, then $m_{a,\mu}^{-}$ is attained by same
$u_{a,\mu,-}$ which is real valued, positive, radially symmetric and
decreasing in $r=|x|$ for all $\mu>0$, and thus, $u_{a,\mu,-}$ is a solution
of (1.5) for all $\mu>0$ with some $\lambda_{a,\mu,-}<0$.

  2. $(2)$

If $q=2+\frac{4}{N}$, then $m_{a,\mu}^{-}$ can not be attained and (1.5) has
no ground states for all $\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$.

###### Proof.

$(1)$ By Lemma 3.3, $0<m_{a,\mu}^{-}<\frac{1}{N}S^{\frac{N}{2}}$ for all
$\mu>0$ in the case of $2+\frac{4}{N}<q<2^{*}$. Now, by following the
arguments in [49, Section 6] step by step, we can show that $m_{a,\mu}^{-}$ is
attained by some $u_{a,\mu,-}$ which is real valued, nonnegative, radially
symmetric and decreasing in $r=|x|$ for all $\mu>0$ in the case of
$2+\frac{4}{N}<q<2^{*}$. By similar arguments as used for [50, Lemma 6.2], we
know that $\mathcal{P}_{a,\mu}=\mathcal{P}_{-}^{a,\mu}\not=\emptyset$ is a
natural constraint in $\mathcal{S}_{a}$ for all $\mu>0$ in the case of
$2+\frac{4}{N}<q<2^{*}$. Thus, $u_{a,\mu,-}$ is a solution of (1.5) for all
$\mu>0$ with some $\lambda_{a,\mu,-}$ in the case of $2+\frac{4}{N}<q<2^{*}$.
As that in the proof of Proposition 3.1, we can show that
$\lambda_{a,\mu,-}<0$ and $u_{a,\mu,-}$ is positive.

$(2)$ Suppose the contrary that $m_{a,\mu}^{-}$ is attained by some
$u_{a,\mu,-}$ for $\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$, then by Lemma 3.3
and (3.14), $\|u_{a,\mu,-}\|_{2^{*}}^{2^{*}}=0$. It is impossible since
$u_{a,\mu,-}\in\mathcal{S}_{a}$. Thus, $m_{a,\mu}^{-}$ can not be attained for
$\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$. It follows that (1.5) has no ground
state for all $\mu\geq a^{q\gamma_{q}-q}\alpha_{N,q}$. ∎

##  4\. The asymptotic behavior of $u_{a,\mu,-}$

In this section, we shall mainly study the question $(Q_{3})$ and give a
precisely description of the asymptotic behavior of $u_{a,\mu,-}$ as $\mu\to
0^{+}$. Since we consider $\mu\to 0^{+}$ now, the assumptions of [49, Theorem
1.1], [34, Theorem 1.6] and Proposition 3.1 always hold and thus,
$u_{a,\mu,-}$, which is a minimizer of $\mathcal{E}|_{\mathcal{S}_{a}}(u)$ on
$\mathcal{P}_{-}^{a,\mu}$, exists for all $N\geq 3$, $2<q<2^{*}$ for $\mu>0$
sufficiently small.

######  Proposition 4.1.

Let $N\geq 3$, $2<q<2^{*}$ and $u_{a,\mu,-}$ is a critical point of
$\mathcal{E}|_{\mathcal{S}_{a}}(u)$ of mountain pass type. If $N\geq 5$ then
$u_{a,\mu,-}\to U_{\varepsilon_{0}}$ strongly in $H^{1}(\mathbb{R}^{N})$ as
$\mu\to 0^{+}$, where $U_{\varepsilon_{0}}$ is the Aubin-Talanti babble such
that $U_{\varepsilon_{0}}\in\mathcal{S}_{a}$. Moreover, if $N\geq 9$, then up
to translations and rotations, $u_{a,\mu,-}$ is the unique minimizer of
$\mathcal{E}|_{\mathcal{S}_{a}}(u)$ on $\mathcal{P}_{-}^{a,\mu}$ for $\mu>0$
sufficiently small.

###### Proof.

By [49, Theorem 1.4], $m_{a,\mu}^{+}\to 0$ as $\mu\to 0^{+}$. Moreover, by
[49, Theorem 1.4] again, we know that
$m_{a,\mu}^{-}\to\frac{1}{N}S^{\frac{N}{2}}$ as $\mu\to 0^{+}$, and

$\displaystyle\|\nabla
u_{a,\mu,-}\|_{2}^{2},\|u_{a,\mu,-}\|_{2^{*}}^{2^{*}}\to
S^{\frac{N}{2}}\quad\text{as }\mu\to 0^{+}$

for $2+\frac{4}{N}\leq q<2^{*}$. On the other hand, by [49, Lemma 4.2] and
similar arguments as used for [50, Lemma 5.7], we also have
$m_{a,\mu}^{-}\gtrsim 1$ for $\mu>0$ sufficiently small in the case of
$2<q<2+\frac{4}{N}$. Thus, by adapting similar arguments as used in the proof
of [49, Theorem 1.1] for the case of $2+\frac{4}{N}\leq q<2^{*}$ to the case
of $2<q<2+\frac{4}{N}$, we can also show that
$m_{a,\mu}^{-}\to\frac{1}{N}S^{\frac{N}{2}}$ as $\mu\to 0^{+}$, and

$\displaystyle\|\nabla
u_{a,\mu,-}\|_{2}^{2},\|u_{a,\mu,-}\|_{2^{*}}^{2^{*}}\to
S^{\frac{N}{2}}\quad\text{as }\mu\to 0^{+}$

for $2<q<2+\frac{4}{N}$ (see also [34, Theorem 1.7]). It follows that, up to a
subsequence, $\\{u_{a,\mu,-}\\}$ is a minimizing sequence of the following
minimizing problem:

$\displaystyle S=\inf_{u\in
D^{1,2}(\mathbb{R}^{N})\backslash\\{0\\}}\frac{\|\nabla
u\|_{2}^{2}}{\|u\|_{2^{*}}^{2}}.$ (4.1)

Since $N\geq 5$, $U_{\varepsilon}\in L^{2}(\mathbb{R}^{N})$ for all
$\varepsilon>0$. We then choose $\varepsilon_{0}>0$ such that
$U_{\varepsilon_{0}}\in\mathcal{S}_{a}$. By [49, Lemma 4.2], there exists
$t(\mu)>0$ such that
$(U_{\varepsilon_{0}})_{t(\mu)}\in\mathcal{P}_{-}^{a,\mu}$ for $\mu>0$
sufficiently small, that is,

$\displaystyle[t(\mu)]^{2}S^{\frac{N}{2}}=\mu\gamma_{q}\|U_{\varepsilon_{0}}\|_{q}^{q}[t(\mu)]^{q\gamma_{q}}+[t(\mu)]^{2^{*}}S^{\frac{N}{2}}.$

Clearly, by the implicit function theorem, $t(\mu)$ is of class $C^{1}$ for
$|\mu|<<1$ such that $t(0)=1$. It follows from
$S^{\frac{N}{2}}(1-[t(\mu)]^{2^{*}-2})=\mu\gamma_{q}\|U_{\varepsilon_{0}}\|_{q}^{q}[t(\mu)]^{q\gamma_{q}-2}$
that

$\displaystyle
t(\mu)=1-\frac{\gamma_{q}\|U_{\varepsilon_{0}}\|_{q}^{q}}{(2^{*}-2)S^{\frac{N}{2}}}\mu+o(\mu),$
(4.2)

which implies

$\displaystyle m_{a,\mu}^{-}$ $\displaystyle\leq$
$\displaystyle\mathcal{E}_{\mu}((U_{\varepsilon_{0}})_{t(\mu)})$ (4.3)
$\displaystyle=$
$\displaystyle\frac{1}{N}S^{\frac{N}{2}}-\frac{\mu\gamma_{q}\|U_{\varepsilon_{0}}\|_{q}^{q}}{2^{*}}-\frac{\mu}{q}(1-\frac{q\gamma_{q}}{2^{*}})\|U_{\varepsilon_{0}}\|_{q}^{q}+o(\mu)$
$\displaystyle=$
$\displaystyle\frac{1}{N}S^{\frac{N}{2}}-\frac{\mu\|U_{\varepsilon_{0}}\|_{q}^{q}}{q}+o(\mu)$

for $N\geq 5$. Since we have $m_{a,\mu}^{-}=\frac{1}{N}\|\nabla
u_{a,\mu,-}\|_{2}^{2}-\frac{\mu}{q}(1-\frac{q\gamma_{q}}{2^{*}})\|u_{a,\mu,-}\|_{q}^{q}$,
by (4.3),

$\displaystyle\frac{1}{N}\|\nabla
u_{a,\mu,-}\|_{2}^{2}-\frac{\mu}{q}(1-\frac{q\gamma_{q}}{2^{*}})\|u_{a,\mu,-}\|_{q}^{q}\leq\frac{1}{N}S^{\frac{N}{2}}-\frac{\mu\|U_{\varepsilon_{0}}\|_{q}^{q}}{q}+o(\mu).$
(4.4)

On the other hand, by (4.1) and $u_{a,\mu,-}\in\mathcal{P}_{-}^{a,\mu}$,

$\displaystyle S$ $\displaystyle\leq$ $\displaystyle\frac{\|\nabla
u_{a,\mu,-}\|_{2}^{2}}{\|u_{a,\mu,-}\|_{2^{*}}^{2}}$ $\displaystyle=$
$\displaystyle\frac{\|\nabla u_{a,\mu,-}\|_{2}^{2}}{(\|\nabla
u_{a,\mu,-}\|_{2}^{2}-\mu\gamma_{q}\|u_{a,\mu,-}\|_{q}^{q})^{\frac{2}{2^{*}}}}$
$\displaystyle=$ $\displaystyle(\|\nabla
u_{a,\mu,-}\|_{2}^{2}-\mu\gamma_{q}\|u_{a,\mu,-}\|_{q}^{q})^{\frac{2}{N}}+\frac{\mu\gamma_{q}\|u_{a,\mu,-}\|_{q}^{q}}{S^{\frac{N-2}{2}}}+o(\mu\|u_{a,\mu,-}\|_{q}^{q}).$

It follows that

$\displaystyle\|\nabla u_{a,\mu,-}\|_{2}^{2}\geq
S^{\frac{N}{2}}-\frac{N-2}{2}\mu\gamma_{q}\|u_{a,\mu,-}\|_{q}^{q}+o(\mu\|u_{a,\mu,-}\|_{q}^{q}).$
(4.5)

Combining (4.4) and (4.5), we have

$\displaystyle\|u_{a,\mu,-}\|_{q}^{q}\geq\|U_{\varepsilon_{0}}\|_{q}^{q}+o(1).$
(4.6)

Since $\\{u_{a,\mu,-}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$,
$u_{a,\mu,-}\rightharpoonup u_{0,-}$ weakly in $H^{1}(\mathbb{R}^{N})$ as
$\mu\to 0^{+}$ up to a subsequence. Since $u_{a,\mu,-}$ is radial and
decreasing for $r=|x|$,

$\displaystyle\sup_{y\in\mathbb{R}^{N}}\int_{B_{1}(y)}|u_{a,\mu,-}|^{2}dx=\int_{B_{1}(0)}|u_{a,\mu,-}|^{2}dx.$

Thus, by (4.6), Lions’ lemma [56, Lemma 1.21] and the Sobolev embedding
theorem, $u_{0,-}\not=0$. Note that it is standard to show that $u_{0,-}$ is a
weak solution of the following equation,

$\displaystyle-\Delta U=U^{2^{*}-1},\quad\text{in }\mathbb{R}^{N},$

thus, we must have $\|\nabla u_{0,-}\|_{2}^{2}\geq S^{\frac{N}{2}}$. It
follows from $\|\nabla u_{a,\mu,-}\|_{2}^{2}\to S^{\frac{N}{2}}$ as $\mu\to
0^{+}$ that $u_{a,\mu,-}\to u_{0,-}$ strongly in $D^{1,2}(\mathbb{R}^{N})$ as
$\mu\to 0^{+}$ up to a subsequence, which implies $u_{0,-}=U_{\varepsilon}$
for some $\varepsilon>0$. Since $\|U_{\varepsilon_{0}}\|_{2}^{2}=a^{2}$, by
the Fatou lemma and (4.6),

$\displaystyle\|U_{\varepsilon}\|_{q}^{q}\geq\|U_{\varepsilon_{0}}\|_{q}^{q}\quad\text{and}\quad\|U_{\varepsilon}\|_{2}\leq\|U_{\varepsilon_{0}}\|_{2}^{2}.$

Hence, we must have $\varepsilon=\varepsilon_{0}$ and thus,
$\|u_{0,-}\|_{2}=\|U_{\varepsilon_{0}}\|_{2}^{2}=a^{2}$, which implies
$u_{a,\mu,-}\to U_{\varepsilon_{0}}$ strongly in $H^{1}(\mathbb{R}^{N})$ as
$\mu\to 0^{+}$ up to a subsequence. Since $U_{\varepsilon_{0}}$ is the unique
Aubin-Talanti babble in $\mathcal{S}_{a}$, we have $u_{a,\mu,-}\to
U_{\varepsilon_{0}}$ strongly in $H^{1}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$.
Moreover, since $u_{a,\mu,-}$ is a solution of (1.5), by the Pohozaev identity
and $u_{a,\mu,-}\in\mathcal{P}_{a,\mu}$, we have

$\displaystyle-\lambda_{a,\mu,-}\|u_{a,\mu,-}\|_{2}^{2}=(1-\gamma_{q})\mu\|u_{a,\mu,-}\|_{q}^{q},$
(4.7)

which implies $\lambda_{a,\mu,-}\to 0$ as $\mu\to 0^{+}$. It remains to prove
that $u_{a,\mu,-}$ is the unique minimizer of
$\mathcal{E}|_{\mathcal{S}_{a}}(u)$ on $\mathcal{P}_{-}^{a,\mu}$ for $\mu>0$
sufficiently small up to translations and rotations. For this, let us first
claim that

$\displaystyle
u_{a,\mu,-}\lesssim\bigg{(}\frac{1}{1+r^{2}}\bigg{)}^{\frac{N-2}{2}}$ (4.8)

for all $r\geq 0$ in the case of $\mu>0$ sufficiently small. Indeed, since
$u_{a,\mu,-}$ is a positive and radially decreasing solution of (1.5), by
Struss’s radial lemma (cf. [12, Lemma A.IV, Theorem A.I’] or [44, Lemma 3.1]),
$u_{a,\mu,-}\lesssim r^{-\frac{N-1}{2}}$ for $r\geq 1$. Thus, by (4.7),
$u_{a,\mu,-}$ satisfies

$\displaystyle-
u_{a,\mu,-}^{\prime\prime}-\frac{N-1}{r}u_{a,\mu,-}^{\prime}\lesssim
u_{a,\mu,-}^{\frac{4}{N-2}}u_{a,\mu,-}\lesssim
r^{-(2+\delta)}u_{a,\mu,-}\quad\text{for }r\gtrsim 1$ (4.9)

for some $\delta>0$. By bootstrapping we obtain the desired decaying estimate
(4.8).

Now, let us consider the following system:

$\displaystyle\left\\{\begin{aligned} &\mathcal{F}(w,\alpha,\mu)=\Delta
w-\alpha w+\mu w^{q-1}+w^{2^{*}-1},\\\
&\mathcal{G}(w,\alpha,\mu)=\|w\|_{2}^{2}-a^{2},\end{aligned}\right.$ (4.10)

where $\alpha,\mu>0$ are parameters. It is easy to see that
$\mathcal{F}(U_{\varepsilon_{0}},0,0)=0$ and
$\mathcal{G}(U_{\varepsilon_{0}},0,0)=0$. Let

$\displaystyle\mathcal{L}(U_{\varepsilon_{0}},0,0)=\left(\begin{aligned}
\partial_{w}\mathcal{F}(U_{\varepsilon_{0}},0,0)\quad\partial_{\alpha}\mathcal{F}(U_{\varepsilon_{0}},0,0)\\\
\partial_{w}\mathcal{G}(U_{\varepsilon_{0}},0,0)\quad\partial_{\alpha}\mathcal{G}(U_{\varepsilon_{0}},0,0)\end{aligned}\right)$

be the linearization of the system (4.10) at $(U_{\varepsilon_{0}},0,0)$ in
$H^{1}(\mathbb{R}^{N})\times\mathbb{R}$, that is,

$\displaystyle\partial_{w}\mathcal{F}(U_{\varepsilon_{0}},0,0)=\Delta+(2^{*}-1)U_{\varepsilon_{0}}^{2^{*}-2},\quad\partial_{\alpha}\mathcal{F}(U_{\varepsilon_{0}},0,0)=-U_{\varepsilon_{0}}$

and

$\displaystyle\partial_{w}\mathcal{G}(U_{\varepsilon_{0}},0,0)=2U_{\varepsilon_{0}},\quad\partial_{\alpha}\mathcal{G}(U_{\varepsilon_{0}},0,0)=0.$

Then $\mathcal{L}(U_{\varepsilon_{0}},0,0)[(\phi,\tau)]=0$ if and only if

$\displaystyle\left\\{\begin{aligned}
&\Delta\phi+(2^{*}-1)U_{\varepsilon_{0}}^{2^{*}-2}\phi-\tau
U_{\varepsilon_{0}}=0,\\\
&\int_{\mathbb{R}^{N}}U_{\varepsilon_{0}}\phi=0.\end{aligned}\right.$ (4.11)

We claim that in $H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$,
$\mathcal{L}(U_{\varepsilon_{0}},0,0)[(\phi,\tau)]=0$ if and only if
$(\phi,\tau)=(0,0)$. Let $\mathcal{L}(U_{\varepsilon_{0}},0,0)[(\phi,\tau)]=0$
for some $(\phi,\tau)\in H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$. Since
it is well-known (cf. [14]) that
$W=\frac{N-2}{2}U_{\varepsilon_{0}}+U_{\varepsilon_{0}}^{\prime}r$ is the
unique radial solution of the following equation

$\displaystyle\Delta\phi+(2^{*}-1)U_{\varepsilon_{0}}^{2^{*}-2}\phi=0$

in $H^{1}_{rad}(\mathbb{R}^{N})$, by multiplying the first equation of (4.11)
with $W$ and integrating by parts, we have

$\displaystyle
0=\tau\int_{\mathbb{R}^{N}}WU_{\varepsilon_{0}}=-\tau\int_{\mathbb{R}^{N}}U_{\varepsilon_{0}}^{2}.$

It follows that $\tau=0$ and thus, $\phi=CW$ for some constant
$C\in\mathbb{R}$. By the second equation of (4.11),

$\displaystyle
0=\int_{\mathbb{R}^{N}}U_{\varepsilon_{0}}\phi=-C\int_{\mathbb{R}^{N}}U_{\varepsilon_{0}}^{2},$

which implies that $\phi=0$. Thus, the kernel of the linearization of the
system (4.10) at $(U_{\varepsilon_{0}},0,0)$ in
$H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$ is trivial, which implies that
the linear operator
$\mathcal{L}(U_{\varepsilon_{0}},0,0):H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}\to
H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$ is injective. On the other hand,
by similar arguments as used for Proposition 2.1, we know that all minimizers
of $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ on $\mathcal{P}_{-}^{a,\mu}$ are
real valued, positive, radially symmetric and radially decreasing up to
translations and rotations. Now, suppose that there are at least two
minimizers of $\mathcal{E}_{\mu}(u)|_{\mathcal{S}_{a}}$ on
$\mathcal{P}_{-}^{a,\mu}$, say $u_{\mu}^{*}$ and $u_{\mu}^{**}$, then without
loss of generality, we may assume that they are all real valued, positive,
radially symmetric and radially decreasing. The corresponding Lagrange
multipliers are $\lambda_{\mu}^{*}$ and $\lambda_{\mu}^{**}$, respectively.
Let

$\displaystyle
w_{\mu}=\frac{u_{\mu}^{*}-u_{\mu}^{**}}{\|u_{\mu}^{*}-u_{\mu}^{**}\|_{H^{1}}+|\lambda_{\mu}^{*}-\lambda_{\mu}^{**}|}\quad\text{and}\quad\varsigma_{\mu}=\frac{\lambda_{\mu}^{*}-\lambda_{\mu}^{**}}{\|u_{\mu}^{*}-u_{\mu}^{**}\|_{H^{1}}+|\lambda_{\mu}^{*}-\lambda_{\mu}^{**}|},$

where $\|\cdot\|_{H^{1}}$ is the usual norm in $H^{1}(\mathbb{R}^{N})$. It is
easy to see that $\\{w_{\mu}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ and
$\\{\varsigma_{\mu}\\}$ is bounded. Moreover, by (1.5), we also have

$\displaystyle-\Delta
w_{\mu}-\lambda_{\mu}^{*}w_{\mu}-\varsigma_{\mu}u_{\mu}^{**}$ $\displaystyle=$
$\displaystyle\mu(q-1)\bigg{(}u_{\mu}^{*}+\theta_{\mu}(u_{\mu}^{*}-u_{\mu}^{**})\bigg{)}^{q-2}w_{\mu}$
(4.12)
$\displaystyle+(2^{*}-1)\bigg{(}u_{\mu}^{*}+\theta_{\mu}^{\prime}(u_{\mu}^{*}-u_{\mu}^{**})\bigg{)}^{2^{*}-2}w_{\mu},$

where $\theta_{\mu},\theta_{\mu}^{\prime}\in(0,1)$. Since $u_{\mu}^{*}$ and
$u_{\mu}^{**}$ belong to $\mathcal{S}_{a}$, we also have

$\displaystyle
2\int_{\mathbb{R}^{N}}u_{\mu}^{*}w_{\mu}=-\|u_{\mu}^{*}-u_{\mu}^{**}\|_{2}\|w_{\mu}\|_{2}.$

Since the linear operator
$\mathcal{L}(U_{\varepsilon_{0}},0,0):H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}\to
H_{rad}^{1}(\mathbb{R}^{N})\times\mathbb{R}$ is injective, it is standard to
prove that $(w_{\mu},\varsigma_{\mu})\rightharpoonup(0,0)$ weakly in
$D^{1,2}(\mathbb{R}^{N})\times\mathbb{R}$ as $\mu\to 0^{+}$. Now, by
multiplying (4.12) with $w_{\mu}$ and integrating by parts, we can use fact
that $u_{\mu}^{**},u_{\mu}^{*}\to U_{\varepsilon_{0}}$ strongly in
$H^{1}(\mathbb{R}^{N})$ as $\mu\to+\infty$ to show that
$(w_{\mu},\varsigma_{\mu})\to(0,0)$ strongly in
$D^{1,2}(\mathbb{R}^{N})\times\mathbb{R}$ as $\mu\to 0^{+}$. Moreover, by
(4.7), we also have

$\displaystyle\varsigma_{\mu}\sim\mu\int_{\mathbb{R}^{N}}\bigg{(}u_{\mu}^{*}+\theta_{\mu}(u_{\mu}^{*}-u_{\mu}^{**})\bigg{)}^{q-1}w_{\mu}=o(\mu).$

By (4.7), (4.8) and (4.12),

$\displaystyle-\Delta
w_{\mu}-\frac{1}{2}\lambda_{\mu}^{*}w_{\mu}\lesssim\frac{\mu}{r^{N-2}}\quad\text{for
}r\gtrsim\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}.$

By (4.7), we also have

$\displaystyle-\Delta(r^{2-N})-\frac{1}{2}\lambda_{\mu}^{*}r^{2-N}=-\frac{1}{2}\lambda_{\mu}^{*}r^{2-N}\gtrsim\frac{\mu}{r^{N-2}}\quad\text{for
}r\gtrsim\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}.$

Since $w_{\mu}$ is radial and $\\{w_{\mu}\\}$ is bounded in
$H^{1}(\mathbb{R}^{N})$, by [12, Lemma A.2],

$\displaystyle|w_{\mu}|\lesssim r^{-\frac{N-1}{2}}\quad\text{for }r\gtrsim 1.$
(4.13)

Thus, by the maximum principle,

$\displaystyle|w_{\mu}|\lesssim r^{2-N}\quad\text{for
}r\gtrsim\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}.$ (4.14)

For $1\lesssim r\lesssim\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}$, by
(4.8), (4.12) and (4.13),

$\displaystyle-\Delta
w_{\mu}\lesssim\frac{1}{r^{4}}(\frac{1}{r^{\frac{N-1}{2}}}+\frac{1}{r^{N-2}})\lesssim
r^{-\frac{7+N}{2}}\lesssim r^{-\frac{\alpha+N}{2}}$

in the case of $N\geq 5$, where $\alpha=\frac{9}{2}$. Recall that for $N\geq
5$, $r^{2-\frac{\alpha+N}{2}}$ is also a superharmornic function. Thus, by the
maximum principle,

$\displaystyle|w_{\mu}|\lesssim r^{2-\frac{\alpha+N}{2}}\quad\text{for
}1\lesssim r\lesssim\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}.$ (4.15)

Note that by $w_{\mu}\to 0$ strongly in $D^{1,2}(\mathbb{R}^{N})$ as $\mu\to
0^{+}$ and $\|w_{\mu}\|_{H^{1}}^{2}=1$, we know that
$\|w_{\mu}\|_{2}^{2}=1+o_{\mu}(1)$. Thus, by $w_{\mu}\to 0$ strongly in
$D^{1,2}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$, the Sobolev embedding theorem and
(4.14) and (4.15),

$\displaystyle 1\sim\int_{\mathbb{R}^{N}}|w_{\mu}|^{2}\lesssim
o_{\mu}(1)+\int_{r_{0}}^{\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}}r^{3-\alpha}+\int_{\frac{1}{|\lambda_{\mu}^{*}|^{\frac{1}{4}}}}^{+\infty}r^{3-N}=o_{\mu}(1)+\frac{1}{2}r_{0}^{-\frac{1}{2}},$

which is a contradiction by taking $r_{0}>0$ sufficiently large. It follows
that $u_{a,\mu,-}$ is the unique minimizer of
$\mathcal{E}|_{\mathcal{S}_{a}}(u)$ on $\mathcal{P}_{-}^{a,\mu}$ for $\mu>0$
sufficiently small up to translations and rotations if $N\geq 5$. ∎

For $N=3,4$, The Aubin-Talanti babbles $U_{\varepsilon}\not\in
L^{2}(\mathbb{R}^{N})$. Thus, we need to modify the arguments for Proposition
4.1 to give a precise description of $u_{a,\mu,-}$ as $\mu\to 0^{+}$ in these
two cases. As in the proof of Proposition 4.1, $\\{u_{a,\mu,-}\\}$ is also a
minimizing sequence of the minimizing problem (4.1) in the cases $N=3,4$.
Since $u_{a,\mu,-}$ is radially symmetric for $r=|x|$, by Lions’ result (cf.
[56, Theorem 1.41]), up to subsequence, there exists $\sigma_{\mu}>0$ such
that for some $\varepsilon_{*}>0$,

$\displaystyle
v_{a,\mu,-}(x)=\sigma_{\mu}^{\frac{N-2}{2}}u_{a,\mu,-}(\sigma_{\mu}x)\to
U_{\varepsilon_{*}}\text{ strongly in }D^{1,2}(\mathbb{R}^{N})\text{ as
}\mu\to 0^{+}.$ (4.16)

We also remark that since $U_{\varepsilon_{*}}\not\in L^{2}(\mathbb{R}^{N})$
for $N=3,4$ and $\|v_{a,\mu,-}\|_{2}^{2}=\frac{a^{2}}{\sigma_{\mu}^{2}}$, by
the Fatou lemma, we have $\sigma_{\mu}\to 0$ as $\mu\to 0^{+}$.

######  Lemma 4.1.

Let $N=3,4$ and $2<q<2^{*}$. Then

$\displaystyle 1\sim\left\\{\begin{aligned}
&\frac{\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}}{-\lambda_{a,\mu,-}},\quad\frac{N}{N-2}<q<2^{*},\\\
&\frac{\mu\sigma_{\mu}^{\frac{3}{2}}}{-\lambda_{a,\mu,-}}\ln\bigg{(}\frac{1}{\sqrt{-\lambda_{a,\mu,-}}\sigma_{\mu}}\bigg{)},\quad
N=3,q=3,\\\
&\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}\bigg{(}\sqrt{-\lambda_{a,\mu,-}}\sigma_{\mu}\bigg{)}^{q-3}}{-\lambda_{a,\mu,-}},\quad
N=3,2<q<3.\end{aligned}\right.$

###### Proof.

By the equation (1.5), we know that $v_{a,\mu,-}$ satisfies

$\displaystyle-\Delta
v_{a,\mu,-}-\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}=\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}v_{a,\mu,-}^{q-1}+v_{a,\mu,-}^{2^{*}-1}\quad\text{in
}\mathbb{R}^{N}.$ (4.17)

It follows from (4.7) that

$\displaystyle-\lambda_{a,\mu,-}\sigma_{\mu}^{2}\|v_{a,\mu,-}\|_{2}^{2}=(1-\gamma_{q})\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}\|v_{a,\mu,-}\|_{q}^{q}.$
(4.18)

Recall that
$\|u_{a,\mu,-}\|_{2}^{2}=\sigma_{\mu}^{2}\|v_{a,\mu,-}\|_{2}^{2}=a^{2}$ and
$\|\nabla u_{a,\mu,-}\|_{2}^{2}\to S^{\frac{N}{2}}$ as $\mu\to 0^{+}$, by
(4.7) and the Hölder inequality, $\lambda_{a,\mu,-}\to 0$ as $\mu\to 0^{+}$.
Clearly,

$\displaystyle\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}\to 0\quad\text{as }\mu\to
0^{+}.$ (4.19)

By the Hölder inequality once more,

$\displaystyle|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}\|v_{a,\mu,-}\|_{2}^{2}\lesssim\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}\|v_{a,\mu,-}\|_{2}^{N-\frac{N-2}{2}q}.$

Since $q>2$, $N-\frac{N-2}{2}q<2$. It follows from
$\|v_{a,\mu,-}\|_{2}^{2}\sim\sigma_{\mu}^{-2}\to+\infty$ as $\mu\to 0^{+}$
that

$\displaystyle|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}=o(\mu\sigma_{\mu}^{N-\frac{N-2}{2}q})\quad\text{as
}\mu\to 0^{+}.$ (4.20)

Recall that $v_{a,\mu,-}\to U_{\varepsilon_{*}}$ strongly in
$D^{1,2}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$ up to a subsequence, by
(4.19)-(4.20), adapting the Moser iteration (cf. [53, B.3 Lemma]) and the
$L^{p}$ theory of elliptic equations to (4.17) and the Sobolev embedding
theorem,

$\displaystyle v_{a,\mu,-}\to U_{\varepsilon_{*}}\quad\text{strongly in
$L^{\infty}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$ up to a subsequence}$ (4.21)

In what follows, we follow the ideas in [7] (see also [24, 38]) to drive a
uniformly upper bound of $v_{a,\mu,-}$. We define

$\displaystyle\widetilde{v}_{a,\mu,-}=\frac{1}{v_{a,\mu,-}(0)}v_{a,\mu,-}(\sqrt{v_{a,\mu,-}(0)}x).$

Since $\widetilde{v}_{a,\mu,-}$ is radial, $\widetilde{v}_{a,\mu,-}$ satisfies

$\displaystyle-\widetilde{v}_{a,\mu,-}^{\prime\prime}-\frac{N-1}{r}\widetilde{v}_{a,\mu,-}^{\prime}=f(\widetilde{v}_{a,\mu,-})\quad\text{in
}\mathbb{R}^{N}.$ (4.22)

where

$\displaystyle f(\widetilde{v}_{a,\mu,-})$ $\displaystyle=$
$\displaystyle\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}(0)\widetilde{v}_{a,\mu,-}+\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}[v_{a,\mu,-}(0)]^{q-1}\widetilde{v}_{a,\mu,-}^{q-1}$
$\displaystyle+[v_{a,\mu,-}(0)]^{2^{*}-1}\widetilde{v}_{a,\mu,-}^{2^{*}-1}.$

Let

$\displaystyle
H(r)=r^{N}(\widetilde{v}_{a,\mu,-}^{\prime})^{2}+(N-2)r^{N-1}\widetilde{v}_{a,\mu,-}\widetilde{v}_{a,\mu,-}^{\prime}+\frac{N-2}{N}r^{N}\widetilde{v}_{a,\mu,-}f(\widetilde{v}_{a,\mu,-}).$

Then by direct calculations and using (4.20)-(4.22),

$\displaystyle H^{\prime}(r)$ $\displaystyle=$
$\displaystyle\frac{r^{N}\widetilde{v}_{a,\mu,-}^{\prime}}{N}(4|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}-(N-2)(2^{*}-q)\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}v_{a,\mu,-}^{q-2})v_{a,\mu,-}$
$\displaystyle=$
$\displaystyle\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}\frac{r^{N}\widetilde{v}_{a,\mu,-}^{\prime}}{N}(o_{\mu}(1)-(N-2)(2^{*}-q)U_{\varepsilon_{*}}^{q-2})v_{a,\mu,-}.$

Since $v_{a,\mu,-}>0$, $\widetilde{v}_{a,\mu,-}^{\prime}<0$ and $v_{a,\mu,-}$
exponentially decays to zero as $r\to+\infty$, there exists $r_{\mu}>0$,
$H^{\prime}(r)>0$ for $0<r<r_{\mu}$ and $H^{\prime}(r)<0$ for $r>r_{\mu}$.
Thus, $H(r)>H(0)=0$ for all $r>0$. Let

$\displaystyle\Psi(r)=\frac{-\widetilde{v}_{a,\mu,-}^{\prime}}{r\widetilde{v}_{a,\mu,-}^{\frac{N}{N-2}}}.$

Then by direct calculations and using (4.22),

$\displaystyle\Psi^{\prime}(r)=\frac{N}{N-2}r^{-(1+N)}\widetilde{v}_{a,\mu,-}^{-\frac{2N-2}{N-2}}H(r)>0$

for all $r>0$. It follows from (4.22) once more that

$\displaystyle\Psi(r)>\Psi(0)=-\widetilde{v}_{a,\mu,-}^{\prime\prime}(0)=\frac{d_{\mu}}{N}$

where

$\displaystyle
d_{\mu}=\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}(0)+\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}[v_{a,\mu,-}(0)]^{q-1}+[v_{a,\mu,-}(0)]^{2^{*}-1}.$

Let

$\displaystyle
Z_{\mu}(r)=\frac{1}{(1+\frac{d_{\mu}}{N(N-2)}r^{2})^{\frac{N-2}{2}}}.$

Then it is easy to check that
$\frac{-Z_{\mu}^{\prime}(r)}{[Z_{\mu}(r)]^{\frac{N}{N-2}}}=\frac{d_{\mu}}{N}r$.
It follows that

$\displaystyle\frac{\widetilde{v}_{a,\mu,-}^{\prime}}{\widetilde{v}_{a,\mu,-}^{\frac{N}{N-2}}}\leq\frac{Z_{\mu}^{\prime}(r)}{[Z_{\mu}(r)]^{\frac{N}{N-2}}}\quad\text{for
all }r>0,$

which together with (4.21), implies

$\displaystyle
v_{a,\mu,-}\lesssim\frac{1}{(1+r^{2})^{\frac{N-2}{2}}}\quad\text{for all }r>0$
(4.23)

uniformly for $\mu>0$ sufficiently small. Now, for the cases
$\frac{N}{N-2}<q<2^{*}$,

$\displaystyle\|v_{a,\mu,-}\|_{q}^{q}\lesssim\int_{0}^{+\infty}\frac{1}{(1+r^{2})^{\frac{(N-2)q}{2}}}r^{N-1}dr\lesssim
1.$

By the Fatou lemma,
$\|v_{a,\mu,-}\|_{q}^{q}\geq\|U_{\varepsilon_{*}}\|_{q}^{q}+o_{\mu}(1)\gtrsim
1$. Thus, by (4.18),

$\displaystyle\frac{\mu\sigma_{\mu}^{N-\frac{N-2}{2}q}}{-\lambda_{a,\mu,-}}\sim
1\quad\text{for }\frac{N}{N-2}<q<2^{*}.$

For $N=3$ and $q=3$, we need to drive the uniformly exponential decay of
$v_{a,\mu,-}$ at infinitely both from below and above to obtain the
conclusions. Let

$\displaystyle\Phi=r^{-1}e^{-\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}.$

Then it is easy to check (cf. [44]) that
$-\Delta\Phi-\lambda_{a,\mu,-}\sigma_{\mu}^{2}\Phi\leq 0$ for $r\geq 1$ in the
case of $N=3$. Since $v_{a,\mu,-}\to U_{\varepsilon_{*}}$ strongly in
$L^{\infty}(\mathbb{R}^{N})$ as $\mu\to 0^{+}$ up to a subsequence, by the
maximum principle,

$\displaystyle v_{a,\mu,-}\gtrsim
r^{-1}e^{-\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}\quad\text{for }r\geq 1$
(4.24)

in the case of $N=3$. On the other hand, let

$\displaystyle\Upsilon=r^{-1}e^{-\frac{1}{2}\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}.$

Then it is also easy to check that
$-\Delta\Upsilon-\frac{1}{2}\lambda_{a,\mu,-}\sigma_{\mu}^{2}\Upsilon\geq 0$
for $r\geq 1$. Since $\mu\sigma_{\mu}^{3-\frac{q}{2}}\to 0$ as $\mu\to 0^{+}$,
by (4.23), for

$\displaystyle r\gtrsim\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}},$

we have $-\Delta
v_{a,\mu,-}-\frac{1}{2}\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}\leq 0$ in
the case of $N=3$. Thus, by the maximum principle and (4.21) once more,

$\displaystyle v_{a,\mu,-}\lesssim
r^{-1}e^{-\frac{1}{2}\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}\quad\text{for
}r\gtrsim\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}.$ (4.25)

For $q=3$ and $N=3$, by (4.23) and (4.25),

$\displaystyle\|v_{a,\mu,-}\|_{3}^{3}$ $\displaystyle\lesssim$
$\displaystyle\int_{1}^{\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}}r^{-1}+\int_{\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}}^{+\infty}e^{-\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}$
$\displaystyle\lesssim$
$\displaystyle\ln\bigg{(}\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}\bigg{)}.$

By (4.24), for $q=3$ and $N=3$, we also have

$\displaystyle\|v_{a,\mu,-}\|_{3}^{3}\gtrsim\int_{1}^{\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}}r^{-1}\gtrsim\ln\bigg{(}\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}\bigg{)}.$

Thus, the conclusion for $q=3$ and $N=3$ then follows (4.18). For $N=3$ and
$2<q<3$, we need to construct a newly upper bound of $v_{a,\mu,-}$ by adapting
ideas in [20]. By (4.23), we have

$\displaystyle-\Delta
v_{a,\mu,-}-\frac{1}{2}\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}\lesssim\mu\sigma_{\mu}^{3-\frac{q}{2}}r^{1-q}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}.$

On the other hand, let
$\phi_{\mu}\sim\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}r^{1-q}$,
then by direct calculations,

$\displaystyle-\Delta\phi_{\mu}-\frac{1}{2}\lambda_{a,\mu,-}\sigma_{\mu}^{2}\phi_{\mu}\gtrsim\mu\sigma_{\mu}^{3-\frac{q}{2}}r^{1-q}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}.$

By (4.23) and the maximum principle,

$\displaystyle
v_{a,\mu,-}\lesssim\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}r^{1-q}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}.$ (4.26)

Now, using (4.26) as a new barrier and by (4.23), we know that

$\displaystyle-\Delta
v_{a,\mu,-}-\frac{1}{2}\lambda_{a,\mu,-}\sigma_{\mu}^{2}v_{a,\mu,-}\lesssim\mu\sigma_{\mu}^{3-\frac{q}{2}}\bigg{(}\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}r^{1-q}\bigg{)}^{q-1}$

for $r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}$. Thus, by
similar comparisons, we have

$\displaystyle
v_{a,\mu,-}\lesssim\bigg{(}\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{q}r^{-(q-1)^{2}}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}.$

By iterating the above arguments $n$ times for a sufficiently large $n$ such
that $q(q-1)^{n}-3>0$, we have

$\displaystyle
v_{a,\mu,-}\lesssim\bigg{(}\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{s_{n}}r^{-(q-1)^{n}}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}},$ (4.27)

where $s_{n}=s_{n-1}(q-1)+1$ which implies

$\displaystyle s_{n}=\frac{(q-1)^{n+1}-1}{q-2}.$

By (4.24), we have

$\displaystyle\|v_{a,\mu,-}\|_{q}^{q}\gtrsim\int_{1}^{\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}}r^{2-q}e^{-q\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}\gtrsim\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}}$
(4.28)

and

$\displaystyle\|v_{a,\mu,-}\|_{2}^{2}\gtrsim\int_{1}^{\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}}e^{-2\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}r}\gtrsim\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{1}{2}}.$

It follows from (4.18) that

$\displaystyle|\lambda_{a,\mu,-}|\gtrsim\mu\sigma_{\mu}^{3-\frac{q}{2}}\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}}\quad\text{and}\quad\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{1}{2}}|\lambda_{a,\mu,-}|\lesssim\mu$
(4.29)

which implies

$\displaystyle\frac{\mu\sigma_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\lesssim|\lambda_{a,\mu,-}|^{\frac{3-q}{2}}\sigma_{\mu}^{1-q}\lesssim\sigma_{\mu}^{2(2-q)}.$

Now, by (4.27) and (4.29),

$\displaystyle\|v_{a,\mu,-}\|_{q}^{q}$ $\displaystyle\lesssim$
$\displaystyle\int_{1}^{\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}}r^{2-q}+(\sigma_{\mu})^{-2((q-1)^{n+1}-1)}\int_{\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}}^{+\infty}r^{2-q(q-1)^{n}}$
(4.30) $\displaystyle\lesssim$
$\displaystyle\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}}+\sigma_{\mu}^{2(q(q-1)^{n}-3)-2((q-1)^{n+1}-1)}$
$\displaystyle\lesssim$
$\displaystyle\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}}+\sigma_{\mu}^{(q-1)^{n}(2+o_{n}(1))}$
$\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\sigma_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}}.$

The conclusion for $N=3$ and $2<q<3$ follows from (4.18), (4.28) and (4.30). ∎

With Lemma 4.1 in hands, we can obtain the following.

######  Proposition 4.2.

Let $N=3,4$ and $2<q<2^{*}$. Then

$\displaystyle
w_{\mu,-}=\varepsilon_{\mu}^{\frac{N-2}{2}}u_{a,\mu,-}(\varepsilon_{\mu}x)\to
U_{\varepsilon_{*}}\quad\text{strongly in }D^{1,2}(\mathbb{R}^{N})\text{ as
}\mu\to 0^{+}$

up to a subsequence for some $\varepsilon_{*}>0$, where $\varepsilon_{\mu}>0$
satisfies

$\displaystyle\mu\sim\left\\{\begin{aligned}
&\varepsilon_{\mu}^{6-q}e^{-2\varepsilon_{\mu}^{-2}},\quad N=4,2<q<2^{*},\\\
&\varepsilon_{\mu}^{\frac{q}{2}-1},\quad N=3,3<q<2^{*},\\\
&\frac{\varepsilon_{\mu}^{\frac{1}{2}}}{\ln(\frac{1}{\varepsilon_{\mu}})},\quad
N=3,q=3,\\\ &\varepsilon_{\mu}^{5-\frac{3q}{2}},\quad
N=3,2<q<3.\end{aligned}\right.$ (4.31)

###### Proof.

Let $\\{V_{\varepsilon}\\}$ be the family given by (2.4). Since
$2\geq\frac{N}{N-2}$ for $N\geq 4$, By [44, (4.2)–(4.5)],

$\displaystyle\|V_{\varepsilon}\|_{q}^{q}\sim\left\\{\begin{aligned}
&\varepsilon^{N-\frac{N-2}{2}q},\quad N=3,4,\frac{N}{N-2}<q<2^{*},\\\
&\varepsilon^{\frac{3}{2}}\ln(R_{\varepsilon}\varepsilon^{-1}),\quad
N=3,q=3,\\\
&\varepsilon^{3-\frac{q}{2}}(R_{\varepsilon}\varepsilon^{-1})^{3-q},\quad
N=3,2<q<3\end{aligned}\right.$ (4.32)

for $\varepsilon>0$ sufficiently small. By [49, Lemmas 4.2, 5.1 and 6.1],
there exist $t_{\mu,\varepsilon}>0$ such that

$\displaystyle\|\nabla
V_{\varepsilon}\|_{2}^{2}=\mu\gamma_{q}\|V_{\varepsilon}\|_{q}^{q}t_{\mu,\varepsilon}^{q\gamma_{q}-2}+\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}t_{\mu,\varepsilon}^{2^{*}-2}$

and

$\displaystyle 2\|\nabla V_{\varepsilon}\|_{2}^{2}<\mu
q\gamma_{q}^{2}\|V_{\varepsilon}\|_{q}^{q}t_{\mu,\varepsilon}^{q\gamma_{q}-2}+2^{*}\|V_{\varepsilon}\|_{2^{*}}^{2^{*}}t_{\mu,\varepsilon}^{2^{*}-2}.$

Thus, $\\{t_{\mu,\varepsilon}\\}$ is uniformly bounded and bounded from below
away from $0$ for all $\varepsilon,\mu>0$ sufficiently small. By (2.5) and
(2.6),

$\displaystyle
S^{\frac{N}{2}}(1-t_{\mu,\varepsilon_{\mu}}^{2^{*}-2})=\mu\gamma_{q}\|V_{\varepsilon}\|_{q}^{q}t_{\mu,\varepsilon}^{q\gamma_{q}-2}+O((R_{\varepsilon}\varepsilon^{-1})^{2-N})).$

Then we can use similar arguments as used for (4.2) to show that

$\displaystyle
t_{\mu,\varepsilon}=1-(1+o(1))\frac{\mu\gamma_{q}\|V_{\varepsilon}\|_{q}^{q}+O((R_{\varepsilon}\varepsilon^{-1})^{2-N}))}{(2^{*}-2)S^{\frac{N}{2}}}$

and thus by similar arguments as used for (4.6),

$\displaystyle\|u_{a,\mu,-}\|_{q}^{q}\geq(1+o(1))(\|V_{\varepsilon}\|_{q}^{q}-C\mu^{-1}(R_{\varepsilon}\varepsilon^{-1})^{2-N}))),$

which together with (2.5) and (4.32), implies

$\displaystyle\|u_{a,\mu,-}\|_{q}^{q}\gtrsim\left\\{\begin{aligned}
&\varepsilon^{4-q}-C\mu^{-1}e^{-2\varepsilon^{-2}},\quad N=4,2<q<2^{*},\\\
&\varepsilon^{3-\frac{q}{2}}-C\mu^{-1}\varepsilon^{2},\quad N=3,3<q<2^{*},\\\
&\varepsilon^{\frac{3}{2}}\ln(\frac{1}{\varepsilon})-C\mu^{-1}\varepsilon^{2},\quad
N=3,q=3,\\\ &\varepsilon^{\frac{3q}{2}-3}-C\mu^{-1}\varepsilon^{2},\quad
N=3,2<q<3.\end{aligned}\right.$

By choosing $\varepsilon_{\mu}$ such that the right hand sides of the above
estimate take the maximum, we have (4.31) and

$\displaystyle\|u_{a,\mu,-}\|_{q}^{q}\gtrsim\left\\{\begin{aligned}
&\varepsilon_{\mu}^{N-\frac{N-2}{2}q},\quad N=3,4,\frac{N}{N-2}<q<2^{*},\\\
&\varepsilon_{\mu}^{\frac{3}{2}}\ln(\frac{1}{\varepsilon_{\mu}}),\quad
N=3,q=3,\\\ &\varepsilon_{\mu}^{\frac{3q}{2}-3},\quad
N=3,2<q<3.\end{aligned}\right.$ (4.33)

We define
$w_{\mu,-}=\varepsilon_{\mu}^{\frac{N-2}{2}}u_{a,\mu,-}(\varepsilon_{\mu}x)$,
then $\|w_{\mu,-}\|_{2^{*}}^{2^{*}},\|\nabla w_{\mu,-}\|_{2}^{2}\sim 1$ and by
(4.33),

$\displaystyle\|w_{\mu,-}\|_{q}^{q}\gtrsim\left\\{\begin{aligned} &1,\quad
N=3,4,\frac{N}{N-2}<q<2^{*},\\\ &\ln(\frac{1}{\varepsilon_{\mu}}),\quad
N=3,q=3,\\\ &\varepsilon_{\mu}^{2q-6},\quad N=3,2<q<3.\end{aligned}\right.$
(4.34)

It is easy to see that

$\displaystyle\sigma_{\mu}^{N-\frac{N-2}{2}q}\|v_{a,\mu,-}\|_{q}^{q}=\|u_{a,\mu,-}\|_{q}^{q}=\varepsilon_{\mu}^{N-\frac{N-2}{2}q}\|w_{\mu,-}\|_{q}^{q}.$
(4.35)

Then by Lemma 4.1, (4.18) and (4.34), we have

$\displaystyle\sigma_{\mu}^{N-\frac{N-2}{2}q}\gtrsim\varepsilon_{\mu}^{N-\frac{N-2}{2}q}$
(4.36)

for $\frac{N}{N-2}<q<2^{*}$ and $N=3,4$. On the other hand, we know that

$\displaystyle
w_{\mu,-}(x)=\bigg{(}\frac{\varepsilon_{\mu}}{\sigma_{\mu}}\bigg{)}^{\frac{N-2}{2}}v_{a,\mu,-}\bigg{(}\frac{\varepsilon_{\mu}}{\sigma_{\mu}}x\bigg{)}$
(4.37)

and $\widetilde{w}_{\mu,-}$ satisfies

$\displaystyle-\Delta\widetilde{w}_{\mu,-}=g(\widetilde{w}_{\mu,-})\quad\text{in
}\mathbb{R}^{N}.$

where

$\displaystyle\widetilde{w}_{\mu,-}=\frac{1}{w_{\mu,-}(0)}w_{\mu,-}([w_{\mu,-}(0)]^{s}x)$

with $s\in\mathbb{R}$ and

$\displaystyle g(\widetilde{w}_{\mu,-})$ $\displaystyle=$
$\displaystyle\lambda_{a,\mu,-}\varepsilon_{\mu}^{2}[w_{\mu,-}(0)]^{2s}\widetilde{w}_{\mu,-}+\mu\varepsilon_{\mu}^{N-\frac{N-2}{2}q}[w_{\mu,-}(0)]^{2s+q-2}\widetilde{w}_{\mu,-}^{q-1}$
$\displaystyle+[w_{\mu,-}(0)]^{2s+2^{*}-2}\widetilde{w}_{\mu,-}^{2^{*}-1}.$

By similar arguments as used for (4.23), we have

$\displaystyle
w_{\mu,-}\lesssim\frac{w_{\mu,-}(0)}{(1+b_{\mu}r^{2})^{\frac{N-2}{2}}}\quad\text{for
all }r>0,$ (4.38)

where

$\displaystyle b_{\mu}$ $\displaystyle=$
$\displaystyle[w_{\mu,-}(0)]^{2s-1}(\lambda_{a,\mu,-}\varepsilon_{\mu}^{2}w_{\mu,-}(0)+\mu\varepsilon_{\mu}^{N-\frac{N-2}{2}q}[w_{\mu,-}(0)]^{q-1}$
$\displaystyle+[w_{\mu,-}(0)]^{2^{*}-1}).$

We recall that $\mu,\sigma_{\mu},\lambda_{a,\mu,-}\to 0$ as $\mu\to 0^{+}$,
and by (4.7), we have $\lambda_{a,\mu,-}\lesssim\mu$. Thus, by Lemma 4.1,
(4.21) and (4.37),

$\displaystyle b_{\mu}$ $\displaystyle\sim$
$\displaystyle\bigg{(}\frac{\varepsilon_{\mu}}{\sigma_{\mu}}\bigg{)}^{(N-2)s+2}.$
(4.39)

Now, take $s=-1$ and by (4.38), we can use similar arguments in the proof of
Lemma 4.1 to show that

$\displaystyle\|w_{\mu,-}\|_{q}^{q}\lesssim\bigg{(}\frac{\varepsilon_{\mu}}{\sigma_{\mu}}\bigg{)}^{\frac{N-2}{2}q-\frac{N}{2}(4-N)}$

for $\frac{N}{N-2}<q<2^{*}$ and $N=3,4$, which together with (4.35), implies
that $\sigma_{\mu}\lesssim\varepsilon_{\mu}$ for $\frac{N}{N-2}<q<2^{*}$ and
$N=3,4$. It follows from (4.36) that $\sigma_{\mu}\sim\varepsilon_{\mu}$ for
$\frac{N}{N-2}<q<2^{*}$ and $N=3,4$. For the case $N=3$ and $q=3$, by
$\|w_{\mu,-}\|_{2}^{2}\sim\varepsilon_{\mu}^{-2}$, Struss’s radial lemma (cf.
[12, Lemma A.IV, Theorem A.I’] or [44, Lemma 3.1]) and similar arguments as
used for (4.25),

$\displaystyle
w_{\mu,-}\lesssim\varepsilon_{\mu}^{-2}r^{-1}e^{-\frac{1}{2}\sqrt{|\lambda_{a,\mu,-}|}\varepsilon_{\mu}r}\quad\text{for
}r\gtrsim\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\varepsilon_{\mu}}.$ (4.40)

It follows from (4.38) and (4.39) that

$\displaystyle\|w_{\mu,-}\|_{q}^{q}\lesssim\bigg{(}\frac{\varepsilon_{\mu}}{\sigma_{\mu}}\bigg{)}^{\frac{3}{2}-\frac{3}{2}(s+2)}\ln(\frac{1}{\sqrt{|\lambda_{a,\mu,-}|}\sigma_{\mu}}).$

By Lemma 4.1, taking $s=-1$ and (4.35), we have
$\varepsilon_{\mu}\gtrsim\sigma_{\mu}$. By Lemma 4.1, taking $s=2$ and (4.35),
we have $\varepsilon_{\mu}\lesssim\sigma_{\mu}$. Thus, for $N=3$ and $q=3$, we
also have $\varepsilon_{\mu}\sim\sigma_{\mu}$. For the case $N=3$ and $2<q<3$,
by (4.7), (4.33), Lemma 4.1 and $\mu\sim\varepsilon_{\mu}^{5-\frac{3q}{2}}$,

$\displaystyle\sigma_{\mu}^{\frac{q}{5-q}}\gtrsim\varepsilon_{\mu}^{\frac{q}{5-q}}\bigg{(}\frac{\mu}{\varepsilon_{\mu}^{5-\frac{3q}{2}}}\bigg{)}^{\frac{3-q}{5-q}}\sim\varepsilon_{\mu}^{\frac{q}{5-q}}$

which implies $\sigma_{\mu}\gtrsim\varepsilon_{\mu}$. Thus, by (4.21), we can
adapt the maximum principle as that in the proof of Proposition 4.2 to show
that

$\displaystyle w_{a,\mu,-}\gtrsim
r^{-1}e^{-\sqrt{|\lambda_{a,\mu,-}|}\varepsilon_{\mu}r}\quad\text{for }r\geq
1.$ (4.41)

By (4.41), we can see that the estimates for (4.29) works for
$\varepsilon_{\mu}$ and thus, we have

$\displaystyle|\lambda_{a,\mu,-}|\gtrsim\mu\varepsilon_{\mu}^{3-\frac{q}{2}}\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\varepsilon_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}},\quad\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\varepsilon_{\mu}^{2}}\bigg{)}^{\frac{1}{2}}|\lambda_{a,\mu,-}|\lesssim\mu$

and

$\displaystyle\frac{\mu\varepsilon_{\mu}^{3-\frac{q}{2}}}{|\lambda_{a,\mu,-}|\varepsilon_{\mu}^{2}}\lesssim|\lambda_{a,\mu,-}|^{\frac{3-q}{2}}\varepsilon_{\mu}^{1-q}\lesssim\varepsilon_{\mu}^{2(2-q)}.$

Now, we can follow similar arguments as used in the proof of Lemma 4.1 to show
that

$\displaystyle\|w_{\mu,-}\|_{q}^{q}$ $\displaystyle\lesssim$
$\displaystyle\bigg{(}\frac{1}{|\lambda_{a,\mu,-}|\varepsilon_{\mu}^{2}}\bigg{)}^{\frac{3-q}{2}},$

which, together with Lemma 4.1 and (4.35), implies that
$\sigma_{\mu}\lesssim\varepsilon_{\mu}$. Thus, we also have
$\sigma_{\mu}\sim\varepsilon_{\mu}$ as $\mu\to 0^{+}$ in the case of $N=3$ and
$2<q<3$. ∎

We are ready to give the proofs of Theorem 1.1 and 1.2.

Proof of Theorem 1.1: It follows immediately from Lemma 3.1, Propositions 3.1
and 3.2. $\Box$

Proof of Theorem 1.2: It follows immediately from Propositions 2.1, 4.1 and
4.2. $\Box$

We close this section by

Proof of Theorem 1.3: $(1)$ Since the proof is similar to that of Proposition
2.1, we only sketch it. In the case of $q=2+\frac{4}{N}$, we have
$\|\varphi\|_{2}^{2}=\|\phi_{0}\|_{2}^{2}$ for all minimizers of the
Gagliardo–Nirenberg inequality (1.9), where $\phi$ is the unique solution of
(1.10). Thus, we choose
$\varphi=\frac{a}{\|\phi_{0}\|_{2}}\phi\in\mathcal{S}_{a}$ as a test function
of $m_{a,\mu}^{-}$. By using similar arguments as used in the proof of
Proposition 2.1 and direct calculations,

$\displaystyle
m_{a,\mu}^{-}\leq\frac{1}{N}(1-\frac{\mu}{\alpha_{N,q,a}})^{\frac{2^{*}}{2^{*}-2}}\bigg{(}\frac{\|\nabla\phi_{0}\|_{2}}{\|\phi_{0}\|_{2^{*}}}\bigg{)}^{N}.$

It follows from $u_{a,\mu,-}\in\mathcal{P}_{a,\mu}$, the Gagliardo–Nirenberg
and Sobolev inequalities that

$\displaystyle S^{\frac{N}{2}}\leq\frac{\|\nabla
u_{a,\mu,-}\|_{2}^{2}}{(1-\frac{\mu}{\alpha_{N,q,a}})^{\frac{2}{2^{*}-2}}}\leq\bigg{(}\frac{\|\nabla\phi_{0}\|_{2}}{\|\phi_{0}\|_{2^{*}}}\bigg{)}^{N},$
(4.42)

which, together with $u_{a,\mu,-}\in\mathcal{P}_{a,\mu}$ once more and the
Pohozaev identity satisfied by $u_{a,\mu,-}$, implies that

$\displaystyle-\lambda_{\mu,-}=\frac{1-\gamma_{q}}{a^{2}}\mu\|u_{a,\mu,-}\|_{q}^{q}\geq(1+o_{\mu}(1))\frac{1-\gamma_{q}}{a^{2}\gamma_{q}}S^{\frac{N}{2}}(1-\frac{\mu}{\alpha_{N,q,a}})^{\frac{2}{2^{*}-2}}$

and

$\displaystyle-\lambda_{\mu,-}=\frac{1-\gamma_{q}}{a^{2}}\mu\|u_{a,\mu,-}\|_{q}^{q}\leq(1-\frac{\mu}{\alpha_{N,q,a}})^{\frac{2}{2^{*}-2}}\bigg{(}\frac{\|\nabla\phi_{0}\|_{2}}{\|\phi_{0}\|_{2^{*}}}\bigg{)}^{N}.$

Thus, $\\{v_{a,\mu,-}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, where

$\displaystyle
v_{a,\mu,-}=(\frac{a}{\|\phi_{0}\|_{2}})^{\frac{N-2}{2}}s_{\mu}^{\frac{N}{2}}u_{a,\mu,-}(\frac{a}{\|\phi_{0}\|_{2}}s_{\mu}x)$

and $s_{\mu}=(1-\frac{\mu}{\alpha_{N,q,a}})^{-\frac{N-2}{4}}$. Clearly,
$v_{a,\mu,-}$ satisfies

$\displaystyle-\Delta
v_{a,\mu,-}=\lambda_{\mu,-}\frac{a^{2}}{\|\phi_{0}\|_{2}^{2}}s_{\mu}^{2}v_{a,\mu,-}+\mu(\frac{a}{\|\phi_{0}\|_{2}})^{\frac{4}{N}}v_{a,\mu,-}^{q-1}+s_{\mu}^{2-\frac{N}{2}(2^{*}-2)}v_{a,\mu,-}^{2^{*}-1}$

By (3.13) and [55, (I.3)], we know that
$\alpha_{N,q,a}(\frac{a}{\|\phi_{0}\|_{2}})^{\frac{4}{N}}=1$ for
$q=2+\frac{4}{N}$. On the other hand, since $v_{a,\mu,-}$ is radial, it is
standard to show that $v_{a,\mu,-}\to\psi_{\nu_{a}^{\prime},1}$ strongly in
$H^{1}(\mathbb{R}^{N})$ as $\mu\to\alpha_{N,q,a}^{-}$ up to a subsequence for
some $\nu_{a}^{\prime}>0$.

$(2)$ In the cases of $2+\frac{4}{N}<q<2^{*}$,
$\frac{2}{q-2}-\frac{N}{2}\not=0$. Thus, we can choose $\nu_{a}>0$, as that in
(2.10), such that $\|\psi_{\nu_{a},1}\|_{2}^{2}=a^{2}$. Again, we use
$\psi_{\nu_{a},1}\in\mathcal{S}_{a}$ as a test function of $m_{a,\mu}^{-}$. By
using similar arguments as used in the proof of Proposition 2.1 and direct
calculations, $m_{a,\mu}^{-}\lesssim\mu^{-\frac{2}{q\gamma_{q}-2}}$ as
$\mu\to+\infty$. It follows that $u_{a,\mu,-}\to 0$ strongly in
$D^{1,2}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})$ as $\mu\to+\infty$. This,
together with $u_{a,\mu,-}\in\mathcal{P}_{a,\mu}$ and the Gagliardo–Nirenberg
and Sobolev inequalities, implies

$\displaystyle\|\nabla
u_{a,\mu,-}\|_{2}^{2}\geq(1+o_{\mu}(1))(\mu\gamma_{q}a^{q-q\gamma_{q}}C_{N,q}^{q})^{-\frac{2}{q\gamma_{q}-2}}.$

On the other hand, for the test function $\psi_{\nu_{a},1}$, it satisfies

$\displaystyle\|\nabla\psi_{\nu_{a},1}\|_{2}^{2}=\mu\gamma_{q}\|\psi_{\nu_{a},1}\|_{q}^{q}t_{\mu}^{q\gamma_{q}-2}+\|\psi_{\nu_{a},1}\|_{2^{*}}^{2^{*}}t_{\mu}^{2^{*}-2},$

where $(\psi_{\nu_{a},1})_{t_{\mu}}\in\mathcal{P}_{a,\mu}$. It follows that

$\displaystyle t_{\mu}\|\nabla\psi_{\nu_{a},1}\|_{2}\leq\bigg{(}\frac{1}{\mu
a^{q-q\gamma_{q}}\gamma_{q}C_{N,q}^{q}}\bigg{)}^{\frac{1}{q\gamma_{q}-2}}.$

Thus,

$\displaystyle\mathcal{E}_{\mu}((\psi_{\nu_{a},1})_{t_{\mu}})$
$\displaystyle=$
$\displaystyle(1+o_{\mu}(1))(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla(\psi_{\nu_{a},1})_{t_{\mu}}\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle(1+o_{\mu}(1))(\frac{1}{2}-\frac{1}{q\gamma_{q}})\bigg{(}\frac{1}{\mu
a^{q-q\gamma_{q}}\gamma_{q}C_{N,q}^{q}}\bigg{)}^{\frac{2}{q\gamma_{q}-2}}.$

Note that $\mathcal{E}_{\mu}((\psi_{\nu_{a},1})_{t_{\mu}})\geq m_{a,\mu}^{-}$
and

$\displaystyle
m_{a,\mu}^{-}=\mathcal{E}_{\mu}(u_{a,\mu,-})=(1+o_{\mu}(1))(\frac{1}{2}-\frac{1}{q\gamma_{q}})\|\nabla
u_{a,\mu,-}\|_{2}^{2}$

as $\mu\to+\infty$, we must have

$\displaystyle\|\nabla
u_{a,\mu,-}\|_{2}^{2}=(1+o_{\mu}(1))(\mu\gamma_{q}a^{q-q\gamma_{q}}C_{N,q}^{q})^{-\frac{2}{q\gamma_{q}-2}}.$
(4.43)

As in $(1)$, $\\{v_{a,\mu,-}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, where
$v_{a,\mu,-}=s_{\mu}^{\frac{N}{2}}u_{a,\mu,-}(s_{\mu}x)$ and
$s_{\mu}=\mu^{\frac{1}{q\gamma_{q}-2}}$. Again, $v_{a,\mu,-}$ satisfies

$\displaystyle-\Delta
v_{a,\mu,-}=\lambda_{\mu,-}s_{\mu}^{2}v_{a,\mu,-}+v_{a,\mu,-}^{q-1}+s_{\mu}^{2-\frac{N}{2}(2^{*}-2)}v_{a,\mu,-}^{2^{*}-1}.$

Using (4.43), the Pohozaev identity satisfied by $u_{a,\mu,-}$ and
$u_{a,\mu,-}\in\mathcal{P}_{a,\mu}$ once more, we have

$\displaystyle-\lambda_{\mu,-}=(1+o_{\mu}(1))\frac{1-\gamma_{q}}{a^{2}}(\mu\gamma_{q}a^{q-q\gamma_{q}}C_{N,q}^{q})^{-\frac{2}{q\gamma_{q}-2}}.$

Now, by similar arguments as used in $(1)$,
$v_{a,\mu,-}\to\psi_{\nu_{a}^{\prime},1}$ strongly in $H^{1}(\mathbb{R}^{N})$
as $\mu\to+\infty$ up to a subsequence for some $\nu_{a}^{\prime}>0$. Since in
the cases of $2+\frac{4}{N}<q<2^{*}$, $\frac{2}{q-2}-\frac{N}{2}\not=0$. By
$\|v_{a,\mu,-}\|_{2}^{2}=a^{2}$, we must have $\nu_{a}^{\prime}=\nu_{a}$. By
the uniqueness of $\psi_{\nu_{a},1}$ in $\mathcal{S}_{a}$, we know that
$v_{a,\mu,-}\to\psi_{\nu_{a},1}$ strongly in $H^{1}(\mathbb{R}^{N})$ as
$\mu\to+\infty$. Using the uniqueness of $\psi_{\nu_{a},1}$ in
$\mathcal{S}_{a}$ and the nondegenerate of $\psi_{\nu_{a},1}$, we can prove
the local uniqueness of $u_{a,\mu,-}$ for $\mu>0$ sufficiently large by
adapting similar arguments as used for $u_{a,\mu,+}$ in the proof of
Proposition 2.1. $\Box$

##  5\. Acknowledgements

The research of J. Wei is partially supported by NSERC of Canada and the
research of Y. Wu is supported by NSFC (No. 11701554, No. 11771319, No.
11971339).

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