Existence and Multiplicity of Solutions for Fractional $p$-Laplacian Equation
Involving Critical Concave-convex Nonlinearities

Weimin Zhang

School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering
Applications (Ministry of Education) & Shanghai Key Laboratory of PMMP, East
China Normal University, Shanghai 200241, China

000 AMS Subject Classification: 35A15, 35J60, 58E05.
E-mail addresses<EMAIL_ADDRESS>(W. Zhang).

###### Abstract

We investigate the following fractional $p$-Laplacian equation

$\begin{cases}\begin{aligned}
(-\Delta)_{p}^{s}u&=\lambda|u|^{q-2}u+|u|^{p_{s}^{*}-2}u&&\text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega,\end{aligned}\end{cases}$

where $s\in(0,1)$, $p>q>1$, $n>sp$, $\lambda>0$, $p_{s}^{*}=\frac{np}{n-sp}$
and $\Omega$ is a bounded domain (with $C^{1,1}$ boundary). Firstly, we get a
dichotomy result for the existence of positive solution with respect to
$\lambda$. For $p\geq 2$, $p-1<q<p$, $n>\frac{sp(q+1)}{q+1-p}$, we provide two
positive solutions for small $\lambda$. Finally, without sign constraint, for
$\lambda$ sufficiently small, we show the existence of infinitely many
solutions.

Keywords: Critical Sobolev exponent, Fractional $p$-Laplacian, Convex-concave,
Multiplicity of solutions.

##  1 Introduction

In this paper, we are interested in the following fractional $p$-Laplacian
equation

$\begin{cases}\begin{aligned}
(-\Delta)_{p}^{s}u&=\lambda|u|^{q-2}u+|u|^{p_{s}^{*}-2}u&&\text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega,\end{aligned}\end{cases}$ $None$

where $s\in(0,1)$, $p>q>1$, $n>sp$, $\lambda>0$, $p_{s}^{*}=\frac{np}{n-sp}$
is called the critical Sobolev exponent, and $\Omega\subset{\mathbb{R}}^{n}$
is a bounded domain (with $C^{1,1}$ boundary). $(-\Delta)_{p}^{s}$ denotes the
fractional $p$-Laplacian operator, and when $u$ is sufficiently smooth, it can
be represented pointwisely by

$(-\Delta)_{p}^{s}u(x)=2\leavevmode\nobreak\ \underset{\varepsilon\rightarrow
0^{+}}{\lim}\int_{\mathbb{R}^{n}\setminus
B_{\varepsilon}(x)}{\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}}dy,$

up to a normalization constant depending on $n$ and $s$, which is consistent
with the usual linear fractional Laplacian $(-\Delta)^{s}$ when $p=2$, see
[13]. Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory mapping,
consider the general fractional $p$-Laplacian equation

$\begin{cases}\begin{aligned} (-\Delta)_{p}^{s}u&=f(x,u)\leavevmode\nobreak\
\leavevmode\nobreak\ \,\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega.\end{aligned}\end{cases}$ (1.1)

To give a weak formulation of (1.1), we denote the Gagliardo seminorm by

$[u]_{s,p}:=\left(\int_{\mathbb{R}^{2n}}{\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}}dxdy\right)^{1/p}.$

Let

$W^{s,p}(\mathbb{R}^{n}):=\\{u\in L^{p}(\mathbb{R}^{n}):[u]_{s,p}<\infty\\}$

be endowed with the norm

$\|u\|_{W^{s,p}}:=(|u|_{p}^{p}+[u]_{s,p}^{p})^{1/p},$

where $|\cdot|_{p}$ denotes the usual norm of $L^{p}(\mathbb{R}^{n})$. Denote
the subspace

$W_{0}^{s,p}(\Omega):=\left\\{u\in
W^{s,p}(\mathbb{R}^{n}):u=0\;\text{a.e.\leavevmode\nobreak\
in}\leavevmode\nobreak\ \mathbb{R}^{n}\setminus\Omega\right\\},$

equivalently renormed with $\|u\|=[u]_{s,p}$ (see [13, Theorem 7.1]), it is
well known that $W_{0}^{s,p}(\Omega)$ is a uniformly convex Banach space.
Furthermore, the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{r}(\Omega)$
is continuous for $r\in[1,p_{s}^{*}]$ and compact for $r\in[1,p_{s}^{*})$, see
[13, Theorems 6.5, 7.1]. $(-\Delta)_{p}^{s}$ can be variationally regarded as
an operator from $W_{0}^{s,p}(\Omega)$ into its dual space
$W_{0}^{s,p}(\Omega)^{*}$ as follows,

$\langle(-\Delta)_{p}^{s}u,v\rangle=\int_{\mathbb{R}^{2n}}{\frac{J_{u}(x,y)(v(x)-v(y))}{|x-y|^{n+sp}}}dxdy,\quad\forall\;v\in
W_{0}^{s,p}(\Omega),$

where $J_{u}(x,y)=|u(x)-u(y)|^{p-2}(u(x)-u(y))$.

We call $u\in W_{0}^{s,p}(\Omega)$ a weak solution (respectively weak
subsolution, weak supersolution) of (1.1) if $f(x,u)\in
W_{0}^{s,p}(\Omega)^{*}$ and

$\langle(-\Delta)_{p}^{s}u,v\rangle=(\text{respectively}\leavevmode\nobreak\
\leq,\,\geq)\int_{\Omega}f(x,u)vdx,\quad\forall\;v\in
W_{0}^{s,p}(\Omega),v\geq 0.$

For simplicity, from now on, we omit the term weak in the rest of our paper.
If $f$ satisfies the growth condition:

$|f(x,t)|\leq C_{0}(1+|t|^{r-1})\quad\mbox{for }\;C_{0}>0,1<r\leq p_{s}^{*}$
(1.2)

and a.e. $x\in\Omega$, $t\in\mathbb{R}$. Thus solutions of problem (1.1)
coincide with critical points of the $C^{1}$ functional

$E(u)=\frac{1}{p}\|u\|^{p}-\int_{\Omega}\int_{0}^{u}f(x,t)dtdx,\quad u\in
W_{0}^{1,p}(\Omega).$ (1.3)

In order to find critical points of $E$, the eigenvalues of
$(-\Delta)_{p}^{s}$ based on $\mathbb{Z}_{2}$-cohomological index introduced
in [25] are usually used to carry out some linking constructions, see for
instance [31, 22, 30]. When $f$ satisfies the subcritical growth condition
(i.e. (1.2) with $1<r<p_{s}^{*}$), many results about existence and
multiplicity of solutions of (1.1) have been established, see for example [20,
19, 22].

In the special case $p=2$, Ros-Oton and Serra [32] gave a Pohozaev identity
for solutions of (1.1), and when $p\neq 2$, a similar identity for $u\in
W_{0}^{s,p}(\Omega)$ solution to (1.1) was conjectured in [22, Section 7]. It
results that if $f(x,t)=|t|^{p_{s}^{*}-2}t$, there may not exist non trivial
solutions of (1.1) when $\Omega$ is star-shaped. This motivates people to
consider the perturbation problem

$\begin{cases}\begin{aligned} (-\Delta)_{p}^{s}u&=\lambda
g(x,u)+|u|^{p_{s}^{*}-2}u&&\text{in}\leavevmode\nobreak\ \Omega,\\\
u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega,\end{aligned}\end{cases}$ (1.4)

where $g:\Omega\times\mathbb{R}\to\mathbb{R}$ satisfies the subcritical growth
(1.2) with $1<r<p_{s}^{*}$. This type of problems bring new difficulties due
to the fact that the associated energy of (1.4) cannot satisfy Palais-Smale
condition globally because the embedding $W_{0}^{s,p}(\Omega)\subset
L^{p_{s}^{*}}(\Omega)$ is not compact. However, the Palais-Smale condition can
hold true in suitable thresholds related to the best Sobolev constant

$S_{s,p}:=\underset{u\in
D^{s,p}(\mathbb{R}^{n})\backslash\\{0\\}}{\text{inf}}\frac{[u]_{s,p}^{p}}{|u|_{p_{s}^{*}}^{p}},$

where $D^{s,p}(\mathbb{R}^{n}):=\big{\\{}u\in
L^{p^{*}_{s}}(\mathbb{R}^{n}):[u]_{s,p}<\infty\big{\\}}.$

  * •

Mahwin and Molica Bisci [28] proved that there exists an interval
$\mathcal{V}\subset(0,\infty)$ such that for every $\lambda\in\mathcal{V}$,
(1.4) admits at least one solution, which is a local minimizer of the
corresponding energy. The main strategy in [28] is to check the sequentially
weakly lower semicontinuity of the functional

$H(u)=\frac{1}{p}\|u\|^{p}-\frac{1}{p^{*}}|u|_{p_{s}^{*}}^{p_{s}^{*}}$

restricted in a sufficiently small ball of $W_{0}^{s,p}(\Omega)$. As a special
case, they showed in [28, Theorem 1.2] that

$\begin{cases}\begin{aligned}
(-\Delta)_{p}^{s}u&=\lambda(|u|^{q-2}u+|u|^{r-2}u)+|u|^{p_{s}^{*}-2}u&&\text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega,\end{aligned}\end{cases}$ (1.5)

has a positive solution for $\lambda\in\mathcal{V}$, provided $2\leq
q<p<r<p_{s}^{*}$.

  * •

In [30], Mosconi et al. proved that when $g(x,u)=|u|^{p-2}u$, (1.4) has a non
trivial solution in the following cases:

    * $\mathrm{(i)}$

$n=sp^{2}$ and $\lambda\in(0,\lambda_{1})$;

    * $\mathrm{(ii)}$

$n>sp^{2}$ and $\lambda\not\in\\{\lambda_{k}\\}$;

    * $\mathrm{(iii)}$

$\frac{n^{2}}{n+s}>sp^{2}$;

    * $\mathrm{(iv)}$

$\frac{n^{3}+s^{3}p^{3}}{n(n+3)}>sp^{2}$ and $\partial\Omega\in C^{1,1}$,

where $\lambda_{k}$ is the $k$-th eigenvalue of $(-\Delta)_{p}^{s}$ given in
[25].

  * •

Bhakta and Mukherjee [6] obtained that when $p\geq 2$, there exist
$\lambda_{0}>0$, $n_{0}\in\mathbb{N}$ and $q_{0}\in(1,p)$ such that for all
$\lambda\in(0,\lambda_{0})$, $n>n_{0}$ and $q\in(q_{0},p)$, $(P_{\lambda})$
has at least one sign changing solution.

For the convex-concave nonlinearities, there are also some literature
concerning problem $(P_{\lambda})$ with the classical $p$-Laplacian operator,
i.e. $s=1$,

$\begin{cases}\begin{aligned}
-\Delta_{p}u&=\lambda|u|^{q-2}u+|u|^{p^{*}-2}u\quad\text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{on}\leavevmode\nobreak\
\partial\Omega,\end{aligned}\end{cases}$ (1.6)

where $p^{*}=\frac{np}{n-p}$ and $\Delta_{p}u=\text{div}(|\nabla
u|^{p-2}\nabla u)$. For example:

  * •

Ambrosetti, Brezis and Cerami [1] did seminal works for $1<q<p=2$, and they
proved that there exists $\Lambda>0$ such that

    * (i)

problem (1.6) has at least two positive solutions if $0<\lambda<\Lambda$;

    * (ii)

problem (1.6) has at least one positive solution if $\lambda=\Lambda$;

    * (iii)

problem (1.6) has no positive solution if $\lambda>\Lambda$.

  * •

García Azorero, Manfredi and Peral Alonso [17] (see also [16]) generalized the
above results for general $p>1$, provided either $\frac{2n}{n+2}<p<3$, $1<q<p$
or $p\geq 3$, $p>q>\frac{p^{*}-2}{p-1}$.

For the linear fractional Laplacian case, that is $p=2$, $s\in(0,1)$, $n>2s$,
$1<q<2$,  the above (i)-(iii) for $(P_{\lambda})$ were obtained by Barrios,
Colorado, Servadei and Soria [4].

Motivated by the works mentioned above, we are concerned here with the
existence and multiplicity problem for $(P_{\lambda})$. To state our results,
we introduce some definitions. For all $x\in\Omega$, let

$\text{d}_{\Omega}(x):=\text{dist}(x,\partial\Omega),$

and consider the weighted space

$\mathcal{C}_{s}^{0}(\overline{\Omega}):=\big{\\{}u\in
C^{0}(\overline{\Omega}):\frac{u}{\text{d}_{\Omega}^{s}}\leavevmode\nobreak\
\text{admits\leavevmode\nobreak\ a\leavevmode\nobreak\
continuous\leavevmode\nobreak\ extension\leavevmode\nobreak\
to}\leavevmode\nobreak\ \overline{\Omega}\big{\\}}$

endowed with the norm
$\|u\|_{\mathcal{C}_{s}^{0}(\overline{\Omega})}=\|\frac{u}{\text{d}_{\Omega}^{s}}\|_{\infty}$.
Our first result is a dichotomy claim which extends the results in [4] with
$(-\Delta)^{s}$ and [17] with $-\Delta_{p}$.

######  Theorem 1.1.

Let $s\in(0,1)$, $p\in(1,\infty)$, $q\in(1,p)$, $n>sp$, let $\Omega$ be a
bounded domain with $C^{1,1}$ boundary. Then there exists $0<\Lambda<\infty$
such that

  * (i)

$(P_{\lambda})$ has no positive solutions for $\lambda>\Lambda$;

  * (ii)

$(P_{\lambda})$ has a minimal positive solution $u_{\lambda}$ for any
$0<\lambda<\Lambda$; moreover, this family of minimal solutions is increasing
with respect to $\lambda$, and
$\|u_{\lambda}\|_{\mathcal{C}_{s}^{0}(\overline{\Omega})}\to 0$ as $\lambda\to
0^{+}$;

  * (iii)

$(P_{\lambda})$ has at least one positive solution $u_{\Lambda}$ for
$\lambda=\Lambda$, given by the pointwise limit of $u_{\lambda}$ as
$\lambda\to\Lambda^{-}$.

The authors in [1] used supersolution and subsolution method to find a minimal
solution $v_{\lambda}$ for $\lambda\in(0,\Lambda)$, which is a stable solution
(i.e., the second variation of the energy functional at $v_{\lambda}$ is
nonnegative). The stability yields that
$\\{v_{\lambda}\\}_{0<\lambda<\Lambda}$ is bounded in $H_{0}^{1}(\Omega)$, so
the weak limit of $v_{\lambda}$ is a weak solution for (1.6) with
$\lambda=\Lambda$.

In contrast to [1], we cannot apply Picone’s identity due to the presence of
the nonlocal term, and there is no related stability theory for fractional
$p$-Laplacian so far. Due to the nonlinearity of $(-\Delta)_{p}^{s}$, we
cannot derive the strong comparison principle directly, that is

$(-\Delta)_{p}^{s}u_{1}\leq(-\Delta)_{p}^{s}u_{2}\;\;\text{and}\;\;u_{1}\neq
u_{2}\Rightarrow u_{1}<u_{2}.$

In [26], the author derived a version of strong comparison principle, but with
rather restrictive assumptions. Our key observation is that all positive
solutions to (1.1) can be controlled by $\text{d}^{s}_{\Omega}(x)$, see
Proposition 2.8. Our strategy is to use the comparison principle to get the
uniqueness of solution to

$\begin{cases}\begin{aligned} (-\Delta)_{p}^{s}u&=\lambda
u^{q-1}\leavevmode\nobreak\ \quad&&\text{in}\leavevmode\nobreak\ \Omega,\\\
u&>0&&\text{in}\leavevmode\nobreak\ \Omega,\\\
u&=0&&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega.\end{aligned}\end{cases}$ $None$

We will use the iteration method to show that the unique solution to
$(Q_{\lambda})$ is less than any positive solution to $(P_{\lambda})$, we
prove then the existence of a minimal positive solution $u_{\lambda}$ of
$(P_{\lambda})$ for all $\lambda\in(0,\Lambda)$, which is increasing with
respect to $\lambda$. For any $\lambda\in(0,\Lambda)$, let
$0<\lambda^{\prime\prime}<\lambda<\lambda^{\prime}<\Lambda$ and

$\Sigma=\big{\\{}u\in
W_{0}^{1,p}(\Omega)\cap\mathcal{C}_{s}^{0}(\overline{\Omega}):u_{\lambda^{\prime\prime}}<u<u_{\lambda^{\prime}}\big{\\}}.$
(1.7)

Moreover, we use the variational method to find a local minimum solution
${\widehat{u}}_{\lambda}\in\Sigma$ with respect to the topology of
$\mathcal{C}^{0}_{s}(\overline{\Omega})$. In Lemma 3.5, we show the
boundedness of $\\{{\widehat{u}}_{\lambda}\\}_{0<\lambda<\Lambda}$ in
$W_{0}^{s,p}(\Omega)$, which yields the boundedness of the minimal solution
$\\{{u}_{\lambda}\\}_{0<\lambda<\Lambda}$ in $W_{0}^{s,p}(\Omega)$.

######  Remark 1.2.

Theorem 1.1 remains true for (1.4) if $g(x,u)=|u|^{q-2}u+|u|^{r-2}u$ with
$1<q<p$, $r\in(1,p_{s}^{*})$. Hence it works in the frame of [28, Theorem
1.2]. In [17], it was shown that (1.6) has an extremal solution $u_{\Lambda}$
in the distributional sense, but the regularity of $u_{\Lambda}$ was not
mentioned. According to the proof of Theorem 1.1, we can assert that
$u_{\Lambda}$ in [17, Lemma 6.3] belongs to $W_{0}^{1,p}(\Omega)$.

Now we will consider the existence of a second positive solution to
$(P_{\lambda})$.

######  Theorem 1.3.

Let $s\in(0,1)$, $p\geq 2$, $p-1<q<p$, $n>\frac{sp(q+1)}{q+1-p}$, and $\Omega$
be a bounded domain with $C^{1,1}$ boundary. There exists $\lambda^{*}>0$ such
that for all $\lambda\in(0,\lambda^{*})$, problem $(P_{\lambda})$ has at least
two positive solutions.

For equation (1.6) with $p=2$, one of main points in [1] for finding two
positive solutions is a result of Brezis and Nirenberg [11] which connects
variational and nonvariational methods. Roughly speaking, a local minimizer in
$C^{1}$-topology is also a local minimizer in $W_{0}^{1,2}(\Omega)$. For
$p>1$, the equivalence between $C^{1}(\overline{\Omega})$ and
$W_{0}^{1,p}(\Omega)$ local minimizers of the energy functional was proven
respectively in [17] and [21].

However, in fractional operator cases, the space $C^{1}(\overline{\Omega})$
seems to be not suitable for this aim, but
$\mathcal{C}^{0}_{s}(\overline{\Omega})$ can serve as a suitable substitute.
Iannizzotto, Mosconi and Squassina [24] proved that for $p\geq 2$ and a given
$f$ satisfying (1.2), a local minimizer of the energy $E$ (see (1.3)) in
$\mathcal{C}^{0}_{s}(\overline{\Omega})\cap W_{0}^{s,p}(\Omega)$ with respect
to $\mathcal{C}^{0}_{s}(\overline{\Omega})$-topology is also a local minimizer
in $W_{0}^{s,p}(\Omega)$, see also Barrios et al. [4, Proposition 2.5] for the
case $p=2$.

In virtue of the maximum principle [7, Theorem A.1], positive solutions to
$(P_{\lambda})$ coincide with nontrivial critical points of the following
functional defined on $W_{0}^{s,p}(\Omega)$

${\widetilde{I}}_{\lambda}(u)=\frac{1}{p}\|u\|^{p}-\frac{\lambda}{q}\int_{\Omega}{(u^{+})}^{q}dx-\frac{1}{p_{s}^{*}}\int_{\Omega}{(u^{+})}^{p_{s}^{*}}dx,$
(1.8)

where $u^{+}=\max\\{u,0\\}$. As mentioned above, $(P_{\lambda})$ has a minimal
solution $u_{\lambda}$, and ${\widetilde{I}}_{\lambda}$ has a minimizer
${\widehat{u}}_{\lambda}$ in $\Sigma$ (see (1.7)) with respect to
$\mathcal{C}_{s}^{0}(\overline{\Omega})$-topology, which is also a local
minimizer in $W_{0}^{s,p}(\Omega)$ if $p\geq 2$ by [24]. We can assume
$u_{\lambda}={\widehat{u}}_{\lambda}$, otherwise, the theorem naturally holds
true. Under the assumption that ${\widetilde{I}}_{\lambda}$ has only two
critical points 0 and ${u}_{\lambda}$, we will prove in Proposition 4.4 that
${\widetilde{I}}_{\lambda}$ satisfies the Palais-Smale condition for all level

$c<c_{s,p}:={\widetilde{I}}_{\lambda}({u}_{\lambda})+\frac{s}{n}S_{s,p}^{\frac{n}{sp}}.$
(1.9)

It remains to construct a mountain pass geometry of
${\widetilde{I}}_{\lambda}$ around ${u}_{\lambda}$, and check that the
mountain pass level strictly less than $c_{s,p}$.

It has been conjectured in [8] that all minimizers for $S_{s,p}$ are of the
form $cU(|x-x_{0}|/{\varepsilon})$, where

$U(x)=\frac{1}{\big{(}1+|x|^{\frac{p}{p-1}}\big{)}^{(n-sp)/p}},\quad
x\in\mathbb{R}^{n}.$

As far as we are aware, this conjecture remains open. However, the asymptotic
estimates for all minimizers were established by [8]. To estimate the mountain
pass level, we will make use of some truncation functions
$u_{\varepsilon,\delta}$ constructed by Mosconi et al. [30] (see (2.9)) and
some useful integral estimates of $u_{\varepsilon,\delta}$, see subsection 2.1
below.

In contrast to [30], our mountain pass geometry is around ${u}_{\lambda}$,
instead of 0, for which the estimates will be more complex, the nonlocal
integral also brings new difficulties to our computations. To overcome these
obstacles, we will proceed simultaneously the construction of mountain pass
geometry and the estimate for mountain pass level.

To be more precise, we consider $\eta_{\delta}{u}_{\lambda}$ where
$\eta_{\delta}$ is a cut-off function (see (2.6)). Lemma 4.1 leads to
$\eta_{\delta}{u}_{\lambda}\to{u}_{\lambda}$ in $W_{0}^{s,p}(\Omega)$ when
$\delta\to 0$. We choose ${u}_{\lambda}$ to be the starting point of mountain
pass path, with the terminal point

$e=\eta_{\delta}{u}_{\lambda}+t_{0}u_{\varepsilon,\delta},$ (1.10)

where $t_{0}$ depending on $\varepsilon$ and $\delta$ is a positive number
such that
${\widetilde{I}}_{\lambda}(e)<{\widetilde{I}}_{\lambda}({u}_{\lambda})$.
Consider the set of mountain pass paths

$\Gamma_{\varepsilon,\delta}:=\\{\gamma\in
C\left([0,1],W_{0}^{s,p}(\Omega)\right):\gamma(0)=u_{\lambda},\,\gamma(1)=e\\},$
(1.11)

and the mountain pass level

$m_{\varepsilon,\delta}:=\underset{\gamma\in\Gamma_{\varepsilon,\delta}}{\inf}\,\underset{t\in[0,1]}{\max}{\widetilde{I}}_{\lambda}(\gamma(t)).$
(1.12)

We select a special mountain pass path

$\gamma_{\varepsilon,\delta}(t)=\begin{cases}\begin{aligned}
&\eta_{{2t\delta}}u_{\lambda}\quad&&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ 0\leq t\leq\frac{1}{2},\\\
&\eta_{{\delta}}u_{\lambda}+(2t-1)t_{0}u_{\varepsilon,\delta}&&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{1}{2}<t\leq 1.\\\ \end{aligned}\end{cases}$ (1.13)

and check that ${\widetilde{I}}_{\lambda}(\gamma_{{\varepsilon,\delta}}(t))$
tends to ${\widetilde{I}}_{\lambda}(u_{\lambda})<c_{s,p}$ as $\delta\to 0$ for
all $0\leq t\leq\frac{1}{2}$. To reach $m_{\varepsilon,\delta}<c_{s,p}$, it
suffices to prove

$\underset{t\geq
0}{\sup}\,{\widetilde{I}}_{\lambda}(\eta_{{\delta}}u_{\lambda}+tu_{\varepsilon,\delta})<c_{s,p}$
(1.14)

for some $\varepsilon,\delta>0$. In Lemma 4.3, for $p\geq 2$, $p-1<q<p$ and
$n>\frac{sp(q+1)}{q+1-p}$, by taking $\varepsilon=\delta^{k+1}$ with suitable
choice of $k\in(0,p-1)$, we can claim that (1.14) holds whenever $\delta$ is
sufficiently small.

A key point is that $u_{\varepsilon,\delta}$ and $\eta_{\delta}{u}_{\lambda}$
will be chosen to have disjoint support domains, which permits us to handle
many integral estimates. The mountain pass technique provides then a second
critical point. Finally, we prove that the mountain pass level is positive for
$\lambda$ positive but small, which guarantees the non-triviality of this
critical point.

######  Remark 1.4.

In Theorem 1.3, it is clear that $\lambda^{*}\leq\Lambda$, that given in
Theorem 1.1. However, we don’t know how to rule out the triviality of the
critical point when the mountain pass level is just zero, hence we cannot
claim actually $\lambda^{*}=\Lambda$.

Finally, without sign constraint, we obtain the existence of infinitely many
solutions for $(P_{\lambda})$. Denote

$I_{\lambda}(u)=\frac{1}{p}\|u\|^{p}-\frac{\lambda}{q}\int_{\Omega}{|u|}^{q}dx-\frac{1}{p_{s}^{*}}\int_{\Omega}{|u|}^{p_{s}^{*}}dx,\quad
u\in W_{0}^{s,p}(\Omega).$ (1.15)

Notice that actually all the non-zero critical points of
${\widetilde{I}}_{\lambda}$ are the positive critical points of $I_{\lambda}$.

######  Theorem 1.5.

Let $s\in(0,1)$, $p>1$, $q\in(1,p)$, $n>sp$, and $\Omega$ be a bounded domain.
There exists $\lambda^{**}>0$ such that for all $\lambda\in(0,\lambda^{**})$,
$(P_{\lambda})$ has a sequence of solutions $\\{u_{j}\\}$ satisfying
$I_{\lambda}(u_{j})\to 0^{-}$.

García Azorero and Peral Alonso in [15, Theorem 4.5] proved that for $1<q<p$
and $n>p$, (1.6) has infinitely many solutions provided $\lambda$ is small
(see also [1] for $p=2$). They used $\mathbb{Z}_{2}$-genus and Lusternik-
Schnirelman theory (see for instance [2, Theorem 10.9]). Using a dual Fountain
theorem due to Bartsch and Willem [5], when $p=2$, it was showed in [33,
Theorem 3.22] that for $\lambda>0$ small enough, there exists a sequence of
solutions to (1.6), whose energies are negative and tends to 0.

Different from the proof of [33, Theorem 3.22], we use the
$\mathbb{Z}_{2}$-genus to construct a sequence of minimax levels $b_{j}$ for
the functional $I_{\lambda}$ in a small ball $B_{r}(0)$, see (5.1) below. We
will show that $b_{j}$ are negative critical values of $I_{\lambda}$, and
lower bounded by $\widetilde{b}_{j}$ defined in (5.2). By means of weak
convergence argument (see Lemma 5.2), we will prove that $\widetilde{b}_{j}\to
0^{-}$ for small $r$, so does $b_{j}$.

######  Remark 1.6.

In Section 5, we provide a space decomposition method for the reflexible and
separable Banach space. We believe that this can extend the Fountain theorem
[33, Section 3] and the dual Fountain theorem [5] to the general Banach
framework.

The paper is organized as follows. In Section 2, we introduce some notations
and preliminary results. The proof of Theorems 1.1, 1.3 and 1.5 are completed
respectively in Sections 3-5.

##  2 Notations and Preliminaries

In this paper, $C,C^{\prime},C_{1},C_{2},...$ denote always generic positive
constants. $|\cdot|_{p}$ means the usual norm of $L^{p}(\mathbb{R}^{n})$ or
$L^{p}(\Omega)$. $\|\cdot\|$ denotes the norm of $W_{0}^{s,p}(\Omega)$. We use
$\varphi_{1}$ to denote the eigenfunction corresponding to the first
eigenvalue $\lambda_{1}$ of $(-\Delta)_{p}^{s}$ in $W_{0}^{s,p}(\Omega)$ such
that $\varphi_{1}>0$ in $\Omega$ and $|\varphi_{1}|_{\infty}=1$.

###  2.1 Truncations for the minimizers of $S_{s,p}$

From [8], we know that if $s\in(0,1)$, $p>1$, there exists a minimizer $U\in
D^{s,p}(\mathbb{R}^{n})$ for $S_{s,p}$. Up to scaling and translation, we can
assume that $U(0)=1$, $U$ is radially symmetric, nonnegative, radially
decreasing and resolves in ${\mathbb{R}}^{n}$

$(-\Delta)_{p}^{s}U=U^{p_{s}^{*}-1}.$ (2.1)

Without confusion, we denote also $U(x)=U(r)$ with $r=|x|$. For any
$\varepsilon>0$, let

$U_{\varepsilon}(x)=\frac{1}{\varepsilon^{(n-sp)/p}}U\left(\frac{x}{\varepsilon}\right)$
(2.2)

which is also a minimizer for $S_{s,p}$, and satisfies (2.1). We recall some
results shown in Lemmas 2.2, 2.6 and 2.7 of [30] respectively.

######  Lemma 2.1.

There exist constants $c_{1},c_{2}>0$ and $\theta>1$ such that for all $r\geq
1$,

$\frac{c_{1}}{r^{(n-sp)/(p-1)}}\leq U(r)\leq\frac{c_{2}}{r^{(n-sp)/(p-1)}}$
(2.3)

and

$2U(\theta r)\leq U(r).$ (2.4)

Let $\theta$ be as in Lemma 2.1, fix $\eta\in
C^{\infty}(\mathbb{R}^{n},[0,1])$ such that

$\eta(x)=\left\\{\begin{array}[]{lll}0,&\text{if}\,|x|\leq 2\theta,\\\
1,&\text{if}\,|x|\geq 3\theta,\\\ \end{array}\right.$ (2.5)

and for any $\delta>0$, denote

$\eta_{\delta}(x)=\eta\left(\frac{x}{\delta}\right).$ (2.6)

######  Lemma 2.2.

Assume that $0\in\Omega$. Then there exists a constant $C=C(n,\Omega,p,s)>0$
such that for any $v\in W_{0}^{s,p}(\Omega)$, $\delta>0$ satisfying
$(-\Delta)^{s}_{p}v\in L^{\infty}(\Omega)$ and
$B_{5\theta\delta}(0)\subset\Omega,$ there holds

$\lVert v\eta_{\delta}\rVert^{p}\leq\lVert
v\rVert^{p}+C\left|(-\Delta)_{p}^{s}v\right|_{\infty}^{p/(p-1)}\delta^{n-sp}.$

For any $\varepsilon,\delta>0$, we denote

$M_{\varepsilon,\delta}=\frac{U_{\varepsilon}(\delta)}{U_{\varepsilon}(\delta)-U_{\varepsilon}(\theta\delta)}.$
(2.7)

Let

$\displaystyle g_{\varepsilon,\delta}(t)=\begin{cases}0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \,\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ 0\leq t\leq U_{\varepsilon}(\theta\delta),\\\
M_{\varepsilon,\delta}^{p}(t-U_{\varepsilon}(\theta\delta)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
U_{\varepsilon}(\theta\delta)\leq t\leq U_{\varepsilon}(\delta),\\\
t+U_{\varepsilon}(\delta)(M_{\varepsilon,\delta}^{p-1}-1),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if}\leavevmode\nobreak\ t\geq
U_{\varepsilon}(\delta),\end{cases}$

and

$\displaystyle
G_{\varepsilon,\delta}(t)=\int_{0}^{t}g^{\prime}_{\varepsilon,\delta}(\tau)^{1/p}d\tau=\begin{cases}0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\ 0\leq t\leq
U_{\varepsilon}(\theta\delta),\\\
M_{\varepsilon,\delta}\,(t-U_{\varepsilon}(\theta\delta)),\leavevmode\nobreak\
\,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
U_{\varepsilon}(\theta\delta)\leq t\leq U_{\varepsilon}(\delta),\\\
t,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ t\geq U_{\varepsilon}(\delta).\end{cases}$ (2.8)

The functions $g_{\varepsilon,\delta}$ and $G_{\varepsilon,\delta}$ are
decreasing and absolutely continuous. Then the radially nonincreasing function

$u_{\varepsilon,\delta}(r)=G_{\varepsilon,\delta}(U_{\varepsilon}(r))\leq
U_{\varepsilon}(r),$ (2.9)

satisfies

$u_{\varepsilon,\delta}(r)=U_{\varepsilon}(r)\text{ if}\leavevmode\nobreak\
r\leq\delta\quad\mbox{and}\quad{\rm supp}(u_{\varepsilon,\delta})\subset
B_{\theta\delta}(0).$ (2.10)

######  Lemma 2.3.

There exists a constant $C=C(n,p,s)>0$ such that for any
$\varepsilon\leq\frac{\delta}{2}$,

$\displaystyle\lVert u_{\varepsilon,\delta}\rVert^{p}\leq
S_{s,p}^{\frac{n}{sp}}+C\left(\frac{\varepsilon}{\delta}\right)^{(n-sp)/(p-1)},$
(2.11) $\displaystyle|u_{\varepsilon,\delta}|_{p^{*}_{s}}^{p_{s}^{*}}\geq
S_{s,p}^{\frac{n}{sp}}-C{\left(\frac{\varepsilon}{\delta}\right)}^{\frac{n}{p-1}},$
(2.12)
$\displaystyle|u_{\varepsilon,\delta}|_{p}^{p}\geq\begin{cases}\frac{1}{C}\varepsilon^{sp}|\log{\left(\frac{\varepsilon}{\delta}\right)}|,\leavevmode\nobreak\
\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\ \leavevmode\nobreak\
n=sp^{2},\\\ \frac{1}{C}\varepsilon^{sp},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ n>sp^{2}.\\\ \end{cases}$ (2.13)

###  2.2 Further useful results

Consider fractional equation:

$(-\Delta)_{p}^{s}u\geq 0\quad\text{in}\leavevmode\nobreak\ \Omega;\quad
u=0\quad\text{in}\leavevmode\nobreak\ \mathbb{R}^{n}\backslash\Omega.$ (2.14)

Referring to [7, Theorem A.1], we recall the following useful maximum
principle.

######  Lemma 2.4.

Let $s\in(0,1)$, $1<p<\infty$ and $\Omega\subset\mathbb{R}^{n}$ be an open
bounded connected set. If $u\in W_{0}^{s,p}(\Omega)$ is a nonnegative solution
to (2.14) and $u\not\equiv 0$ in $\Omega$, then $u>0$ almost everywhere in
$\Omega$.

The following comparison principle for fractional $p$-Laplacian can be found
in [23, Proposition 2.10] (see also [27, Lemma 9]).

######  Lemma 2.5.

Let $p>1$ and $\Omega$ be an open bounded set. If $u,v\in W_{0}^{s,p}(\Omega)$
satisfy that for all $\varphi\in W_{0}^{s,p}(\Omega)$ with $\varphi\geq 0$ in
$\Omega$,

$\int_{\mathbb{R}^{2n}}{\frac{J_{u}(x,y)(\varphi(x)-\varphi(y))}{|x-y|^{n+sp}}}dxdy\geq\int_{\mathbb{R}^{2n}}{\frac{J_{v}(x,y)(\varphi(x)-\varphi(y))}{|x-y|^{n+sp}}}dxdy,$

then $u\geq v$ in $\Omega$.

Iannizzotto, Mosconi and Squassina [24] proved the following equivalence.

######  Lemma 2.6.

Let $p\geq 2$, $s\in(0,1)$ and $n>sp$. Let $\Omega\subset\mathbb{R}^{n}$ be a
bounded domain with a $C^{1,1}$-boundary,
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory mapping satisfying
(1.2). Then, for any $u_{0}\in W_{0}^{s,p}(\Omega)$, the following are
equivalent:

  * (i)

there exists $\rho>0$ such that $E(u_{0}+v)\geq E(u_{0})$ for all $v\in
W_{0}^{s,p}(\Omega)$, $\|v\|\leq\rho$;

  * (ii)

there exists $\sigma>0$ such that $E(u_{0}+v)\geq E(u_{0})$ for all $v\in
W_{0}^{s,p}(\Omega)\cap\mathcal{C}^{0}_{s}(\overline{\Omega})$,
$\|v\|_{\mathcal{C}^{0}_{s}(\overline{\Omega})}\leq\sigma$,

where $E$ is defined in (1.3).

Next is a Hölder regularity result for solutions of (1.1), which comes from
[12, Theorem 3.3, Remark 3.4] and [23, Theorem 1.1].

######  Proposition 2.7.

Let $\Omega$ be a bounded domain with $C^{1,1}$ boundary, $f$ satisfy (1.2).
Then there exists $\alpha\in(0,s]$ such that if $u\in W_{0}^{s,p}(\Omega)$ is
a solution of (1.1), we have $u\in C^{\alpha}(\overline{\Omega})$.

If we assume $f(x,u)\in L^{\infty}(\Omega)$ and $f(x,u)\geq 0$, the solutions
of (1.1) will satisfy also a kind of Hopf’s lemma [14, Theorem 1.5]. In
particular, the solutions can be controlled by ${\text{d}^{s}_{\Omega}(x)}$.
The decay information of solutions near the boundary is very useful to
construct suitable subsolutions or supersolutions.

######  Proposition 2.8.

Let $\Omega$ be a bounded domain with $C^{1,1}$ boundary and let $u\in
W_{0}^{s,p}(\Omega)$ be a weak solution of the following equation

$\begin{cases}\begin{aligned} (-\Delta)^{s}_{p}u&=f\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \,\leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega,\end{aligned}\end{cases}$ (2.15)

where $f\in L^{\infty}(\Omega)$, $f\geq 0$ and $f\not\equiv 0$. Then there are
two positive numbers $C$, $C^{\prime}$ such that

$C{\text{d}^{s}_{\Omega}(x)}\leq{u(x)}\leq
C^{\prime}{\text{d}^{s}_{\Omega}(x)},\quad\mbox{for a.e.}\;x\in\Omega.$

###### Proof.

By Lemmas 2.4 and 2.5, we have $u>0$ a.e. in $\Omega$. Since $f\in
L^{\infty}(\Omega)$, the Hölder continuity follows from [23, Theorem 1.1].
From the Hopf’s lemma [14, Theorem 1.5], there exists $C$ such that
$C{\text{d}^{s}_{\Omega}(x)}\leq u(x).$ Morover, it follows from [23, Theorem
4.4] that there exists $C^{\prime}$ such that
$C^{\prime}{\text{d}^{s}_{\Omega}(x)}\geq u(x)$. ∎

At last, we summarize the super-subsolution method as follows.

######  Proposition 2.9.

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory mapping
satisfying that $f(x,t)$ is continuous and increasing in $t$ for a.e.
$x\in\Omega$. Let $\underline{u}\in W_{0}^{s,p}(\Omega)$ be a subsolution to
(1.1), and $\overline{u}\in W_{0}^{s,p}(\Omega)$ be a supersolution to (1.1)
such that $\underline{u}\leq\overline{u}$ and

$f(\cdot,\underline{u}(\cdot)),\leavevmode\nobreak\
f(\cdot,\overline{u}(\cdot))\in L^{\frac{p_{s}^{*}}{p_{s}^{*}-1}}(\Omega).$

Then there exists a weak solution $u\in W_{0}^{s,p}(\Omega)$ of (1.1) such
that $\underline{u}\leq u\leq\overline{u}$.

###### Proof.

Let $u_{0}:=\underline{u}$, and by induction we define $u_{j}$ $(j\geq 1)$ as
solutions of

$\begin{cases}\begin{aligned}
(-\Delta)_{p}^{s}u_{j+1}&=f(x,u_{j})&&\text{in}\leavevmode\nobreak\ \Omega,\\\
u_{j+1}&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega.\\\ \end{aligned}\end{cases}$ (2.16)

By comparison principle, then $\underline{u}\leq u_{j}\leq
u_{j+1}\leq\overline{u}$. Consequently,

$|u_{j}|\leq\max\\{|\underline{u}|,|\overline{u}|\\},\quad|f(x,u_{j})|\leq\max\\{|f(x,\underline{u})|,|f(x,\overline{u})|\\}.$

Multiplying (2.16) by $u_{j+1}$,

$\displaystyle\lVert u_{j+1}\rVert^{p}=$
$\displaystyle\int_{\Omega}f(x,u_{j})u_{j+1}dx$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\max\\{|\underline{u}|,|\overline{u}|\\}\max\\{|f(x,\underline{u})|,|f(x,\overline{u})|\\}dx.$

Therefore $\\{u_{j}\\}$ is a bounded sequence in $W_{0}^{s,p}(\Omega)$, and
$u_{j}$ weakly converges to some $u$ in $W_{0}^{s,p}(\Omega)$,
$u_{j}(x)\rightarrow u(x)$ a.e. in $\mathbb{R}^{n}$. Consequently, $u$ is a
weak solution to (1.1) and $\underline{u}\leq u\leq\overline{u}$. ∎

##  3 Proof of Theorem 1.1

Let

$\Lambda:=\sup\\{\lambda>0:(P_{\lambda})\leavevmode\nobreak\ \text{has a
positive solution in }\;W_{0}^{s,p}(\Omega)\\}.$ (3.1)

We first verify that $\Lambda$ is finite and nonzero. For simplicity, we will
not repeat always the regularity assumption for $\Omega$ and we also often
omit to repeat $n>sp$.

######  Lemma 3.1.

For $s\in(0,1)$, $p>q>1$ and $n>sp$, we have $0<\Lambda<\infty$.

###### Proof.

Let $e$ be the solution of the following equation

$\left\\{\begin{split}(-\Delta)_{p}^{s}e&=1\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \Omega,\\\
e&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega.\end{split}\right.$ (3.2)

Since $1<q<p<p_{s}^{*}$, we can find $\lambda_{0},M>0$ small satisfying

$\lambda_{0}M^{q-p}|e|^{q-1}_{\infty}+M^{p_{s}^{*}-p}|e|^{p_{s}^{*}-1}_{\infty}\leq
1,$

which implies that for $0<\lambda\leq\lambda_{0}$,

$\begin{split}(-\Delta)_{p}^{s}(Me)&=M^{p-1}\geq\lambda
M^{q-1}|e|^{q-1}_{\infty}+M^{p_{s}^{*}-1}|e|^{p_{s}^{*}-1}_{\infty}\geq\lambda(Me)^{q-1}+(Me)^{p_{s}^{*}-1}.\end{split}$

So $Me$ is a supersolution of $(P_{\lambda})$. Fix any
$0<\lambda\leq\lambda_{0}$, there exists $\varepsilon_{0}\in(0,1)$ (recall
that $\varphi_{1}\in C(\overline{\Omega})$, $|\varphi_{1}|_{\infty}=1$ and
$\varphi_{1}>0$ in $\Omega$), such that

$\lambda_{1}\leq\lambda(\varepsilon\varphi_{1})^{q-p}\leq\lambda(\varepsilon\varphi_{1})^{q-p}+(\varepsilon\varphi_{1})^{p_{s}^{*}-p},\quad\forall\;0<\varepsilon<\varepsilon_{0}.$

We check readily that $\varepsilon\varphi_{1}$ is a subsolution of
$(P_{\lambda})$. According to Proposition 2.8, we can choose $\varepsilon>0$
small enough such that $\varepsilon\varphi_{1}\leq Me$. By Proposition 2.9,
$(P_{\lambda})$ has a positive solution, which yields $\Lambda>0$.

From [9, Theorem 4.1], we know that $\lambda_{1}$ is an isolated eigenvalue,
so there is $\widetilde{\lambda}>\lambda_{1}$, which is not an eigenvalue of
$(-\Delta)_{p}^{s}$. Assume by contradiction that $\Lambda=\infty$, we can
choose then

$\widehat{\lambda}>\max_{t\geq
0}\big{(}\widetilde{\lambda}t^{p-q}-t^{p_{s}^{*}-q}\big{)}$

such that $(P_{\widehat{\lambda}})$ has a positive solution
$u_{\widehat{\lambda}}$. Hence $u_{\widehat{\lambda}}$ is a supersolution of
the following equation

$\left\\{\begin{split}(-\Delta)_{p}^{s}u&=\widetilde{\lambda}|u|^{p-2}u&&\text{in}\leavevmode\nobreak\
\Omega,\\\ u&=0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &&\text{in}\leavevmode\nobreak\
\mathbb{R}^{n}\setminus\Omega.\end{split}\right.$ (3.3)

Applying again Propositions 2.7–2.8, there is $\varepsilon>0$ small such that
$\varepsilon\varphi_{1}\leq u_{\widehat{\lambda}}$. Since
$\lambda_{1}<\widetilde{\lambda}$, $\varepsilon\varphi_{1}$ is a subsolution
to (3.3). By Proposition 2.9, there exists a positive solution of (3.3), hence
a contradiction occurs with the choice of $\widetilde{\lambda}$. It means that
$0<\Lambda<\infty$. ∎

In the sequel, we are going to find the minimal positive solution of
$(P_{\lambda})$. We claim that the minimal solution can be derived by
iterating from the solution of $(Q_{\lambda})$. The first step is to prove
that the solution of $(Q_{\lambda})$ is unique.

######  Lemma 3.2.

Let $s\in(0,1)$, $p>1$, $n\geq 1$, $q\in(1,p)$ and $\lambda>0$. Then there
exists a unique solution $v_{\lambda}$ to $(Q_{\lambda})$. Moreover, for any
$\lambda>0$, $v_{\lambda}=\lambda^{\frac{1}{p-q}}v_{1}$.

###### Proof.

Consider the functional

$E_{\lambda}(v)=\frac{\lambda}{p}\rVert
v\lVert^{p}-\frac{1}{q}|v^{+}|_{q}^{q},\quad v\in W_{0}^{s,p}(\Omega),$

which is coercive and weakly lower semi-continuous. Therefore, there is a
global minimizer $v_{\lambda}\in W_{0}^{s,p}(\Omega)$ for $E_{\lambda}$. By
$p>q>1$, we obtain $E_{\lambda}(v_{\lambda})<0$ hence $v_{\lambda}\neq 0$. By
maximum principle, we have $v_{\lambda}>0$ in $\Omega$. We claim that
$v_{\lambda}$ is the unique positive solution to $(Q_{\lambda})$. Let $w$ be
another positive solution, using Proposition 2.7, $v_{\lambda}$, $w\in
L^{\infty}(\Omega)$, then by Proposition 2.8, there are two positive constants
$C$, $C^{\prime}$ such that

$Cw(x)\leq v_{\lambda}(x)\leq C^{\prime}w(x),\quad\mbox{for a.e.
$x\in\Omega$}.$ (3.4)

Let $\beta_{0}:=\sup\\{\ell:v_{\lambda}\geq\ell w\leavevmode\nobreak\
\text{a.e. in}\leavevmode\nobreak\ \Omega\\}$, by (3.4), we have
$0<\beta_{0}<\infty$. Clearly

$(-\Delta)_{p}^{s}v_{\lambda}=\lambda
v_{\lambda}^{q-1}\geq\lambda\beta_{0}^{q-1}w^{q-1}=(-\Delta)_{p}^{s}\big{(}\beta_{0}^{\frac{q-1}{p-1}}w\big{)}.$

By comparison principle, we get $v_{\lambda}\geq\beta_{0}^{\frac{q-1}{p-1}}w$,
so that $\beta_{0}\geq\beta_{0}^{\frac{q-1}{p-1}}$, hence $\beta_{0}\geq 1$,
which means $v_{\lambda}\geq w$ in $\Omega$. Similarly, $w\geq v_{\lambda}$,
so we have $w=v_{\lambda}$.

The uniqueness yields readily the expression of $v_{\lambda}$ by $v_{1}$,
since $\lambda^{\frac{1}{p-q}}v_{1}$ resolves $(Q_{\lambda})$. ∎

Due to the nonlinearity of $(-\Delta)_{p}^{s}$, we cannot derive the strong
comparison principle directly. Here we will show the strict comparison between
minimal solutions, thanks to the following inequality.

######  Lemma 3.3.

Let $f_{\lambda}(t)=\lambda t^{q-1}+t^{p_{s}^{*}-1}$ for $\lambda>0$, $t>0$,
$q<p_{s}^{*}$. Then for any $0<\lambda<\lambda^{\prime}<\infty$ and $M>0$,
there exists $\beta_{0}>1$ such that $f_{\lambda}(\beta_{0}t)\leq
f_{\lambda^{\prime}}(t)$ for $0<t\leq M$.

###### Proof.

Let $1<\beta\leq(\frac{\lambda^{\prime}+\lambda}{2\lambda})^{\frac{1}{q-1}}$,
then

$\begin{split}f_{\lambda^{\prime}}(t)-f_{\lambda}(\beta
t)&=\Big{(}\frac{\lambda^{\prime}+\lambda}{2}-\lambda\beta^{q-1}\Big{)}t^{q-1}+\frac{\lambda^{\prime}-\lambda}{2}t^{q-1}+(1-\beta^{p_{s}^{*}-1})t^{p_{s}^{*}-1}\\\
&\geq\frac{\lambda^{\prime}-\lambda}{2}t^{q-1}+(1-\beta^{p_{s}^{*}-1})t^{p_{s}^{*}-1}.\end{split}$

As $\lambda^{\prime}>\lambda$ and $p_{s}^{*}>q$, there exist $t_{0}>0$,
$\beta_{1}>1$ such that $f_{\lambda}(\beta_{1}t)\leq f_{\lambda^{\prime}}(t)$
for all $0<t<t_{0}$. On the other hand, there exists some $\beta_{2}>1$ but
close enough such that $f_{\lambda}(\beta_{2}t)\leq f_{\lambda^{\prime}}(t)$
for all $t\in[t_{0},M]$. $\beta_{0}=\min(\beta_{1},\beta_{2})$ will satisfy
the claim. ∎

Now, we are ready to prove the existence of minimal positive solution to
$(P_{\lambda})$.

######  Proposition 3.4.

For any $0<\lambda<\Lambda$, the problem $(P_{\lambda})$ has a minimal
positive solution $u_{\lambda}$ such that $u_{\lambda}<u_{\lambda^{\prime}}$
if $0<\lambda<\lambda^{\prime}<\Lambda$. Moreover,
$\|u_{\lambda}\|_{\mathcal{C}_{s}^{0}(\overline{\Omega})}\to 0$ as $\lambda\to
0^{+}$.

###### Proof.

Let $w_{\lambda}$ be an arbitrary positive solution of $(P_{\lambda})$ with
$\lambda\in(0,\Lambda)$. Let $v_{\lambda}$ be the unique solution of
$(Q_{\lambda})$, we claim that $v_{\lambda}\leq w_{\lambda}$. Indeed,
$w_{\lambda}$ is a supersolution to $(Q_{\lambda})$. Using Propositions
2.7–2.8, we see that for $\varepsilon>0$ small enough, there holds
$\varepsilon\varphi_{1}\leq w_{\lambda}$ and $\varepsilon\varphi_{1}$ is a
subsolution to $(Q_{\lambda})$. As before, we get a solution of
$(Q_{\lambda})$ between $w_{\lambda}$ and $\varepsilon\varphi_{1}$, there
holds $v_{\lambda}\leq w_{{\lambda}}$ seeing Lemma 3.2.

Take any $\lambda\in(0,\Lambda)$. By the definition of $\Lambda$, there exists
$\lambda^{\prime}\in(\lambda,\Lambda)$ such that $(P_{\lambda^{\prime}})$ has
a positive solution, denoted by $w_{\lambda^{\prime}}$. Applying Lemma 3.2 and
the above argument, there holds $v_{\lambda}\leq v_{\lambda^{\prime}}\leq
w_{\lambda^{\prime}}$. As $v_{\lambda}$ and $w_{\lambda^{\prime}}$ are
respectively sub and supersolution to $(P_{\lambda})$, we can use the
classical monotone iteration method or Proposition 2.9, starting with
$v_{\lambda}$, to obtain a positive solution $u_{\lambda}$ for
$(P_{\lambda})$. As $w_{\lambda^{\prime}}$ can be replaced by arbitrary
solution of $(P_{\lambda})$ or $(P_{\lambda^{\prime}})$, we conclude that
$u_{\lambda}$ is the minimal positive solution to $(P_{\lambda})$ and
$u_{\lambda}\leq u_{\lambda^{\prime}}$.

More precisely, for $0<\lambda<\lambda^{\prime}<\Lambda$, using Lemma 3.3,
there exists $\beta_{0}>1$ such that

$\begin{split}(-\Delta)_{p}^{s}u_{\lambda^{\prime}}=f_{\lambda^{\prime}}(u_{\lambda^{\prime}})\geq
f_{\lambda^{\prime}}(u_{\lambda})\geq
f_{\lambda}(\beta_{0}u_{\lambda})=\lambda\beta_{0}^{q-1}u_{\lambda}^{q-1}+\beta_{0}^{p_{s}^{*}-1}u_{\lambda}^{p_{s}^{*}-1}\geq(-\Delta)_{p}^{s}(\beta_{0}^{\frac{q-1}{p-1}}u_{\lambda}).\end{split}$

Applying again the comparison principle, we see that
$u_{\lambda^{\prime}}\geq\beta_{0}^{\frac{q-1}{p-1}}u_{\lambda}>u_{\lambda}$
in $\Omega$.

Finally, coming back to the proof of Lemma 3.1, for any $M>0$ small enough,
there exists $\lambda>0$ small such that $Me$ is a supersolution of
$(P_{\lambda})$. As above, since $u_{\lambda}$ is the minimal solution, we get
$u_{\lambda}\leq Me$. So
$\|u_{\lambda}\|_{\mathcal{C}_{s}^{0}(\overline{\Omega})}\to 0$ as $\lambda\to
0^{+}$. ∎

For any $\lambda\in(0,\Lambda)$, we can find
$0<\lambda^{\prime\prime}<\lambda<\lambda^{\prime}<\Lambda$. By Proposition
3.4, there are three minimal solutions corresponding to
$\lambda^{\prime\prime}$, $\lambda$, $\lambda^{\prime}$, denoted by
$u_{1},u_{\lambda},u_{2}$ respectively. Let us define for every $x\in\Omega$,

$\widehat{f}_{\lambda}(x,t)=\left\\{\begin{array}[]{lll}f_{\lambda}(u_{2}(x)),&\mbox{if
}\;t\geq u_{2}(x);\\\ f_{\lambda}(t),&\mbox{if }\;u_{2}(x)>t\geq 0.\\\
\end{array}\right.$

Consider the following equation

$\begin{cases}\begin{aligned}
(-\Delta)_{p}^{s}u&=\widehat{f}_{\lambda}(x,u)&&\mbox{in }\;\Omega,\\\ u&\geq
0&&\mbox{in }\;\Omega,\\\ u&=0&&\mbox{in }\;\mathbb{R}^{n}\setminus\Omega,\\\
\end{aligned}\end{cases}$ (3.5)

with the associated energy functional

$\widehat{I}_{\lambda}(u)=\frac{1}{p}\lVert
u\rVert^{p}-\int_{\Omega}{\widehat{F}}_{\lambda}(x,u)dx,\quad\mbox{where
}\;{\widehat{F}}_{\lambda}(x,u)=\int_{0}^{u^{+}}\widehat{f}_{\lambda}(x,t)dt.$

One can prove that $\widehat{I}_{\lambda}$ is weakly lower continuous and
coercive in $W_{0}^{s,p}(\Omega)$, so $\widehat{I}_{\lambda}$ can achieve its
global minimum at some ${\widehat{u}}_{\lambda}\in W_{0}^{s,p}(\Omega)$ such
that $0<{\widehat{u}}_{\lambda}\leq u_{2}$. We can find a subsolution
$\varepsilon\varphi_{1}$ of $(P_{\lambda^{\prime\prime}})$ such that
$\varepsilon\varphi_{1}\leq{\widehat{u}}_{\lambda}$ and use super-subsolution
method to conclude ${\widehat{u}}_{\lambda}\geq u_{1}$.

In virtue of Proposition 3.4, $u_{1}<u_{2}$ in $\Omega$. Let $\Sigma$ be as in
(1.7), i.e. $\Sigma=\\{u\in
W_{0}^{1,p}(\Omega)\cap\mathcal{C}_{s}^{0}(\overline{\Omega}):u_{1}<u<u_{2}\\}.$
We shall prove that ${\widehat{u}}_{\lambda}$ is a interior point of $\Sigma$
with respect to $\mathcal{C}_{s}^{0}(\overline{\Omega})$-topology.

######  Lemma 3.5.

${\widehat{u}}_{\lambda}$ is a solution to $(P_{\lambda})$, and
${\widehat{u}}_{\lambda}\in\Sigma^{o}$ is a local minimizer of
$\widetilde{I}_{\lambda}$ given in (1.8). In other words, there exists
$\sigma>0$ such that for any $h\in
W_{0}^{s,p}(\Omega)\cap\mathcal{C}_{s}^{0}(\overline{\Omega})$ with $\lVert
h\rVert_{\mathcal{C}_{s}^{0}(\overline{\Omega})}<\sigma$,

${\widehat{u}}_{\lambda}+h\in\Sigma\quad\mbox{and}\quad\widetilde{I}_{\lambda}({\widehat{u}}_{\lambda}+h)\geq\widetilde{I}_{\lambda}({\widehat{u}}_{\lambda}).$

Moreover, we have ${\widetilde{I}}_{\lambda}({\widehat{u}}_{\lambda})<0$,

$\begin{split}(p-1)\|\widehat{u}_{\lambda}\|^{p}-(q-1)\lambda|\widehat{u}_{\lambda}|_{q}^{q}-({p_{s}^{*}}-1)|\widehat{u}_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}\geq
0,\end{split}$ (3.6)

and there exists $C>0$ such that $\lVert\widehat{u}_{\lambda}\rVert<C$ for all
$0<\lambda<\Lambda$.

###### Proof.

Using Lemma 3.3, there exists $\beta>1$ such that

$\begin{split}(-\Delta)_{p}^{s}u_{2}&=f_{\lambda^{\prime}}(u_{2})\geq
f_{\lambda^{\prime}}({\widehat{u}}_{\lambda})\geq
f_{\lambda}(\beta{\widehat{u}}_{\lambda})={\lambda}\beta^{q-1}{\widehat{u}}_{\lambda}^{q-1}+\beta^{p_{s}^{*}-1}{\widehat{u}}_{\lambda}^{p_{s}^{*}-1}\geq(-\Delta)_{p}^{s}(\beta^{\frac{q-1}{p-1}}\widehat{u}_{\lambda}),\end{split}$

which together with the comparison principle implies

$u_{2}\geq\gamma_{2}\widehat{u}_{\lambda},\quad\text{where
}\leavevmode\nobreak\ \gamma_{2}:=\beta^{\frac{q-1}{p-1}}>1.$ (3.7)

Similarly, there exists $\gamma_{1}>1$ such that

${\widehat{u}}_{\lambda}\geq\gamma_{1}u_{1}.$ (3.8)

Moreover, by Proposition 2.8, there exists $\beta^{\prime}>0$ such that

$u_{1}(x)\geq\beta^{\prime}\text{d}^{s}_{\Omega}(x).$ (3.9)

Choose now $\sigma:=\left[\min(\gamma_{1},\gamma_{2})-1\right]\beta^{\prime}$.
Let $h\in W_{0}^{s,p}(\Omega)\cap\mathcal{C}_{s}^{0}(\overline{\Omega})$
satisfy $\lVert h\rVert_{\mathcal{C}_{s}^{0}(\overline{\Omega})}<\sigma$, then
by (3.7)–(3.9), there holds

${\widehat{u}}_{\lambda}+h>{\widehat{u}}_{\lambda}-(\gamma_{1}-1)\beta^{\prime}{\text{d}^{s}_{\Omega}(x)}>{\widehat{u}}_{\lambda}-(\gamma_{1}-1)u_{1}\geq
u_{1}$

and similarly ${\widehat{u}}_{\lambda}+h<u_{2}$, so
${\widehat{u}}_{\lambda}+h\in\Sigma$. Furthermore,

$\displaystyle{\widetilde{I}}_{\lambda}({\widehat{u}}_{\lambda}+h)$
$\displaystyle=\widehat{I}_{\lambda}({\widehat{u}}_{\lambda}+h)+\int_{\Omega}{\widehat{F}}_{\lambda}(x,{\widehat{u}}_{\lambda}+h)dx-\frac{\lambda}{q}|{\widehat{u}}_{\lambda}+h|_{q}^{q}-\frac{1}{p_{s}^{*}}|{\widehat{u}}_{\lambda}+h|_{p_{s}^{*}}^{p_{s}^{*}}$
$\displaystyle\geq\widehat{I}_{\lambda}(\widehat{u}_{\lambda})+\int_{\Omega}\int_{0}^{{\widehat{u}}_{\lambda}+h}\widehat{f}_{\lambda}(x,t)dtdx-\frac{\lambda}{q}|{\widehat{u}}_{\lambda}+h|_{q}^{q}-\frac{1}{p_{s}^{*}}|{\widehat{u}}_{\lambda}+h|_{p_{s}^{*}}^{p_{s}^{*}}$
$\displaystyle={\widetilde{I}}_{\lambda}(\widehat{u}_{\lambda}).$

The last equality comes from the definition of $\widehat{f}_{\lambda}$ and
${\widehat{u}}_{\lambda}+h\in\Sigma$, so ${\widehat{u}}_{\lambda}$ is a local
minimizer for $\widetilde{I}_{\lambda}$ with respect to
$C_{s}^{0}(\overline{\Omega})$ topology. Consider
$g(t)={\widetilde{I}}_{\lambda}(t{\widehat{u}}_{\lambda})$,
$g^{\prime\prime}(1)\geq 0$ hence (3.6) holds.

On the other hand, $C_{c}^{\infty}(\Omega)$ is dense in $W_{0}^{s,p}(\Omega)$
(see [29, Theorem 2.6]), so dose $W_{0}^{s,p}(\Omega)\cap
C_{s}^{0}(\overline{\Omega})$. Then $\widehat{u}_{\lambda}$ solves
$(P_{\lambda})$, hence we have

$\|\widehat{u}_{\lambda}\|^{p}-\lambda|\widehat{u}_{\lambda}|_{q}^{q}-|\widehat{u}_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}=0.$
(3.10)

Combining (3.6) with (3.10), we get
$(p_{s}^{*}-q)\lambda|\widehat{u}_{\lambda}|_{q}^{q}\geq(p^{*}_{s}-p)\|\widehat{u}_{\lambda}\|^{p}$.
Using again (3.10),

$\displaystyle\widetilde{I}_{\lambda}(\widehat{u}_{\lambda})=\left(\frac{1}{p}-\frac{1}{p_{s}^{*}}\right)\|\widehat{u}_{\lambda}\|^{p}-\left(\frac{1}{q}-\frac{1}{p_{s}^{*}}\right)\lambda|\widehat{u}_{\lambda}|_{q}^{q}<0.$

Moreover, for $0<\lambda<\Lambda$,

$\|\widehat{u}_{\lambda}\|^{p}\leq C_{1}|\widehat{u}_{\lambda}|_{q}^{q}\leq
C_{2}\|\widehat{u}_{\lambda}\|^{q},$

which deduces that $\|\widehat{u}_{\lambda}\|$ is uniformly bounded. ∎

Now we are in position to prove the existence of positive solution for
$(P_{\Lambda})$.

######  Lemma 3.6.

Let $s\in(0,1)$, $p>1$, $n>sp$. For $\lambda=\Lambda$, there is a positive
solution to $(P_{\Lambda})$.

###### Proof.

Let $\widehat{u}_{\lambda}$ be given in Lemma 3.5, there exists $C>0$ such
that $\|\widehat{u}_{\lambda}\|\leq C$. Notice that $\widehat{u}_{\lambda}$ is
also a critical point of $\widetilde{I}_{\lambda}$, hence a positive solution
to $(P_{\lambda})$. Let $u_{\lambda}$ be the minimal positive solution given
in Lemma 3.4. So

$\|u_{\lambda}\|^{p}=\lambda|u_{\lambda}|_{q}^{q}+|u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}\leq\lambda|\widehat{u}_{\lambda}|_{q}^{q}+|\widehat{u}_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}=\|\widehat{u}_{\lambda}\|^{p}\leq
C^{p}.$

As $u_{\lambda}$ is increasing with respect to $\lambda$, then there exists
$u_{\Lambda}\in W_{0}^{s,p}(\Omega)$ such that $u_{\lambda}$ weakly converges
to $u_{\Lambda}$ and $u_{\lambda}(x)\to u_{\Lambda}(x)$ a.e. in $\Omega$.
Consequently, $u_{\Lambda}$ is a nontrivial positive solution to
$(P_{\Lambda})$. ∎

Clearly, Lemma 3.1, Lemma 3.4 and Lemma 3.6 complete the proof of Theorem 1.1.

##  4 Proof of Theorem 1.3

Now, we consider the existence of second positive solution for
$(P_{\lambda})$. For convenience and without loss of generality, we assume
$0\in\Omega$. For $0<\lambda<\Lambda$, let $u_{\lambda}$ be the minimal
solution given in Proposition 3.4, and ${\widehat{u}}_{\lambda}$ be given in
Lemma 3.5. If $u_{\lambda}\neq{\widehat{u}}_{\lambda}$, we get already two
positive solutions of $(P_{\lambda})$. Therefore, in this section, we always
assume

$u_{\lambda}={\widehat{u}}_{\lambda}.$

When $p\geq 2$, from Lemma 2.6 and Lemma 3.5, it follows that $u_{\lambda}$ is
a local minimizer of ${\widetilde{I}}_{\lambda}$ in $W_{0}^{s,p}(\Omega)$,
that is, there exists $\rho>0$ such that

${\widetilde{I}}_{\lambda}(u)\geq{\widetilde{I}}_{\lambda}(u_{\lambda})\quad\text{for}\leavevmode\nobreak\
\text{any}\leavevmode\nobreak\ \|u-u_{\lambda}\|\leq\rho.$ (4.1)

In order to find a second positive solution, we will show a mountain pass
geometry for ${\widetilde{I}}_{\lambda}$ around $u_{\lambda}$, we will choose
the mountain pass paths from $u_{\lambda}$ to a terminal point $e$ such that
$\|e\|>\rho$ and
${\widetilde{I}}_{\lambda}(e)<{\widetilde{I}}_{\lambda}(u_{\lambda})$. We
denote the set of mountain pass paths by $\Gamma_{\varepsilon,\delta}$, and
the mountain pass level by $m_{\varepsilon,\delta}$ as in (1.11) and (1.12)
respectively. We will use the path $\gamma_{\varepsilon,\delta}$ given in
(1.13). The following three claims will be checked:

  * 1.

$\gamma_{\varepsilon,\delta}\in C([0,1],W_{0}^{s,p}(\Omega))$;

  * 2.

There exist $\delta\geq 2\varepsilon>0$ such that the maximum of
${\widetilde{I}}_{\lambda}$ along the path $\gamma_{\varepsilon,\delta}$ is
strictly less than $c_{s,p}$ with $c_{s,p}$ given in (1.9);

  * 3.

${\widetilde{I}}_{\lambda}$ satisfies the Palais-Smale condition for any level
$c<c_{s,p}$.

Let $\eta_{\delta}$ be given in (2.6). We can get the following lemma.

######  Lemma 4.1.

Let $B_{5\theta\delta}(0)\subset\Omega$ where $\theta$ is given in Lemma 2.1.
If $u\in W_{0}^{s,p}(\Omega)\cap L^{\infty}(\Omega)$, then $\eta_{\delta}u\to
u$ in $W_{0}^{s,p}(\Omega)$ as $\delta\to 0$.

###### Proof.

It is easy to see that there exists $C=C(p)>0$ such that

$\displaystyle\|\eta_{\delta}u-u\|^{p}$ $\displaystyle\leq
C\int_{\mathbb{R}^{2n}}\frac{|1-\eta_{\delta}(x)|^{p}|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}dxdy+C\int_{\mathbb{R}^{2n}}\frac{|(1-\eta_{\delta})(x)-(1-\eta_{\delta})(y)|^{p}|u(y)|^{p}}{|x-y|^{n+sp}}dxdy$
$\displaystyle=K_{1}+K_{2}.$

By the Lebesgue’s dominated convergence theorem, we have $\underset{\delta\to
0}{\lim}K_{1}=0$. Moreover, since $u\in L^{\infty}(\Omega)$,

$K_{2}\leq
C\|1-\eta_{\delta}\|^{p}=C\|\eta_{\delta}\|^{p}=C\|\eta_{1}\|^{p}\delta^{n-sp}.$

Therefore $\lim_{\delta\to 0}\|\eta_{\delta}u-u\|^{p}=0$ as $n>sp$. ∎

By the above lemma, we get
$\gamma_{\varepsilon,\delta}\in\Gamma_{\varepsilon,\delta}$. The second key
observation is

$\underset{u\in\gamma_{\varepsilon,\delta}([0,1])}{\sup}{\widetilde{I}}_{\lambda}(u)<c_{s,p},$
(4.2)

which means then $m_{\varepsilon,\delta}<c_{s,p}$. We begin with the following
basic inequality.

######  Lemma 4.2.

Assume that $p\in[2,\infty)$ and $\gamma\in(0,2]$, then there exists
$C=C(p,\gamma)>0$ such that

$\displaystyle|a-b|^{p}\leq a^{p}+b^{p}-pab^{p-1}+Ca^{\gamma}b^{p-2}$

for all $a,b>0$.

###### Proof.

Let $f(t)=|1-t|^{p}-1-t^{p}+pt$. We see that for $p\geq 2$,
$\limsup_{t\to\infty}f(t)\leq 0$, then for any $\gamma\in(0,2]$,
$\sup_{t>0}t^{-\gamma}f(t)<\infty$, so we are done. ∎

Now we prove the claim (4.2).

######  Lemma 4.3.

Assume that $p\in[2,\infty)$, $p-1<q<p$, $n>\frac{sp(q+1)}{q+1-p}$. Let
$\theta$ be given in Lemma 2.1, $B_{5\theta\delta}(0)\subset\Omega$, and
$m_{\varepsilon,\delta}$ be given in (1.12). Then there exist $\delta\geq
2\varepsilon>0$ such that $m_{\varepsilon,\delta}<c_{s,p}$.

###### Proof.

By Proposition 2.7, $u_{\lambda}\in C^{\alpha}(\overline{\Omega})$ for some
$\alpha\in(0,s]$, which with Lemma 2.2 deduces that

$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda})$
$\displaystyle=\frac{1}{p}\lVert\eta_{\delta}u_{\lambda}\rVert^{p}-\frac{\lambda}{q}|\eta_{\delta}u_{\lambda}|_{q}^{q}-\frac{1}{p_{s}^{*}}|\eta_{\delta}u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}$
(4.3) $\displaystyle\leq\frac{1}{p}\lVert
u_{\lambda}\rVert^{p}+C\left|(-\Delta)_{p}^{s}u_{\lambda}\right|_{\infty}^{p/(p-1)}\delta^{n-sp}-\frac{\lambda}{q}|u_{\lambda}|_{q}^{q}-\frac{1}{p_{s}^{*}}|u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}+C\delta^{n}$
$\displaystyle={\widetilde{I}}_{\lambda}(u_{\lambda})+C\delta^{n-sp}+C\delta^{n}.$

We will estimate the maximum of
${\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta})$
with respect to $R>0$.

Step 1. Estimate for $W_{0}^{s,p}(\Omega)$-norm. We claim that if
$\delta/\varepsilon\geq 2$, the following estimate holds true.

$\displaystyle\lVert\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}\rVert^{p}$
$\displaystyle\leq\lVert\eta_{\delta}u_{\lambda}\rVert^{p}+{R^{p}}\left[S^{\frac{n}{sp}}+C\left(\frac{\varepsilon}{\delta}\right)^{(n-sp)/(p-1)}\right]$
(4.4) $\displaystyle\quad-
C_{1}R^{p-1}\varepsilon^{\frac{n-sp}{p}}+\frac{C_{2}R^{p-2}}{\delta^{sp}}\varepsilon^{n-\frac{(n-sp)(p-2)}{p}}\left(\frac{\delta}{\varepsilon}\right)^{n-\frac{(n-sp)(p-2)}{p-1}}.$

where $C_{1},C_{2},C_{3}$ are independent of $\varepsilon,\delta,R$. Indeed,

$\displaystyle\lVert\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}\rVert^{p}$
$\displaystyle=\int_{\mathbb{R}^{2n}}\frac{|\eta_{\delta}u_{\lambda}(x)+Ru_{\varepsilon,\delta}(x)-\eta_{\delta}u_{\lambda}(y)-Ru_{\varepsilon,\delta}(y)|^{p}}{|x-y|^{n+sp}}dxdy$
(4.5)
$\displaystyle\leq\int_{A_{1}}\frac{|\eta_{\delta}u_{\lambda}(x)-\eta_{\delta}u_{\lambda}(y)|^{p}}{|x-y|^{n+sp}}dxdy+\int_{A_{2}}\frac{|Ru_{\varepsilon,\delta}(x)-Ru_{\varepsilon,\delta}(y)|^{p}}{|x-y|^{n+sp}}dxdy$
$\displaystyle\quad+2\int_{A_{3}}{\frac{|\eta_{\delta}u_{\lambda}(x)-Ru_{\varepsilon,\delta}(y)|^{p}}{|x-y|^{n+sp}}}dxdy$
$\displaystyle=:K_{1}+K_{2}+2K_{3},$

where $A_{1}=B^{c}_{\theta\delta}(0)\times B^{c}_{\theta\delta}(0)$,
$A_{2}=B_{2\theta\delta}(0)\times B_{2\theta\delta}(0)$,
$A_{3}=B^{c}_{2\theta\delta}(0)\times B_{\theta\delta}(0)$. We estimate
$K_{3}$ by Lemma 4.2 with $\gamma=2$. For $p\in[2,\infty)$, there exists
$C=C(p)>0$ such that

$\displaystyle K_{3}$
$\displaystyle\leq\int_{A_{3}}{\frac{|\eta_{\delta}u_{\lambda}(x)|^{p}}{|x-y|^{n+sp}}}dxdy+\int_{A_{3}}{\frac{|Ru_{\varepsilon,\delta}(y)|^{p}}{|x-y|^{n+sp}}}dxdy$
(4.6)
$\displaystyle\quad-p\int_{A_{3}}{\frac{|\eta_{\delta}u_{\lambda}(x)||Ru_{\varepsilon,\delta}(y)|^{p-1}}{|x-y|^{n+sp}}}dxdy+C\int_{A_{3}}{\frac{|\eta_{\delta}u_{\lambda}(x)|^{2}|Ru_{\varepsilon,\delta}(y)|^{p-2}}{|x-y|^{n+sp}}}dxdy$
$\displaystyle=:L_{1}+L_{2}-L_{3}+L_{4}.$

First

$K_{1}+2L_{1}=\lVert\eta_{\delta}u_{\lambda}\rVert^{p}\quad\text{and}\quad
K_{2}+2L_{2}=\lVert Ru_{\varepsilon,\delta}\rVert^{p}.$ (4.7)

For any $y\in B_{\theta\delta}(0)$, $x\in B^{c}_{2\theta\delta}(0)$, we have
$|x-y|\leq|x|+\theta\delta$. By (2.10) and (2.2), we obtain

$\displaystyle\begin{aligned} L_{3}&\geq
pR^{p-1}\int_{B_{5\theta\delta}\backslash
B_{2\theta\delta}}\frac{|\eta_{\delta}u_{\lambda}(x)|}{(|x|+\theta\delta)^{n+sp}}dx\int_{B_{\theta\delta}}u_{\varepsilon,\delta}(y)^{p-1}dy\\\
&\geq\frac{CR^{p-1}}{\delta^{sp}}\int_{B_{\theta\delta}}u_{\varepsilon,\delta}(y)^{p-1}dy\\\
&\geq\frac{CR^{p-1}}{\delta^{sp}}\int_{B_{\delta}}U_{\varepsilon}(y)^{p-1}dy=\frac{CR^{p-1}\varepsilon^{n-\frac{(n-sp)(p-1)}{p}}}{\delta^{sp}}\int_{B_{\delta/\varepsilon}}U(y)^{p-1}dy.\end{aligned}$

Due to Lemma 2.1 and $\delta/\varepsilon\geq 2$, we get

$\displaystyle\int_{B_{\delta/\varepsilon}}U(y)^{p-1}dy$ $\displaystyle\geq
C\int_{1}^{\delta/\varepsilon}U(r)^{p-1}r^{n-1}dr$ $\displaystyle\geq
C\int_{1}^{\delta/\varepsilon}\frac{1}{r^{n-sp}}r^{n-1}dr=\frac{C}{sp}\left[\left(\frac{\delta}{\varepsilon}\right)^{sp}-1\right]\geq\frac{C}{2sp}\left(\frac{\delta}{\varepsilon}\right)^{sp}.$

Thus,

$L_{3}\geq\frac{CR^{p-1}\varepsilon^{n-\frac{(n-sp)(p-1)}{p}}}{\delta^{sp}}\left(\frac{\delta}{\varepsilon}\right)^{sp}=CR^{p-1}\varepsilon^{n-\frac{(n-sp)(p-1)}{p}-sp}=CR^{p-1}\varepsilon^{\frac{n-sp}{p}}.$
(4.8)

For any $x\in B^{c}_{2\theta\delta}(0)$, $y\in B_{\theta\delta}(0)$, we have
$|x-y|\geq|x|-\theta\delta\geq\frac{|x|}{2}$. It follows from (2.9) that

$\displaystyle\begin{aligned}
L_{4}\leq\frac{CR^{p-2}}{\delta^{sp}}\int_{B_{\theta\delta}}u_{\varepsilon,\delta}(y)^{p-2}dy&\leq\frac{CR^{p-2}}{\delta^{sp}}\int_{B_{\theta\delta}}U_{\varepsilon}(y)^{p-2}dy\\\
&=\frac{CR^{p-2}}{\delta^{sp}}\varepsilon^{n-\frac{(n-sp)(p-2)}{p}}\int_{B_{\theta\delta/\varepsilon}}U(y)^{p-2}dy.\end{aligned}$

Using $U\in L^{\infty}(\mathbb{R}^{n})$, Lemma 2.1 and $\delta/\varepsilon\geq
2$, we have

$\displaystyle\int_{B_{\delta/\varepsilon}}U(y)^{p-2}dy$ $\displaystyle\leq
C\int_{1}^{\delta/\varepsilon}U(r)^{p-2}r^{n-1}dr+C$ $\displaystyle\leq
C\int_{1}^{\delta/\varepsilon}\frac{1}{r^{\frac{(n-sp)(p-2)}{p-1}}}r^{n-1}dr+C\leq
C\left(\frac{\delta}{\varepsilon}\right)^{n-\frac{(n-sp)(p-2)}{p-1}}.$

Therefore

$L_{4}\leq\frac{CR^{p-2}}{\delta^{sp}}\varepsilon^{n-\frac{(n-sp)(p-2)}{p}}\left(\frac{\delta}{\varepsilon}\right)^{n-\frac{(n-sp)(p-2)}{p-1}},$

which together with (4.5)–(4.8) and (2.11), implies (4.4).

Step 2. Estimates for power terms. We claim that

$|\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}|_{q}^{q}\geq|\eta_{\delta}u_{\lambda}|_{q}^{q}+CR^{q}\varepsilon^{n-\frac{(n-sp)q}{p}}\quad\mbox{for
all }\;R>0,$ (4.9)

and

$|\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}|_{p_{s}^{*}}^{p_{s}^{*}}\geq|\eta_{\delta}u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}+{R^{p_{s}^{*}}}\left[S^{\frac{n}{sp}}-C{\left(\frac{\varepsilon}{\delta}\right)}^{\frac{n}{p-1}}\right].$
(4.10)

Indeed, because the supports of $\eta_{\delta}u_{\lambda}$ and
$u_{\varepsilon,\delta}$ are disjoint, there holds

$|\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}|_{q}^{q}=|\eta_{\delta}u_{\lambda}|_{q}^{q}+|Ru_{\varepsilon,\delta}|_{q}^{q},$
(4.11)

and

$|\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta}|_{p_{s}^{*}}^{p_{s}^{*}}=|\eta_{\delta}u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}+|Ru_{\varepsilon,\delta}|_{p_{s}^{*}}^{p_{s}^{*}}.$
(4.12)

By a direct computation,

$|u_{\varepsilon,\delta}|_{q}^{q}=\int_{B_{\theta\delta}}u_{\varepsilon,\delta}^{q}(y)dy\geq\int_{B_{\delta}}U_{\varepsilon}(y)^{q}dy=\varepsilon^{n-\frac{(n-sp)q}{p}}\int_{B_{\delta/\varepsilon}}U(y)^{q}dy.$

It follows from Lemma 2.1, $\delta/\varepsilon\geq 2$ and
$n>\frac{spq}{q-p+1}$ that

$\displaystyle\int_{B_{\delta/\varepsilon}}U(y)^{q}dy$ $\displaystyle\geq
C\int_{1}^{\delta/\varepsilon}U(r)^{q}r^{n-1}dr\geq
C\int_{1}^{\delta/\varepsilon}\frac{1}{r^{(n-sp)q/(p-1)}}r^{n-1}dr\geq C.$

Thus,

$|u_{\varepsilon,\delta}|_{q}^{q}\geq C\varepsilon^{n-\frac{(n-sp)q}{p}}.$
(4.13)

By (4.11), (4.13), we have that (4.9) holds. Using (4.12) and (2.12), we get
(4.10).

Step 3. Conclusion.

Combining (4.4), (4.9) with (4.10), we deduce that

$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta})\leq$
$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda})+\frac{R^{p}}{p}\left[S^{\frac{n}{sp}}+C\left(\frac{\varepsilon}{\delta}\right)^{(n-sp)/(p-1)}\right]-\frac{R^{p_{s}^{*}}}{p_{s}^{*}}\left[S^{\frac{n}{sp}}-C{\left(\frac{\varepsilon}{\delta}\right)}^{\frac{n}{p-1}}\right]$
$\displaystyle-
C_{1}R^{q}\varepsilon^{n-\frac{(n-sp)q}{p}}-C_{1}R^{p-1}\varepsilon^{\frac{n-sp}{p}}+\frac{C_{2}R^{p-2}}{\delta^{sp}}\varepsilon^{n-\frac{(n-sp)(p-2)}{p}}\left(\frac{\delta}{\varepsilon}\right)^{n-\frac{(n-sp)(p-2)}{p-1}}.$

Taking $\varepsilon=\delta^{k+1}$ with $k>0$ in the above inequality, then

$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta})=$
$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda})+g_{\delta}(R),$
(4.14)

with

$\displaystyle g_{\delta}(R)=$
$\displaystyle\frac{R^{p}}{p}\left(S_{s,p}^{\frac{n}{sp}}+C\delta^{k(n-sp)/(p-1)}\right)-\frac{R^{p_{s}^{*}}}{p_{s}^{*}}\left(S_{s,p}^{\frac{n}{sp}}-C{\delta}^{\frac{kn}{p-1}}\right)$
$\displaystyle-
C_{1}R^{q}\delta^{(k+1)\left[n-\frac{(n-sp)q}{p}\right]}-C_{1}R^{p-1}\delta^{\frac{(k+1)(n-sp)}{p}}+{C_{2}R^{p-2}}{\delta^{(n-sp)\left[1-\frac{(p-2)(p-k-1)}{p(p-1)}\right]}}.$

Let $R_{\delta}\in\mathbb{R}^{+}$ satisfy

$g_{\delta}(R_{\delta})=\underset{R\in\mathbb{R}^{+}}{\max}\,g_{\delta}(R).$

Clearly, there exists $\delta_{0}>0$ such that when $\delta_{0}>\delta>0$,
$\\{R_{\delta}\\}_{\delta_{0}>\delta>0}$ is bounded, and it has a positive
lower bound $T>0$. Let

$h_{\delta}(R)=\frac{R^{p}}{p}\left(S_{s,p}^{\frac{n}{sp}}+C\delta^{k(n-sp)/(p-1)}\right)-\frac{R^{p_{s}^{*}}}{p_{s}^{*}}\left(S_{s,p}^{\frac{n}{sp}}-C{\delta}^{\frac{kn}{p-1}}\right).$

Obviously, $h_{R}$ is increasing on $[0,R_{\delta}^{*}]$, decreasing on
$[R_{\delta}^{*},\infty)$ with

$R_{\delta}^{*}=\left(\frac{S_{s,p}^{\frac{n}{sp}}+C\delta^{k(n-sp)/(p-1)}}{S_{s,p}^{\frac{n}{sp}}-C{\delta}^{\frac{kn}{p-1}}}\right)^{\frac{1}{p_{s}^{*}-p}}.$

Therefore

$\underset{R\in\mathbb{R}^{+}}{\max}\,h_{\delta}(R)=\frac{s}{n}\frac{\left(S_{s,p}^{\frac{n}{sp}}+C\delta^{k(n-sp)/(p-1)}\right)^{\frac{p_{s}^{*}}{p_{s}^{*}-p}}}{\left(S_{s,p}^{\frac{n}{sp}}-C{\delta}^{\frac{kn}{p-1}}\right)^{\frac{p}{p_{s}^{*}-p}}}=\frac{s}{n}S_{s,p}^{\frac{n}{sp}}+O\left(\delta^{k(n-sp)/(p-1)}\right).$

Thus,

$\displaystyle\underset{R\in\mathbb{R}^{+}}{\max}g_{\delta}(R)\leq$
$\displaystyle\underset{R\in\mathbb{R}^{+}}{\max}h_{\delta}(R)-C_{1}R_{\delta}^{q}\delta^{(k+1)\left(n-\frac{(n-sp)q}{p}\right)}$
(4.15) $\displaystyle-
C_{1}R_{\delta}^{p-1}\delta^{\frac{(k+1)(n-sp)}{p}}+{C_{2}R_{\delta}^{p-2}}{\delta^{(n-sp)\left[1-\frac{(p-2)(p-k-1)}{p(p-1)}\right]}}$
$\displaystyle=$
$\displaystyle\frac{s}{n}S_{s,p}^{\frac{n}{sp}}+O\left(\delta^{k(n-sp)/(p-1)}\right)-C_{1}R_{\delta}^{q}\delta^{(k+1)\left[n-\frac{(n-sp)q}{p}\right]}$
$\displaystyle-
C_{1}R_{\delta}^{p-1}\delta^{\frac{(k+1)(n-sp)}{p}}+{C_{2}R_{\delta}^{p-2}}{\delta^{(n-sp)\left[1-\frac{(p-2)(p-k-1)}{p(p-1)}\right]}}.$

Putting $0<k<p-1$, it follows that

$\frac{k(n-sp)}{p-1}<\min\left\\{(n-sp)\left[1-\frac{(p-2)(p-k-1)}{p(p-1)}\right],\frac{(k+1)(n-sp)}{p}\right\\}.$
(4.16)

Using (4.15) and (4.16), we get

$\underset{R\in\mathbb{R}^{+}}{\max}g_{\delta}(R)\leq\frac{s}{n}S_{s,p}^{\frac{n}{sp}}-C_{1}R_{\delta}^{q}\delta^{(k+1)\left(n-\frac{(n-sp)q}{p}\right)}+O\left(\delta^{k(n-sp)/(p-1)}\right).$
(4.17)

By (4.3), (4.14), (4.17) and $R_{\delta}>T>0$, we obtain

$\displaystyle{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta})=c_{s,p}-C\delta^{(k+1)\left[n-\frac{(n-sp)q}{p}\right]}+O\left(\delta^{k(n-sp)/(p-1)}\right)+O(\delta^{n-sp}).$

By $0<k<p-1$, to reach our aim, it suffices to have

$(k+1)\left[n-\frac{(n-sp)q}{p}\right]<\frac{k(n-sp)}{p-1},$ (4.18)

and it is equivalent to

$n-\frac{(n-sp)q}{p}<k\left[\frac{n-sp}{p-1}-n+\frac{(n-sp)q}{p}\right].$
(4.19)

We want to prove that there exists $0<k<p-1$ such that (4.19) holds. In fact
(4.19) holds true with $k=p-1$, since we have

$n>\frac{sp(q+1)}{q+1-p}.$

We get then

$\underset{R\in\mathbb{R}^{+}}{\max}\,{\widetilde{I}}_{\lambda}(\eta_{\delta}u_{\lambda}+Ru_{\varepsilon,\delta})<c_{s,p}$
(4.20)

if $\delta$ is sufficiently small. Combining Lemma 4.1 with (4.20), we
conclude (4.2) provided $\delta$ is small enough, which means
$m_{\varepsilon,\delta}<c_{s,p}$. ∎

Next, we proceed to check that the functional ${\widetilde{I}}_{\lambda}$
satisfies the Palais-Smale condition at the level $c<c_{s,p}$. Recall that the
functional ${\widetilde{I}}_{\lambda}$ is said satisfying the Palais-Smale
condition at a level $c\in\mathbb{R}$ (for short $(PS)_{c}$) if any sequence
$\\{u_{j}\\}\subset W_{0}^{s,p}(\Omega)$ such that

${\widetilde{I}}_{\lambda}(u_{j})\rightarrow
c\quad\mbox{and}\quad{\widetilde{I}}^{\prime}_{\lambda}(u_{j})\rightarrow
0\;\mbox{ in }\;W_{0}^{s,p}(\Omega)^{*}$

admits a subsequence which is convergent in $W_{0}^{s,p}(\Omega)$.

######  Proposition 4.4.

Let $s\in(0,1)$, $1<q<p$, $n>sp$ and $u_{\lambda}$ be the minimal positive
solution to $(P_{\lambda})$ in Proposition 3.4. Assume that
${\widetilde{I}}_{\lambda}$ has only two critical points $0$ and
$u_{\lambda}$. Then ${\widetilde{I}}_{\lambda}$ satisfies the $(PS)_{c}$
condition for all $c<c_{s,p}$.

###### Proof.

Let $\\{u_{j}\\}\subset W_{0}^{s,p}(\Omega)$ be a $(PS)_{c}$ sequence of
${\widetilde{I}}_{\lambda}$ with $c<c_{s,p}$, i.e.

${\widetilde{I}}_{\lambda}(u_{j})=\frac{1}{p}\lVert
u_{j}\lVert^{p}-\frac{\lambda}{q}|u^{+}_{j}|^{q}_{q}-\frac{1}{p_{s}^{*}}|u^{+}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}=c+o(1)$
(4.21)

and

$\displaystyle\langle{\widetilde{I}}^{\prime}_{\lambda}(u_{j}),v\rangle=$
$\displaystyle\int_{\mathbb{R}^{2n}}{\frac{J_{u_{j}}(x,y)(v(x)-v(y))}{|x-y|^{n+sp}}}dxdy-\lambda\int_{\Omega}(u^{+}_{j})^{q-1}vdx$
(4.22) $\displaystyle-\int_{\Omega}(u^{+}_{j})^{p_{s}^{*}-1}vdx=o(1)\lVert
v\lVert\text{\leavevmode\nobreak\ \leavevmode\nobreak\ for\leavevmode\nobreak\
all\leavevmode\nobreak\ }v\in W_{0}^{s,p}(\Omega).$

Taking $v=u_{j}$ in (4.22), by (4.21) and the Sobolev embedding
$W_{0}^{s,p}(\Omega)\subset L^{q}(\Omega)$, when $j$ goes to $\infty$,

$\begin{split}p_{s}^{*}c+o(1)\|u_{j}\|+o(1)=&p_{s}^{*}{\widetilde{I}}_{\lambda}(u_{j})-\langle{\widetilde{I}}^{\prime}_{\lambda}(u_{j}),u_{j}\rangle\\\
=&\left(\frac{p_{s}^{*}}{p}-1\right)\|u_{j}\|^{p}-\left(\frac{p_{s}^{*}}{q}-1\right)\lambda|u_{j}^{+}|_{q}^{q}\\\
\geq&\left(\frac{p_{s}^{*}}{p}-1\right)\|u_{j}\|^{p}-\left(\frac{p_{s}^{*}}{q}-1\right)\lambda
C\|u_{j}\|^{q}.\\\ \end{split}$

Hence, $\\{u_{j}\\}$ is bounded in $W_{0}^{s,p}(\Omega)$. Therefore there is a
renamed subsequence of $\\{u_{j}\\}$, which converges weakly to some $u\in
W_{0}^{s,p}(\Omega)$ and $u_{j}\rightarrow u$ a.e. in $\mathbb{R}^{n}$. Now we
will study more precisely the behavior of weakly convergent sequence
$\\{u_{j}\\}$ in several steps.

Step 1. We claim that

$\lim_{j\to\infty}(\lVert u_{j}-u\lVert^{p}-\lVert u_{j}\lVert^{p}+\lVert
u\lVert^{p})=0,\quad\mbox{and}\quad\liminf_{j\to\infty}(|u_{j}-u|_{p_{s}^{*}}^{p_{s}^{*}}-|u^{+}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}+|u^{+}|_{p_{s}^{*}}^{p_{s}^{*}})\geq
0.$

Consider

$\Theta_{j}(x,y):=\frac{u_{j}(x)-u_{j}(y)}{|x-y|^{\frac{n}{p}+s}}\quad\mbox{
and }\quad\Theta(x,y):=\frac{u(x)-u(y)}{|x-y|^{\frac{n}{p}+s}}.$

Then $\\{\Theta_{j}\\}$ is bounded in $L^{p}(\mathbb{R}^{2n})$, and
$\Theta_{j}(x,y)\rightarrow\Theta(x,y)$ a.e. in $\mathbb{R}^{2n}$. By Brezis-
Lieb’s lemma (see [33, Lemma 1.32]), the first claim is done. Moreover, as
$|u_{j}(x)-u(x)|\geq|u_{j}^{+}(x)-u^{+}(x)|$ and $u_{j}^{+}(x)\to u^{+}(x)$
a.e. in $\mathbb{R}^{n}$, using again Brezis-Lieb’s lemma,

$|u_{j}-u|_{p_{s}^{*}}^{p_{s}^{*}}\geq|u^{+}_{j}-u^{+}|_{p_{s}^{*}}^{p_{s}^{*}}=|u^{+}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}-|u^{+}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1),$

which gives the second claim.

Step 2. For any $v\in W_{0}^{s,p}(\Omega),$

$\lim_{j\to\infty}\int_{\mathbb{R}^{2n}}{\frac{J_{u_{j}}(x,y)(v(x)-v(y))}{|x-y|^{n+sp}}}dxdy=\int_{\mathbb{R}^{2n}}{\frac{J_{u}(x,y)(v(x)-v(y))}{|x-y|^{n+sp}}}dxdy.$
(4.23)

Indeed, define
$\varPhi_{j}(x,y):=|x-y|^{-\frac{n+sp}{p^{\prime}}}J_{u_{j}}(x,y)$ and
$\varPhi(x,y):=|x-y|^{-\frac{n+sp}{p^{\prime}}}J_{u}(x,y)$ where
$p^{\prime}=\frac{p}{p-1}$. Then $\\{\varPhi_{j}\\}$ is bounded in
$L^{p^{\prime}}(\mathbb{R}^{2n})$, and
$\varPhi_{j}(x,y)\rightarrow\varPhi(x,y)$ a.e. in $\mathbb{R}^{2n}$, so
$\varPhi_{j}$ converges weakly to $\varPhi$ in
$L^{p^{\prime}}(\mathbb{R}^{2n})$. On the other hand,
$|x-y|^{-\frac{n+sp}{p}}|v(x)-v(y)|\in L^{p}(\mathbb{R}^{2n})$, hence (4.23)
holds.

Step 3. By Step 2, it is easy to see that $u$ is a critical point of
${\widetilde{I}}_{\lambda}$, so

$\lVert u\lVert^{p}=\lambda|u^{+}|_{q}^{q}+|u^{+}|_{p_{s}^{*}}^{p_{s}^{*}}.$
(4.24)

Setting $\widehat{u}_{j}=u_{j}-u$, Step 1 implies then

$\lVert\widehat{u}_{j}\lVert^{p}=\lVert u_{j}\lVert^{p}-\lVert
u\lVert^{p}+o(1)\;\;\text{and}\;\;|\widehat{u}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}\geq|u^{+}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}-|u^{+}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1).$
(4.25)

Taking $v=u_{j}$ in (4.22), since $\\{u_{j}\\}$ is bounded in
$W_{0}^{s,p}(\Omega)$ and $u_{j}\rightarrow u$ in $L^{q}(\Omega)$, we get

$\lVert
u_{j}\lVert^{p}=\lambda|u^{+}|_{q}^{q}+|u^{+}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1).$
(4.26)

It follows from (4.25), (4.26) and (4.24) that

$\displaystyle\|\widehat{u}_{j}\|^{p}=|{u}_{j}^{+}|_{p_{s}^{*}}^{p_{s}^{*}}-|{u}^{+}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1)\leq|\widehat{u}_{j}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1)\leq\frac{\lVert\widehat{u}_{j}\lVert^{p_{s}^{*}}}{S_{s,p}^{p_{s}^{*}/p}}+o(1),$
(4.27)

so

$\lVert\widehat{u}_{j}\lVert^{p}(S_{s,p}^{p_{s}^{*}/p}-\lVert\widehat{u}_{j}\lVert^{p_{s}^{*}-p})\leq
o(1).$ (4.28)

By (4.25) and (4.27), there holds

$\displaystyle{\widetilde{I}}_{\lambda}(u_{j})$
$\displaystyle={\widetilde{I}}_{\lambda}(u)+\frac{1}{p}\|\widehat{u}_{j}\|^{p}-\frac{1}{p_{s}^{*}}|{u}_{j}^{+}|_{p_{s}^{*}}^{p_{s}^{*}}+\frac{1}{p_{s}^{*}}|u^{+}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1)$
$\displaystyle={\widetilde{I}}_{\lambda}(u)+\frac{1}{p}\|\widehat{u}_{j}\|^{p}-\frac{1}{p_{s}^{*}}\|\widehat{u}_{j}\|^{p}+o(1)$
$\displaystyle={\widetilde{I}}_{\lambda}(u)+\frac{s}{n}\|\widehat{u}_{j}\|^{p}+o(1).$

Hence

${\widetilde{I}}_{\lambda}(u)+\frac{s}{n}\underset{j\rightarrow\infty}{\text{limsup}}\lVert\widehat{u}_{j}\lVert^{p}=c<c_{s,p}.$

As we assume that ${\widetilde{I}}_{\lambda}$ has only two critical points $0$
and $u_{\lambda}$, it follows that either $u=0$ or $u=u_{\lambda}$. By Lemma
3.5, we know that ${\widetilde{I}}_{\lambda}(u_{\lambda})<0$. Hence,

$\underset{j\rightarrow\infty}{\text{limsup}}\lVert\widehat{u}_{j}\lVert^{p}<S_{s,p}^{\frac{n}{sp}}.$
(4.29)

Using (4.28) and (4.29), we get $\lVert\widehat{u}_{j}\lVert\rightarrow 0$,
which means that $\\{u_{j}\\}$ has a convergent subsequence. ∎

To get the main result, we will apply Ghoussoub-Preiss’ generalized mountain
pass theorem [18, Theorem (1)].

######  Theorem 4.5.

Let $X$ be a Banach space, and $\varphi$ be a $C^{1}$ functional on $X$.
Taking $u,v\in X$ and consider

$c=\underset{\gamma\in\Gamma}{\inf}\Big{[}\underset{0\leq t\leq
1}{\max}\varphi(\gamma(t))\Big{]}$

where $\Gamma=\\{\gamma\in C([0,1],X):\gamma(0)=u,\gamma(1)=v\\}$. Assume that
$F$ is a closed subset of $X$ such that for any $\gamma\in\Gamma$, one has
$\gamma([0,1])\cap\\{x\in F:\varphi(x)\geq c\\}\neq\varnothing$. Then there
exists a sequence $\\{x_{j}\\}\subset X$ satisfying

  * (i)

$\underset{j\to\infty}{\lim}\mbox{dist}(x_{j},F)=0$;

  * (ii)

$\underset{j\to\infty}{\lim}\varphi(x_{j})=c$;

  * (iii)

$\underset{j\to\infty}{\lim}\|\varphi^{\prime}(x_{j})\|_{X^{*}}=0$.

######  Lemma 4.6.

Let $s\in(0,1)$, $p\geq 2$, $p-1<q<p$ and $n>\frac{sp(q+1)}{q+1-p}$. Let
$\Lambda$ be given in (3.1), and $m_{\varepsilon,\delta}$ be the mountain pass
level defined in (1.12) where $\delta$, $\varepsilon$ are given by Lemma 4.3.
For $\lambda\in(0,\Lambda)$, if $m_{\varepsilon,\delta}\neq 0$, then problem
$(P_{\lambda})$ has at least two positive solutions.

###### Proof.

We assume by contradiction that there are only two critical points $0$ and
$u_{\lambda}$ of ${\widetilde{I}}_{\lambda}$. It follows from Proposition 4.4
that ${\widetilde{I}}_{\lambda}$ satisfies $(PS)_{c}$ condition for
$c<c_{s,p}$.

Let $\rho$ be given in (4.1). If there exists $0<\rho_{0}<\rho$ such that
$\inf_{u\in\partial
B_{\rho_{0}}(u_{\lambda})}{\widetilde{I}}_{\lambda}(u)>{\widetilde{I}}_{\lambda}(u_{\lambda})$,
then $m_{\varepsilon,\delta}>{\widetilde{I}}_{\lambda}(u_{\lambda})$. As Lemma
4.1 and Lemma 4.3 showed, $m_{\varepsilon,\delta}<c_{s,p}$. Using mountain
pass theorem in [3], we obtain a mountain pass critical point $w_{\lambda}$ of
${\widetilde{I}}_{\lambda}$. As
$(-\Delta)_{p}^{s}w_{\lambda}=\lambda(w_{\lambda}^{+})^{q-1}+(w_{\lambda}^{+})^{p_{s}^{*}-1}$,
by Lemmas 2.5, $w_{\lambda}$ is also a nonnegative solution to
$(P_{\lambda})$. Since $m_{\varepsilon,\delta}\neq 0$ and
$m_{\varepsilon,\delta}>{\widetilde{I}}_{\lambda}(u_{\lambda})$, then
$w_{\lambda}\notin\\{0,u_{\lambda}\\}$. So problem $(P_{\lambda})$ has at
least two positive solutions $u_{\lambda}$ and $w_{\lambda}$.

If $m_{\varepsilon,\delta}={\widetilde{I}}_{\lambda}(u_{\lambda})$, then for
any $0<\rho_{0}<\rho$, we have $\inf_{u\in\partial
B_{\rho_{0}}(u_{\lambda})}{\widetilde{I}}_{\lambda}(u)={\widetilde{I}}_{\lambda}(u_{\lambda})$.
Applying Theorem 4.5 with

$c=m_{\varepsilon,\delta},\;\;X=W_{0}^{s,p}(\Omega),\;\;F=\partial
B_{\rho_{0}}(u_{\lambda}),\;\;\varphi(x)=\widetilde{I}_{\lambda}(x),\;\;u=u_{\lambda},\;\;v=e,$

we obtain still another critical point $w_{\lambda}\in\partial
B_{\rho_{0}}(u_{\lambda})$ of ${\widetilde{I}}_{\lambda}$. ∎

######  Remark 4.7.

From the above lemma, $(P_{\lambda})$ has two positive solutions for
$0<\lambda<\Lambda$ whenever $m_{\varepsilon,\delta}\neq 0$. However, we
cannot rule out the case that the mountain pass critical point is trivial when
$m_{\varepsilon,\delta}=0$. The following lemma can tell us that
$m_{\varepsilon,\delta}>0$ if $\lambda$ is sufficiently small, which means
that the trivial critical point will not occur.

######  Lemma 4.8.

Let $m_{\varepsilon,\delta}$ be the mountain pass level defined in (1.12).
There exists $\lambda^{*}>0$ such that $m_{\varepsilon,\delta}>0$ for
$\lambda\in(0,\lambda^{*})$.

###### Proof.

By the Sobolev embedding, for any $r\in[1,p_{s}^{*}]$, there is $C>0$ such
that for any $u\in W_{0}^{s,p}(\Omega)$,

${\widetilde{I}}_{\lambda}(u)\geq\frac{1}{p}\|u\|^{p}-\frac{C\lambda}{q}\|u\|^{q}-\frac{C}{p_{s}^{*}}\|u\|^{p_{s}^{*}},\quad\forall\;u\in
W_{0}^{s,p}(\Omega).$

Therefore, there exists $\lambda^{*}>0$ such that if
$\lambda\in(0,\lambda^{*})$, there are $\rho_{0}>0$ and $c>0$ satisfying

${\widetilde{I}}_{\lambda}(u)\geq c,\quad\mbox{for all $\|u\|=\rho_{0}$}.$
(4.30)

We claim that if $\lambda\in(0,\lambda^{*})$, then $\|u_{\lambda}\|<\rho_{0}$.
Indeed, let $g_{\lambda}(t)={\widetilde{I}}_{\lambda}(tu_{\lambda})$, then

$g^{\prime}_{\lambda}(t)=t^{p-1}\|u_{\lambda}\|_{p}^{p}-\lambda
t^{q-1}|u_{\lambda}|_{q}^{q}-t^{p_{s}^{*}-1}|u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}=t^{q-1}f_{\lambda}(t)$

where

$f_{\lambda}(t):=t^{p-q}\|u_{\lambda}\|_{p}^{p}-{\lambda}|u_{\lambda}|_{q}^{q}-t^{p_{s}^{*}-q}|u_{\lambda}|_{p_{s}^{*}}^{p_{s}^{*}}.$

Clearly, $f_{\lambda}$ has a unique maximal point $t_{\max}>0$ such that
$f_{\lambda}$ is increasing on the interval $[0,t_{\max}]$ and decreasing on
$[t_{\max},\infty)$. Moreover, we can conclude $f_{\lambda}(t_{\max})>0$,
otherwise $g_{\lambda}$ is nonincreasing, hence $g_{\lambda}(t)\leq 0$ for all
$t\geq 0$, which contradicts (4.30). Therefore, there are only two positive
critical points $t_{1}$, $t_{2}$ of $g_{\lambda}$ satisfying
$0<t_{1}<t_{\max}<t_{2}<\infty$, and $g_{\lambda}$ is decreasing on the
intervals $(0,t_{1})$ and $(t_{2},\infty)$, and increasing on the interval
$(t_{1},t_{2})$. Since $u_{\lambda}$ is a local minimizer of
${\widetilde{I}}_{\lambda}$, so $t_{1}=1$. This implies $g_{\lambda}(t)<0$ for
$t\in(0,1]$ as ${\widetilde{I}}_{\lambda}(u_{\lambda})<0$. Hence
$\|u_{\lambda}\|<\rho_{0}$. If we set $t_{0}$ in (1.10) large such that
$\|e\|>\rho_{0}$, then for any $\gamma\in\Gamma_{\varepsilon,\delta}$, we have
$\gamma([0,1])\cap\partial B_{\rho_{0}}(0)\neq\varnothing$. As a consequence,
$m_{\varepsilon,\delta}\geq c>0$. ∎

Thanks to Lemmas 4.6, 4.8, we complete the proof of Theorem 1.3.

##  5 Proof of Theorem 1.5

In this section, we will prove the existence of infinitely many solutions of
$(P_{\lambda})$ for $\lambda>0$ small. Consider

$\mathcal{F}=\\{A\subset W_{0}^{s,p}(\Omega)\backslash\\{0\\}:u\in
A\Rightarrow-u\in A\\},$

and

$\mathcal{A}_{j,\,r}=\\{A\in\mathcal{F}:A\leavevmode\nobreak\
\text{is}\leavevmode\nobreak\ \text{compact},A\subset B_{r}(0),\,{\rm
ind}(A)\geq j\\}.$

Here ${\rm ind}(A)$ denotes the $\mathbb{Z}_{2}$-genus of $A$, namely, the
least integer $k$ such that there exists odd functional $\phi\in
C(W_{0}^{s,p}(\Omega),\mathbb{R}^{k})$ satisfying $\phi(u)\neq 0$ for all
$u\in A$. By [3], the $\mathbb{Z}_{2}$-genus possesses the following
properties:

  * (i)

Definiteness: ${\rm ind}(A)=0$ if and only if $A=\varnothing$;

  * (ii)

Monotonicity: If there is an odd continuous map from $A$ to $B$ (in
particular, if $A\subset B$) for $A,B\in\mathcal{F}$, then ${\rm
ind}(A)\leq{\rm ind}(B)$;

  * (iii)

Subadditivity: If $A$ and $B$ are closed set in $\mathcal{F}$, then ${\rm
ind}(A\cup B)\leq{\rm ind}(A)+{\rm ind}(B)$.

Define

$b_{j}:=\underset{A\in\mathcal{A}_{j,r}}{\inf}\,\underset{u\in
A}{\max}\,I_{\lambda}(u).$ (5.1)

We are going to prove that $b_{j}$ is finite, a critical value of
$I_{\lambda}$, and $b_{j}\to 0^{-}$ as $j\to\infty$.

######  Lemma 5.1.

There exists $\lambda^{**}>0$ such that for all $\lambda\in(0,\lambda^{**}]$
there are $r,a>0$ such that

  * (i)

$I_{\lambda}(u)\geq a$ for all $\|u\|=r$;

  * (ii)

$I_{\lambda}$ is bounded from below in $B_{r}(0)\subset W_{0}^{s,p}(\Omega)$;

  * (iii)

$I_{\lambda}$ satisfies (PS) condition in $B_{r}(0)$.

###### Proof.

Assume that $\\{u_{j}\\}\subset B_{r}(0)$ is a (PS) sequence, that is,
$\\{I_{\lambda}(u_{j})\\}$ is bounded in $\mathbb{R}$ and
$I^{\prime}_{\lambda}(u_{j})\to 0$ in $W_{0}^{s,p}(\Omega)^{*}$. Since
$\\{u_{j}\\}$ is bounded in $W_{0}^{s,p}(\Omega)$, there is a subsequence,
denoted still by $\\{u_{j}\\}$, which converges weakly to some
$u\in\overline{B}_{r}(0)$ and $u_{j}(x)\rightarrow u(x)$ a.e. in
$\mathbb{R}^{n}$. Arguing as Proposition 4.4, we have

$\|u_{j}-u\|^{p}=|u_{j}-u|_{p_{s}^{*}}^{p_{s}^{*}}+o(1)\leq\frac{\lVert
u_{j}-u\lVert^{p_{s}^{*}}}{S_{s,p}^{p_{s}^{*}/p}}+o(1),$

which implies that

$\text{either}\quad
S_{s,p}^{\frac{n}{sp^{2}}}\leq\underset{j\to\infty}{\lim\inf}\lVert
u_{j}-u\lVert\quad\text{or}\quad\underset{j\to\infty}{\lim}\|u_{j}-u\|=0.$

Therefore if we fix $0<2r<S_{s,p}^{\frac{n}{sp^{2}}}$, then $u_{j}$ must
converge to $u$ in $W_{0}^{s,p}(\Omega)$.

By the Sobolev embedding $W_{0}^{s,p}(\Omega)\subset L^{m}(\Omega)$ for
$m\in[1,p_{s}^{*}]$, there exists $C>0$ such that for any $u\in
W_{0}^{s,p}(\Omega)$,

$I_{\lambda}(u)\geq\frac{1}{p}\|u\|^{p}-\frac{C\lambda}{q}\|u\|^{q}-\frac{C}{p_{s}^{*}}\|u\|^{p_{s}^{*}},$

which concludes that there exists $\lambda^{**}>0$ such that if
$\lambda\in(0,\lambda^{**}]$, there are
$\frac{1}{2}S_{s,p}^{\frac{n}{sp^{2}}}>r>0$ and $a>0$ such that
$I_{\lambda}(u)\geq a$ for $\|u\|=r$. Obviously, $I_{\lambda}$ is bounded from
below in $B_{r}(0)$. ∎

By Lemma 5.1, the deformation lemma will hold for $I_{\lambda}$ restricted in
$B_{r}(0)$ for some $r>0$. If $b_{j}\in(-\infty,0)$ is not a critical value of
$I_{\lambda}$, it follows from [33, Lemma 3.1] that there exist
$\varepsilon\in(0,-b_{j})$ and a homotopy mapping, odd in $u$

$\eta:[0,1]\times I_{\lambda}^{b_{j}+\varepsilon}\to
I_{\lambda}^{b_{j}+\varepsilon}$

such that $\eta(0,\cdot)$ is the identity map of
$I_{\lambda}^{b_{j}+\varepsilon}$ and
$\eta(1,I_{\lambda}^{b_{j}+\varepsilon})\subset
I_{\lambda}^{b_{j}-\varepsilon}$. Here for any $c\in\mathbb{R}$,
$I_{\lambda}^{c}$ means the sublevel set $\\{u\in B_{r}(0):I_{\lambda}(u)\leq
c\\}$. According to the definition of $b_{j}$, there exists
$A\in\mathcal{A}_{j,r}$ such that $A\subset I_{\lambda}^{b_{j}+\varepsilon}$.
However, by means of the monotonicity of $\mathbb{Z}_{2}$-genus, we have

${\rm ind}(\eta(1,A))\geq{\rm ind}(A)\geq j.$

However, $\eta(1,A)\subset I_{\lambda}^{b_{j}-\varepsilon}$, which contradicts
the definition of $b_{j}$. Hence, the hypothesis was wrong, $b_{j}$ is a
critical value of $I_{\lambda}$.

Next, we aim to prove $b_{j}\to 0$ as $j\to\infty$. For that, we will consider
suitable subsets of ${\mathcal{A}}_{j,r}$. Since $W_{0}^{s,p}(\Omega)$ is
reflexive and separable, by [10, Corollary 3.27], $W_{0}^{s,p}(\Omega)^{*}$ is
also reflexive and separable. Therefore, there exists a sequence
$\\{f_{j}\\}\subset W_{0}^{s,p}(\Omega)^{*}$ such that $\\{f_{j}\\}$ are
linearly independent and ${\rm span}\\{f_{j},j\geq 1\\}$ is dense in
$W_{0}^{s,p}(\Omega)^{*}$. Denote $F_{j}={\rm span}\\{f_{k},1\leq k\leq j\\}$
for any $j\geq 1$.

Let $g_{1}=f_{1}$, let $v_{1}\in W_{0}^{s,p}(\Omega)$ satisfy
$g_{1}(v_{1})=1$. Clearly, there is $g_{2}\in F_{2}\backslash F_{1}$ such that
$g_{2}(v_{1})=0$, and there exists $v_{2}\in W_{0}^{s,p}(\Omega)$ satisfying
$g_{2}(v_{2})=1$. By induction, we get two sequences $g_{j}\in F_{j}\backslash
F_{j-1}$, $v_{j}\in W_{0}^{s,p}(\Omega)$ such that

$\forall\;j\geq 2,\quad g_{j}(v_{k})=0\;\;\mbox{for }1\leq k\leq
j-1\;\;\mbox{and}\;\;g_{j}(v_{j})=1.$

Define

$E_{j}={\rm span}\\{v_{k},1\leq k\leq j\\},\;\;E_{j}^{\perp}=\underset{1\leq
k\leq j}{\cap}{\rm Ker}(g_{k})=\underset{f\in F_{j}}{\cap}{\rm
Ker}(f),\quad\forall\;j\geq 1.$

Clearly, $\\{v_{j}\\}$ are linearly independent, hence ${\rm dim}(E_{j})=j$
for all $j\geq 1$. It’s not difficult to see that for any $j\geq 1$, there
holds $W_{0}^{s,p}(\Omega)=E_{j}\oplus E_{j}^{\perp}$. In other words, for any
$w\in W_{0}^{s,p}(\Omega)$, there is a unique $w_{j}\in E_{j}$ such that
$g_{k}(w_{j})=g_{k}(w)$ for all $1\leq k\leq j$, so $w_{j}^{\perp}=w-w_{j}\in
E_{j}^{\perp}$. We denote the projection from $W_{0}^{s,p}(\Omega)$ into
$E_{j}$ parallel to $E_{j}^{\perp}$ by $P_{j}$, i.e. $P_{j}(w)=w_{j}$. An easy
but very useful observation is

######  Lemma 5.2.

Let $\\{u_{j}\\}\subset W_{0}^{s,p}$ be a bounded sequence such that $u_{j}\in
E_{j}^{\perp}$ for any $j$, then $u_{j}\to 0$ weakly in $W_{0}^{s,p}(\Omega)$.

###### Proof.

Let $f\in\cup_{j}F_{j}$. From the definition of $E_{j}^{\perp}$, $f(u_{j})=0$
for $j$ large enough, so $\lim_{j\to\infty}f(u_{j})=0$. The same conclusion
holds for any $f\in W_{0}^{s,p}(\Omega)^{*}$, since $\cup_{j}F_{j}$ is dense
in $W_{0}^{s,p}(\Omega)^{*}$ and $\\{u_{j}\\}$ is bounded. ∎

###### Proof of Theorem 1.5.

Let $\lambda^{**},\,r$ be as in Lemma 5.1 and $\lambda\in(0,\lambda^{**})$.
Let $A_{j}=E_{j}\cap\partial B_{1}(0)$, clearly $A_{j}\in{\mathcal{A}}_{j,r}$
since ${\rm ind}(A_{j})=j$ and $A_{j}$ is compact. Moreover, as ${\rm
dim}(E_{j})<\infty$ there is $C_{j}>0$ such that for any $v\in E_{j}$,
$\|u\|\leq C_{j}|u|_{q}$. Consequently, for any $u\in A_{j}$ and any $t>0$,

$\displaystyle I_{\lambda}(tu)\leq\frac{t^{p}}{p}\|u\|^{p}-\frac{\lambda
t^{q}}{q}|u|_{q}^{q}\leq\frac{t^{p}}{p}\|u\|^{p}-\frac{C\lambda
t^{q}}{q}\|u\|^{q}.$

There exists then $\varepsilon\in(0,r)$ satisfying

$\varepsilon A_{j}\subset
B_{r}(0)\quad\mbox{and}\quad\underset{u\in\varepsilon
A_{j}}{\max}I_{\lambda}(u)<0.$

It means that $b_{j}$ is finite and negative. It follows that there exists a
critical point $u_{j}\in B_{r}(0)$ of $I_{\lambda}$ with
$I_{\lambda}(u_{j})=b_{j}<0$. Let

$\widetilde{b}_{j}=\underset{A\in\mathcal{A}_{j,r}}{\inf}\,\underset{u\in
A\cap E_{j-1}^{\perp}}{\sup}\,I_{\lambda}(u).$ (5.2)

Note that $\widetilde{b}_{j}$ is well defined because $A\cap
E_{j-1}^{\perp}\neq\varnothing$ for any $A\in\mathcal{A}_{j,r}$. In fact, if
$P_{j-1}u\neq 0$ for all $u\in A$, it follows from the property of
$\mathbb{Z}_{2}$-genus that ${\rm ind}(A)\leq{\rm ind}(P_{j-1}(A))\leq j-1$,
which is a contradiction. Obviously $\widetilde{b}_{j}\leq b_{j}$.

Now we claim $b_{j}\to 0^{-}$ as $j\to\infty$. Suppose the contrary:
$b_{j}\leq\alpha<0$ for all $j\in{\mathbb{N}}^{+}$. By the definition of
$\widetilde{b}_{j}$, there is a sequence $\\{u_{j}\\}$ such that

$u_{j}\in E_{j-1}^{\perp}\cap B_{r}(0)\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }|I_{\lambda}(u_{j})-\widetilde{b}_{j}|<\frac{1}{j}.$

By Lemma 5.2, $u_{j}\to 0$ weakly in $W_{0}^{s,p}(\Omega)$, hence $u_{j}\to 0$
in $L^{m}(\Omega)$ for all $m\in[1,p_{s}^{*})$. As
$\widetilde{b}_{j}\leq\alpha$, we have

$\underset{j\to\infty}{\lim\sup}\,I_{\lambda}(u_{j})\leq\alpha<0.$ (5.3)

On the other hand,

$\displaystyle
I_{\lambda}(u_{j})=\frac{1}{p}\|u_{j}\|^{p}-\frac{1}{p_{s}^{*}}|u_{j}|_{p_{s}^{*}}^{p_{s}^{*}}+o(1)$
$\displaystyle\geq\frac{1}{p}\|u_{j}\|^{p}-\frac{S_{s,p}^{-p_{s}^{*}/p}}{p_{s}^{*}}\|u_{j}\|^{p_{s}^{*}}+o(1)$
$\displaystyle=\|u_{j}\|^{p}\left(\frac{1}{p}-\frac{S_{s,p}^{-p_{s}^{*}/p}}{p_{s}^{*}}\|u_{j}\|^{p_{s}^{*}-p}\right)+o(1).$

By $\frac{1}{2}S_{s,p}^{\frac{n}{sp^{2}}}>r>0$, there holds

$\frac{1}{p}-\frac{S_{s,p}^{-p_{s}^{*}/p}}{p_{s}^{*}}\|u_{j}\|^{p_{s}^{*}-p}\geq\frac{1}{p}-\frac{S_{s,p}^{-p_{s}^{*}/p}}{p_{s}^{*}}r^{p_{s}^{*}-p}\geq
0.$

Hence $I_{\lambda}(u_{j})\geq o(1)$, which contradicts (5.3). This implies
$b_{j}\to 0^{-}$ and problem $(P_{\lambda})$ has infinitely many solutions. ∎

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