\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2019 (2019), No. 13, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2019 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2019/13\hfil 
Fractional  Kirchhoff-Schr\"odinger-Poisson systems]
{Existence of infinitely many small solutions \\ for sublinear fractional
 Kirchhoff-Schr\"odinger-Poisson systems}

\author[J. C. de Albuquerque, R. Clemente, D. Ferraz \hfil EJDE-2019/13\hfilneg]
{Jos\'e Carlos de Albuquerque, Rodrigo Clemente, Diego Ferraz}

\address{Jos\'e Carlos de Albuquerque \newline
Institute of Mathematics and Statistics,
Federal University of Goi\'as,
74001-970, Goi\^ania, Goi\'as, Brazil}
\email{joserre@gmail.com}

\address{Rodrigo Clemente \newline
Department of Mathematics,
Rural Federal University of Pernambuco,
52171-900, Recife, Pernambuco, Brazil}
\email{rodrigo.clemente@ufrpe.br}

\address{Diego Ferraz \newline
Department of Mathematics,
Federal University of Rio Grande do Norte,
59078-970, Natal, Rio Grande do Norte, Brazil}
\email{diego.ferraz.br@gmail.com}

\thanks{Submitted June 25, 2018. Published January 25, 2019.}
\subjclass[2010]{35A15, 35J50, 35R11, 45G05}
\keywords{Kirchhoff-Schr\"odinger-Poisson equation; fractional Laplacian;
\hfill\break\indent variational method}

\begin{abstract}
 We study the Kirchhoff-Schr\"odinger-Poisson system
 \begin{gather*}
 m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi u = f(x,u), \quad
 x\in\mathbb{R}^3,\\
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3,
 \end{gather*}
 where $[\cdot]_{\alpha}$ denotes the Gagliardo semi-norm,
 $(-\Delta)^{\alpha}$ denotes the fractional Laplacian operator with
 $\alpha,\beta\in (0,1]$, $4\alpha+2\beta\geq 3$ and
 $m:[0,+\infty)\to[0,+\infty)$ is a Kirchhoff function satisfying
 suitable assumptions. The functions $V(x)$ and $k(x)$ are nonnegative and
 the nonlinear term $f(x,s)$ satisfies certain local conditions.
 By using a variational approach, we use a Kajikiya's version of the
 symmetric mountain pass lemma and Moser iteration method to prove the
 existence of infinitely many small solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

In recent years, systems of the form
\begin{equation}\label{16}
\begin{gathered}
-\Delta u+V(x)u+\phi u = f(x,u), \quad x\in\mathbb{R}^3,\\
-\Delta \phi = u^2, \quad x\in\mathbb{R}^3,
\end{gathered}
\end{equation}
have been widely studied by many researchers.
In system \eqref{16}, the first equation is a nonlinear Schr\"{o}dinger
equation in which the potential $\phi$ satisfies a nonlinear Poisson equation.
In this context, it is well known the study of existence of solutions for
system \eqref{16} by using variational methods, under suitable conditions.
For instance, we refer the readers to \cite{ambrosetti_SP,bao,batkam,lu,ruiz}
and the references given there. Particularly, we call attention to the work by
Bao \cite{bao}, where it was studied the existence of infinitely many small
solutions for \eqref{16} with sign-changing potential $V(x)$ and without require
any global growth condition on the nonlinearity $f(x,s)$.

We mention that a great attention has been focused on the study of problems
involving fractional Sobolev spaces and corresponding nonlocal equations,
both from a pure mathematical point of view and their concrete applications.
In fact, fractional Schr\"{o}dinger equations naturally arise in many different
contexts, such as, obstacle problems, flame propagation, minimal surfaces,
conservation laws, financial market, optimization, crystal dislocation,
phase transition and water waves. The literature is quite large, here we
just refer the reader to the important works \cite{guia,caffa} and references
therein.

There are some works concerned with the existence of solutions for the
following class of nonlinear fractional Schr\"odinger-Poisson systems,
 \begin{equation}\label{eq_conversa}
 \begin{gathered}
 (-\Delta)^\alpha u+V(x)u+k(x)\phi u = f(x,u), \quad x\in\mathbb{R}^3,\\
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3,
 \end{gathered}
\end{equation}
where $\alpha,\beta\in (0,1]$. For instance, Liu \cite{liu} studied the
case when $\alpha,\beta\in(0,1)$, $V(x)\equiv 1$, $f(x,u)=|u|^{p-1}u$,
 $k(x)=V(|x|)$ and $1<p<(3+2\alpha)/(3-2\alpha)$.
The author obtained the existence of infinitely many nonradial positive
solutions for \eqref{eq_conversa}, based on Lyapunov-Schmidt reduction.
By considering a general nonlinear term, Li \cite{fsp1}, studied the case when
$k(x), V(x)\equiv1$ and $\alpha,\beta\in (0,1]$ with $4\alpha+2\beta > 3$.
The author has obtained the existence of non-trivial solutions based on the
perturbation method and the mountain pass theorem, supposing that $f(x,s)$
is a subcritical nonlinearity satisfying an Ambrosetti-Rabinowitz type condition,
 precisely, there exists $\mu>4$ such that
\begin{equation}\label{AR_conversa}
 0<\mu F(x,s):=\mu \int_0 ^s f(x,\tau) dx \leq f(x,s)s, \quad \text{for all }
 (x,s) \in \mathbb{R}^N\times \mathbb{R}.
\end{equation}
In a similar fashion, Duarte et al.\ \cite{marco} studied \eqref{eq_conversa}
under more general conditions, where it is assumed a positive potential $V(x)$
is bounded away from zero, and a general autonomous nonlinearity with
$4$-superlinear growth, namely $\inf _{x \in \mathbb{R}^3 }V(x) > 0$,
\begin{equation}\label{AR_conversa_extra}
\lim_{s \to \infty} \frac{F(s)}{s^4} = \infty
\text{ and the function }s \mapsto \frac{f(s)}{|s|^3} \text{ is increasing for }
|s|\neq 0.
\end{equation}
For more works in this direction, we refer the readers to
 \cite{kteng,shangxu,zhang_JM}.
To the best of our knowledge, there are few works concerned with the class
of fractional Schr\"odinger-Poisson equations \eqref{eq_conversa} in the
presence of Kirchhoff term with general $\alpha\in (0,1]$.
 Here we cite \cite{ambrosio2018}, where the author used a minimax type argument
 to prove the existence of a non-trivial solution for a
fractional Kirchhoff-Schr\"odinger-Poisson system in $\mathbb{R}^3$
involving a Berestycki-Lions type nonlinearity.

Motivated by the above discussion, we study the existence of infinitely many
small solutions for the following class of fractional
Kirchhoff-Schr\"{o}dinger-Poisson equations
\begin{equation}\label{01}
 \begin{gathered}
 m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi u = f(x,u), \quad
 x\in\mathbb{R}^3,\\
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3,
 \end{gathered}
\end{equation}
where $\alpha,\beta\in (0,1]$ such that $4\alpha+2\beta\geq 3$ and
$(-\Delta)^{\alpha}$ denotes the fractional Laplacian operator which can be
represented by the singular integral
 \[
 (-\Delta )^{\alpha}u(x)=C(\alpha)\operatorname{P.V.}
\int_{\mathbb{R}^3}\frac{u(x)-u(y)}{|x-y|^{3+2\alpha}}\,\mathrm{d}y,
 \]
for $u$ sufficiently smooth (see \cite{guia}). Henceforth, we omit the
normalization constant $C(\alpha)$. The term
$$
[u]_{\alpha}=\Big(\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\frac{|u(x)-u(y)|^2}{|x-y|^{3+2\alpha}}\,\mathrm{d} x\mathrm{d} y\Big)^{1/2}
$$
is the so-called \textit{Gagliardo semi-norm} of the function $u$.
In Section \ref{s_prelimi} we give more details about the fractional setting.
In the present paper, $k(x)$ and $V(x)$ are nonnegative functions, where
the potential $V(x)$ is locally integrable. In addition, we assume the
following hypotheses:
\begin{itemize}
 \item[(H1)] $k \in L^{r} (\mathbb{R}^3)\cup L^\infty(\mathbb{R}^3)$ such that
 \begin{gather*}
 r > r_\ast := \frac{6}{4\alpha+2\beta-3}, \quad \text{if } 4\alpha+2\beta > 3,\\
 r=r_\ast = \infty, \quad \text{if } 4\alpha+2\beta=3.
 \end{gather*}
 
 \item[(H2)] There exists $\delta_0 >0$ such that for the level set 
$\mathcal{G}_{\delta_0 } := \{ x \in \mathbb{R}^3 : V(x) < \delta_0 \}$,
 we have $0 < |\mathcal{G}_{\delta_0 }| < \infty$, where $|\cdot|$ denotes 
the Lebesgue measure.

 \item[(H3)]For each $\delta > 0 $ and level set 
$\mathcal{G}_{\delta}:= \{ x \in \mathbb{R}^3 : V(x) < \delta\}$, we have 
$0 \leq |\mathcal{G}_{\delta}| < \infty$.


\item[(H4)]  $m(t)\geq m_0>0$, for all $t\in[0,+\infty)$.

\item[(H5)] There exist constants $a_1, a_2>0$ and $t_0>0$ such that for 
some $\sigma\geq0$
 \[
 M(t):=\int_{0}^{t}m(\tau)\,\mathrm{d}\tau\leq a_{1}t+\frac{a_{2}}{2}t^{\sigma+2},
\quad \text{for all } t\leq t_0.
 \]

 \item[(H6)]  $f\in C(\mathbb{R}^3\times [-\delta_{1},\delta_{1}],\mathbb{R})$
 for some $\delta_{1}>0$ and there exist $\nu\in(1,2)$, 
$\mu\in(3/(2\alpha),2/(2-\nu))$ and a nonnegative function 
$\xi\in L^{\mu}(\mathbb{R}^3)$ such that
 \[
 |f(x,s)|\leq \nu \xi(x)|s|^{\nu-1}, \quad \text{for all } 
 (x,s)\in \mathbb{R}^3\times[-\delta_{1},\delta_{1}].
 \]

 \item[(H7)]  There exist $x_{0}\in\mathbb{R}^3$ and a constant $r_{0}>0$ such that
 \begin{gather*}
 \liminf_{s\to0}\Big( \inf_{x\in B_{r_{0}}(x_{0})}\frac{F(x,s)}{s^2}\Big) >-\infty,\\
 \limsup_{s\to0}\Big( \inf_{x\in B_{r_{0}}(x_{0})}\frac{F(x,s)}{s^2}\Big) =+\infty,
 \end{gather*}
 where $F(x,s):=\int_{0}^{s}f(x,\tau)\,\mathrm{d}\tau$.

\item[(H8)]
 There exists $\delta_{2}>0$ such that $f(x,-s)=-f(x,s)$, for all 
$(x,s)\in \mathbb{R}^3\times[-\delta_{2},\delta_{2}]$.
\end{itemize}

From the nature of the problem, it is well known that system \eqref{01} 
can be reduced to a nonlinear Schr\"{o}dinger equation with an additional 
nonlocal term (see Section \ref{s_prelimi}). 
This new term has forth order homogeneity and it is usual to apply 
variational arguments for nonlinearities which behave like $|s|^{p-2}s$, 
for $4<p<2_\alpha ^\ast := 6/(3-2\alpha)$ by considering hypothesis 
\eqref{AR_conversa} or \eqref{AR_conversa_extra} 
(see \cite{chen, marco,fsp1,kteng} 
and the references therein). In order to get the strictly inequality 
$4<2^{*}_{\alpha}$, it is necessary to impose the lower bound $\alpha>3/4$ 
in the fractional Laplacian operator. Differently from this case, and 
similar ones, our assumptions (H6)-(H8) allow the fractional
 parameter $\alpha$ to vary in $(0,1]$ submitted only to condition (H1) 
(see \cite{marco}).

Another interesting feature of our assumptions is that the function $\xi(x)$ 
in (H6) may not be bounded (see Remark \ref{remark} (iii) below).
Thus, the nonlinear term $f(x,s)$ may not be uniformly bounded in $x$. 
For this reason, unlike \cite{bao}, we consider general nonnegative potentials.
 However, we mention that our arguments also permit to consider sign-changing 
potentials provided that $\xi(x)$ is bounded. In fact, in this case, 
under (H2) and(H3), we can assume $\inf _{x \in \mathbb{R}^3 }V(x)> - \infty$ 
and $V_0 >0$ such that $\tilde{V}(x) = V(x)+V_0 >0$, in order to apply 
our approach to the equivalent problem
 \begin{gather*}
 m([u]_{\alpha}^2)(-\Delta)^\alpha u+\tilde{V}(x)u+k(x)\phi u
= f(x,u)+V_0u, \quad x\in\mathbb{R}^3,\\
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3.
 \end{gather*}
In this new framework, it is possible to follow the arguments contained 
in the proof of \cite[Theorem 1.1]{bao} and \cite[Lemma 3.3]{zhou}, 
to get a suitable $L^{\infty}$-estimate, which is an important part of our 
main result. To do this, it is crucial the use of a cut-off type argument 
and the boundedness of $\xi(x)$ to conclude that the truncated nonlinear 
term $f_h(x,s):=(f(x,s)+V_{0}s)h(s)$ is uniformly bounded.
The main result of this work can now be stated as follows.

 \begin{theorem}\label{B}
 Suppose {\rm (H1)--(H8)} hold. Then, system \eqref{01} has infinitely many 
non-trivial solutions $(u_n,\phi_n)_{n\in\mathbb{N}}$ such that
 \[
 \frac{1}{2}M([u_n]_{\alpha}^2)
+\frac{1}{2}\int_{\mathbb{R}^3}V(x)u_n^2\,\mathrm{d}x
+\frac{1}{4}\int_{\mathbb{R}^3}k(x)\phi_{u_n}u_n^2\,\mathrm{d}x
-\int_{\mathbb{R}^3}F(x,u_n)\,\mathrm{d}x\leq0.
 \]
 Moreover, $u_n\to0$ as $n\to+\infty$.
 \end{theorem}

We mention that our result extends some papers in the literature, 
since we are considering a general class of fractional 
Kirchhoff-Schr\"odinger-Poisson systems. 
Precisely, we deal with a class of potentials $V(x)$ under assumptions 
which induce compactness of the corresponding Sobolev embedding, 
the nonnegative term $k(x)$ is bounded or belongs to a suitable 
Lebesgue space and we are assuming that the nonlinear term $f(x,s)$ 
satisfies only local conditions. To prove the existence of infinitely many 
small solutions to ystem \eqref{01}, we use a Kajikiya's version of the 
symmetric mountain pass lemma (see \cite{kaj}). One shall also notice that 
the novelty of our result also provides a regularity type result 
for ystem \eqref{01}, showing that the solutions have a priori $L^{\infty }$-bound 
(see Lemma \ref{it_satan}), which is crucial  to obtain more regularity for 
solutions of elliptic problems involving the fractional Laplacian 
(see \cite{cabre}). For this purpose, we use the $\alpha$-harmonic extension 
jointly with a Moser iteration method. To the best of our knowledge, 
there seems to be no similar results in the current literature for the 
class of equations studied here, even in the local case $\alpha=\beta=1$.

\begin{remark}\label{remark} \rm
Now we give some remarks and examples of functions which satisfy our assumptions:

(i) It is important to mention that the potential considered here may null
 in nonempty interior sets of $\mathbb{R}^3$. This class of potentials is
 somehow inspired by \cite{sira,BW}, where it first appeared for the local case.
 Examples of potentials which satisfy (H2) and (H3) are given by
 $V_1(x) = |x| +1/|x| - 2$ and $V_2(x) = |x|$, if $|x| >1$, and $V_2(x) = 0$, 
if $|x|\leq 1$. We emphasize that our arguments are general and thus, it allow
many other classes of nonnegative potentials whose may go to infinity as 
$|x|\to \infty$ (see the local case \cite{sirakov_outro}).

(ii) A typical example of $m:[0,+\infty)\to[0,+\infty)$ verifying (H4) 
and (H5) is given by
$ m(t)=m_{0}+a_2t$, $a_{2}\geq0$,
which is the one considered in the classical Kirchhoff equation, 
see \cite{kirchhoff}. More generally, the following function
\[
 m(t)=m_{0}+a_2t+ \sum_{i=1}^{k}b_{i}t^{d_{i}},
\]
with $b_{i}\geq 0$ and $d_{i}\in (0,1)$ for all $i\in \{1,2,\ldots, k\}$ satisfies
 assumptions (H4) and (H5).

(iii) One can see that the following function satisfies conditions 
(H6)--(H8). More precisely, consider
\begin{equation*}
F(x,s)=
 \begin{cases}
\xi(x) |s|^{\theta } \sin ^2(|s| ^{\varepsilon }), & \text{if } 
 x = (x_1,x_2,x_3) \in \mathbb{R}^3 \text{ and } 0 < |s| \leq 1,\\
 0, &\text{if }  s=0,
 \end{cases}
\end{equation*}
a primitive of the function $f(x,s)$, where $\xi(x) = |x|^{-d}$ for 
$0<d < 3 / \mu$ if $|x| \leq 1$, and $\xi (x) = 0$, if $|x| >1$. 
We take $\varepsilon >0$ small enough, $\theta \in (1+\varepsilon , 2)$,
 $\delta =1$ and $\nu =\theta - \varepsilon$. Notice that 
$\xi \in L ^\mu (\mathbb{R}^3)$, for $\mu \in (3/2\alpha, 2/(2- \nu))$.
\end{remark}

The remainding of the paper is organized as follows: 
In the forthcoming section we present some preliminary results and we set 
up the variational framework to our problem. 
In Section~\ref{s_canjica}, we prove the existence of a sequence of solutions 
for the modified problem associated to \eqref{01}.
In Section~\ref{sec4}, we introduce the $\alpha$-harmonic extension and we 
apply Moser iteration method in order to prove that our sequence of solutions 
converges to zero in $L^\infty$-norm. Throughout this paper, the symbols 
$C$, $C_1$, $C_2$, \dots  represent several (possibly different) positive 
constants.

\section{Preliminary results}\label{s_prelimi}

In this Section we collect some basic results of fractional Sobolev spaces and 
we introduce the variational framework of system \eqref{01}. For $0<\alpha<1$, 
the fractional Sobolev space is defined as
\[
H^\alpha(\mathbb{R}^3):=\big\{ u\in L^2(\mathbb{R}^3):
\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)-u(y)|^2}{|x-y|^{3+2\alpha}}
\,\mathrm{d}x\,\mathrm{d}y <+\infty\big\}.
\]
For $u,v\in H^{\alpha}(\mathbb{R}^3)$, we define
 \[
 (u,v)_{\alpha}:=\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{3+2\alpha}}\,\mathrm{d}x\,\mathrm{d}y.
 \]
It is well known that $H^{\alpha}(\mathbb{R}^3)$ is a Hilbert space when
endowed with the standard inner product
 \[
 \langle u,v \rangle = (u,v)_{\alpha}+\int_{\mathbb{R}^3}uv\,\mathrm{d}x,
 \]
and the correspondent induced norm
\[
\| u \|_{H^\alpha(\mathbb{R}^3)}
=\big( [u]_{\alpha}^2+\| u \|_{L^2(\mathbb{R}^3)}^2 \big)^{1/2}.
\]
To introduce a variational approach to our problem we define the suitable 
subspace of $H^{\alpha}(\mathbb{R}^3)$,
\[
E:=\big\{u\in H^{\alpha}(\mathbb{R}^3):\int_{\mathbb{R}^3}V(x)u^2<+\infty \big\}.
\]
In view of assumptions (H2) and (H3) it is not hard to check that $E$ 
is a Hilbert space when endowed with the inner product
\[
\langle u,v \rangle_E=(u,v)_{\alpha}+\int_{\mathbb{R}^3}V(x)uv\,\mathrm{d}x,
\]
and the corresponding induced norm $\| u \|^2=\langle u,v \rangle_E$
(see Proposition \ref{p_imerscomp}). For $u\in E$ and a subset 
$\Omega\subset\mathbb{R}^3$ we denote
\[
\|u\|_{\Omega}^2:=\int_{\Omega}\int_{\Omega}
\frac{|u(x)-u(y)|^2}{|x-y|^{3+2\alpha}}\,\mathrm{d}x\,\mathrm{d}y
+\int_{\Omega}V(x)u^2\,\mathrm{d}x.
\]
For any $\beta\in(0,1)$, we recall the homogeneous fractional Sobolev space 
$\mathcal{D}^{\beta,2}(\mathbb{R}^3)$ as
\[
\mathcal{D}^{\beta,2}(\mathbb{R}^3)
=\{ u\in L^{2_\beta^*}(\mathbb{R}^3):[u]_{\beta}<+\infty\},
\]
which is the completion of $C_{0}^{\infty}(\mathbb{R}^3)$ with respect 
to the norm
\[
\| u \|_{\mathcal{D}^{\beta,2} (\mathbb{R}^3)}
=\Big(\int_{\mathbb{R}^3}\vert (-\Delta)^{\beta/2}u \vert^2\,\mathrm{d}x
 \Big)^{1/2}.
\]
We recall that $2^{*}_{\beta}:=6/(3-2\beta)$ is the critical Sobolev exponent 
for $H^{\beta}(\mathbb{R}^3)$.

\begin{lemma}\label{ds2}
 For any $\beta\in(0,1)$, the space $\mathcal{D}^{\beta,2}(\mathbb{R}^3)$ is 
continuously embedded into $L^{2^{*}_{\beta}}(\mathbb{R}^3)$;
 that is, there exists $S_{\beta}>0$ such that
 \[
 \Big(\int_{\mathbb{R}^3}|u|^{2^{*}_{\beta}}\,\mathrm{d}x\Big)^{2/2^{*}_{\beta}}
\leq S_{\beta} \int_{\mathbb{R}^3}|(-\Delta)^{\beta/2}u|^2\,\mathrm{d}x, \quad
\text{for all }   u\in \mathcal{D}^{\beta,2}(\mathbb{R}^3).
 \]
\end{lemma}

For a more information about fractional Sobolev spaces we refer the readers
 to \cite{guia}. Next we prove an embedding result involving our space of functions.

\begin{proposition}\label{p_imerscomp}
 If {\rm (H2)} holds, then $E$ is continuously embeeded into 
$H^{\alpha}(\mathbb{R}^3)$. In addition, if {\rm (H3)} holds, then $E$
 is compactly embedded into $L^p(\mathbb{R}^3)$, for $p\in[2,2^{*}_{\alpha})$.
\end{proposition}

\begin{proof}
 We start by proving that $E \hookrightarrow H^\alpha (\mathbb{R}^3)$, 
i.e., there exists $C>0$ such that 
$\| u \|_{H^\alpha (\mathbb{R}^3) } ^2 \leq C \| u \| ^2$, for all $u \in E$. 
To do that, we use H\"{o}lder inequality and Lemma \ref{ds2} to see that
 \begin{equation*}
 \int_{\mathcal{G}_{\delta_0 }} u^2 \,\mathrm{d}x 
\leq |\mathcal{G}_{\delta_0 }|^{\frac{2\alpha}{N}} S _\alpha [u]^2_\alpha, 
\quad \text{for all } u \in H^\alpha (\mathbb{R}^3).
 \end{equation*}
By using this estimate, assumption (H2) and the fact that $V(x)$ is nonnegative, 
we have that
 \begin{align*}
\|u \|^2 &\geq \frac{1}{2} [u]^2_\alpha
 + \frac{1}{2} |\mathcal{G}_{\delta_0 }|^{-\frac{2\alpha}{N}} 
 S_\alpha ^{-1}\int_{\mathcal{G}_{\delta_0 }} u^2 \,\mathrm{d}x
  + \delta_0 \int_{\mathbb{R}^3 \setminus \mathcal{G}_{\delta_0 } }
  u^2 \, \mathrm{d}x + \int_{\mathcal{G}_{\delta_0 }} V(x) u^2 \,\mathrm{d}x\\
&\geq C \|u \|_{H^{\alpha}(\mathbb{R}^3)}^2,
\end{align*}
 where $C = \min\{1/2, (1/2) |\mathcal{G}_{\delta_0 }|^{-2 \alpha /N}
 S_\alpha ^{-1} , \delta_0 \}, $ which implies the continuous embedding.

 Now we prove the compact embedding $E \lhook\joinrel\twoheadrightarrow
 L^p(\mathbb{R}^3)$, for 
$2\leq p < 2^\ast_\alpha$. Let $(u_n)_{n\in\mathbb{N}} \subset E$ be such that
 $u_n \rightharpoonup u$ weakly in $E$. In view of an interpolation inequality, 
it suffices to show that $u_n \to u$ strongly in $L^2(\mathbb{R}^3)$, up to 
subsequence. To prove this fact, we claim that for any $\varepsilon>0$, 
there exists $R>0$ such that
 \begin{equation}\label{ep_compact}
 \int_{\mathbb{R}^3 \setminus B_R } u^2_n \, \mathrm{d}x
 < \varepsilon, \quad \text{for all }  n \in \mathbb{N}  
 \text{ (uniformly in $n$)},
 \end{equation}
 where $B_R$ denotes the open ball with radius $R$ centered at zero. 
In fact, let us consider $p \in (1, 3/(3-2 \alpha))$ and constants 
$M, \mathcal{C} >0$ satisfying
 \begin{equation*}
 \frac{1}{M} \sup _{n\in\mathbb{N}} \| u_n \|^2 <\frac{\varepsilon}{2}\quad
 \text{and} \quad \sup _{u \in E \setminus \{0 \} } 
\big[\frac{ \| u \|^2 _{L^{2p} (\mathbb{R}^3)}}{\| u \|^2 }\big]\leq \mathcal{C}.
 \end{equation*}
 On the other hand, note that there exits $R>0$ such that
 \begin{equation*}
 | \{ x \in \mathbb{R}^3\setminus B_R : V(x) <M \} | 
\leq [\frac{\varepsilon}{2 \mathcal{C} \sup _{n\in\mathbb{N}} \| u_n \|^2 }]^{p'},
 \quad \text{where }  \frac{1}{p} + \frac{1}{p'} = 1.
 \end{equation*}
 If $A = \{ x \in \mathbb{R}^3 \setminus B_R : V(x) \geq M\}$ and 
$B = \{ x \in \mathbb{R}^3 \setminus B_R : V(x) < M\}$, then one has
\begin{gather*}
 \int_A u^2_n \,\mathrm{d}x \leq \frac{1}{M} \int_{A} V(x) u^2_n \,\mathrm{d}x 
\leq \frac{1}{M} \sup _{n\in\mathbb{N}} \|u_n \|^2 < \frac{\varepsilon}{2}, \\
  \int_B u^2_n \, \mathrm{d}x \leq |B|^{\frac{1}{p'}} \|u_n \|^2 _{L^{2p} (B_R)} 
\leq \mathcal{C} |B|^{\frac{1}{p'}} \sup_{n\in\mathbb{N}} \|u_n\|^2 
< \frac{\varepsilon}{2},
\end{gather*}
which proves \eqref{ep_compact}. 
Let $\theta = \lim _{n \to \infty} \| u _n \|^2_{L^2(\mathbb{R}^3)}$. 
By the semicontinuity of the norm we have $\| u\|^2_{L^2(\mathbb{R}^3)} \leq \theta$. 
On the other hand, using \eqref{ep_compact} and the fact that $u_n \to u$ strongly 
in $L^2(B_R)$, we see that
 \begin{align*}
 \|u\|^2 _{L^2 (\mathbb{R}^3)} 
&= \|u\|^2 _{L^2 (B_R ) } + \|u\|^2 _{ L^2 (\mathbb{R}^3 \setminus B_R ) } \\
& \geq \lim_{n \to \infty} \big[ \| u_n \|^2 _{L^2 (\mathbb{R}^3) } 
 - \|u_n \| _{L^2 (\mathbb{R}^3 \setminus B_R) } \big] \\
& \geq \theta - \varepsilon.
 \end{align*}
 Therefore, $\| u\|^2_{L^2(\mathbb{R}^3)} \geq \theta$, which implies that 
$\| u _n \|^2_{L^2(\mathbb{R}^3)} \to\| u\|^2_{L^2(\mathbb{R}^3)}$, up to a 
subsequence.
\end{proof}

For any $u\in H^\alpha(\mathbb{R}^3)$, let 
$L_u: \mathcal{D}^{\beta,2}(\mathbb{R}^3)\to\mathbb{R}$ be the linear functional 
defined by
\[
L_u(v)=\int_{\mathbb{R}^3}k(x)u^2v\,\mathrm{d}x.
\]
By using (H1), Lemma~\ref{ds2} and H\"older inequality we deduce that
\begin{equation}\label{phi}
|L_u (v)| \leq \begin{cases}
 \|k(x)\|_{L^{\infty}(\mathbb{R}^3)} \|u\|^2_{L^{l_\infty}(\mathbb{R}^3)}
 \|v\|_{L^{2_\beta ^\ast}(\mathbb{R}^3)}, 
& \text{if }  k\in L^\infty(\mathbb{R}^3),\\
 \|k(x)\|_{L^r(\mathbb{R}^3)} \|u\|^2 _{L^{l_r}(\mathbb{R}^3)}
\|v\|_{L^{2_\beta ^\ast}(\mathbb{R}^3)}, & \text{if } 
 k \in L^r (\mathbb{R}^3),
\end{cases}
\end{equation}
for all $v\in\mathcal{D}^{\beta,2}(\mathbb{R}^3)$, where 
$l_\infty = 2\cdot 2_\alpha ^\ast/(2_\alpha ^\ast-1)$ and 
$l_r:=12r/((3+2\alpha)r-6)$. Condition $4\alpha+3\beta\geq 3$ implies that 
$2\leq l_\infty ,l_r \leq 2_{\alpha}^{*}$. It follows from \eqref{phi} that
 $L_{u}$ is continuous. Thus, in light of Lax-Milgram Theorem, there exists a 
unique $\phi_u\in\mathcal{D}^{\beta,2}(\mathbb{R}^3)$ such that
\begin{equation}\label{nonnegative}
\int_{\mathbb{R}^3}(-\Delta)^{\beta/2}\phi_u(-\Delta)^{\beta/2}v\,\mathrm{d}x
=\int_{\mathbb{R}^3}k(x)u^2v\,\mathrm{d}x, \quad \text{for all } 
  v\in\mathcal{D}^{\beta,2}(\mathbb{R}^3),
\end{equation}
that is, $\phi_u$ is a weak solution of the problem
 \[
 (-\Delta)^\beta\phi_u=k(x)u^2, \quad x\in\mathbb{R}^3.
 \]
It is well known that the following representation formula holds
\[
\phi_u(x)=c_{\beta}\int_{\mathbb{R}^3}\frac{k(y)u^2(y)}{|x-y|^{3-2\beta}}\,\mathrm{d}y, \quad \text{for all}   x\in\mathbb{R}^3,
\]
which is called $\beta$-Riesz potential, where
 \[
 c_{\beta}=\frac{\Gamma(3-2\beta)}{\pi^{3/2}2^{2\beta}\Gamma(\beta)}.
 \]

Since we only required local assumptions on the nonlinear term $f(x,s)$, 
we use a cut-off argument similar to the one introduced in \cite{kaj}. 
Let us consider $0<r<(1/2)\min\{ \delta_1,\delta_2,1\}$. 
We define an even function $h\in C^\infty(\mathbb{R},\mathbb{R}^+)$ such that 
$0\leq h(t)\leq 1$, $h(t)=1$ for $|t|\leq r$, $h(t)=0$ for $|t|\geq 2r$ and $h$ 
is decreasing in $[r,2r]$. Let $f_h(x,u)=f(x,u)h(u)$ and 
$F_h(x,u)=\int_{0}^{u}f_h(x,t)\,\mathrm{d}t$. We introduce the modified problem
 \begin{equation}\label{03}
\begin{gathered}
 m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi u = f_{h}(x,u),
\quad x\in\mathbb{R}^3,\\
 (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3.
 \end{gathered} %\tag{$\mathcal{\tilde{P}}_{h}$}
\end{equation}
Replacing $\phi$ by $\phi_u$ in the first equation of \eqref{03},
we obtain the fractional Kirchhoff-Schr\"odinger equation
\begin{equation}\label{15}
m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi_u u
= f_{h}(x,u),\quad x\in\mathbb{R}^3.
\end{equation}
Problem \eqref{15} admits a variational formulation and its solutions are 
the critical points of the energy functional
\[
I_h(u)= \frac{1}{2}M([u]_{\alpha}^2)
+\frac{1}{2}\int_{\mathbb{R}^3}V(x)u^2\,\mathrm{d}x
+\frac{1}{4}\int_{\mathbb{R}^3}k(x)\phi_uu^2\,\mathrm{d}x
-\int_{\mathbb{R}^3}F_h(x,u)\,\mathrm{d}x.
\]
It follows from (H6) that
\begin{equation}\label{growth1}
 |f_{h}(x,s)|\leq \nu\xi(x)|s|^{\nu-1}, \quad \text{for all } 
(x,s)\in\mathbb{R}^3\times\mathbb{R},
\end{equation}
which implies that
\begin{equation}\label{growth2}
 |F_{h}(x,s)|\leq \xi(x)|s|^{\nu}, \quad \text{for all }  
 (x,s)\in\mathbb{R}^3\times\mathbb{R}.
\end{equation}
Let us define $\mu^{*}:=\nu\mu/(\mu-1)$. Since $\nu\in(1,2)$ and 
$\mu\in(3/(2\alpha),2/(2-\nu))$ we have that $\mu^{*}\in(2,2^{*}_{\alpha})$. 
Hence, for any $u\in E$, it follows from \eqref{growth2}, H\"{o}lder 
inequality and Sobolev embedding that
\[
\int_{\mathbb{R}^3}|F_{h}(x,u)|\,\mathrm{d}x
\leq \|\xi\|_{L^{\mu }(\mathbb{R}^3)}\|u \|_{L^{\mu^{*}}(\mathbb{R}^3)}^{\nu}
\leq C(\mu^{*},\nu)\|\xi\|_{L^{\mu }(\mathbb{R}^3)}\|u\|^{\nu}<+\infty.
\]
Therefore, $I_h$ is well defined.

\begin{definition} \rm
 We say that $(u,\phi_u)\in H^\alpha(\mathbb{R}^3)\times 
\mathcal{D}^{\beta,2}(\mathbb{R}^3)$ is a solution of \eqref{03} if $u$ 
is a weak solution of \eqref{15}; that is,
 \begin{align*}
 m([u]_{\alpha}^2)(u,v)_{\alpha} + \int_{\mathbb{R}^3}V(x)uv\,\mathrm{d}x
+\int_{\mathbb{R}^3}k(x)\phi_{u}uv\,\mathrm{d}x 
= \int_{\mathbb{R}^3}f_{h}(x,u)v\,\mathrm{d}x,
 \end{align*}
for all $v\in E$.
\end{definition}

 Note that if $u$ is a critical point of the functional $I_{h}$ and 
$\|u\|_{L^{\infty}(\mathbb{R}^3)}\leq r$, then $u$ is a solution of \eqref{01}.

\section{Kajikiya symmetric mountain pass lemma}\label{s_canjica}

Let $X$ be a Banach space and $\Gamma$ be the family of sets 
$A\subset X\setminus \left\{0\right\}$ which are closed in $X$ and symmetric 
with respect to the origin, i.e. $x\in A$ implies $-x\in A$. For $A\in\Gamma$, 
the genus $\gamma(A)$ is defined as
\[
\gamma(A)=\inf\big\{N\in\mathbb{N}:\exists\psi\in C(A,\mathbb{R}^N
\setminus\{0\})\text{ with }\psi(-z)=-\psi(z), \text{ for all } z\in A\big\}.
\]
If there is no mapping as above for any $N\in\mathbb{N}$, then $\gamma(A)=+\infty$. 
Here we summarize the properties of genus whose will be used in the proof of 
Theorem \ref{B}. A detailed proof can be found in \cite{rabino}.

\begin{proposition}\label{12}
 Let $A,B\in \Gamma\subset X\backslash\{0\}$. Then, the following properties hold:
\begin{itemize}
 \item[(a)] If there is an odd homeomorphism from $A$ to $B$, then 
$\gamma(A)=\gamma(B)$.
 \item[(b)] If $\mathbb{S}^{N-1}$ is the unit sphere in $\mathbb{R}^{N}$, 
then $\gamma(\mathbb{S}^{N-1})=N$.
\end{itemize}
\end{proposition}

\begin{definition} \rm
 Let $X$ be a Banach space, $(u_n)_{n\in\mathbb{N}}\subset X$ be a sequence
and $J:X\to \mathbb{R}$ be a $C^1$ functional. We say that $(u_n)_{n\in\mathbb{N}}$ 
is a Palais-Smale sequence at level $c\in\mathbb{R}$, if
\begin{equation}\label{ps}
 J(u_n)\to c \quad \text{and} \quad J'(u_n)\to 0.
\end{equation}
We say that $J$ satisfies the Palais-Smale condition at level $c\in\mathbb{R}$, 
whenever any Palais-Smale sequence at level $c\in\mathbb{R}$ admits a 
convergent subsequence.
\end{definition}

To prove the existence of infinitely many solutions for system \eqref{01}, 
we use the following version of the symmetric mountain pass lemma which 
is due to Kajikiya, see \cite{kaj}.

\begin{theorem}\label{09}
 Let $X$ be an infinite dimensional Banach space, $\Gamma_k$ be the family of 
closed symmetric subsets $A\subset X$ such that $0\notin A$ and the genus 
$\gamma(A)\geq k$, $J\in C^1(X)$ be an even functional such that $J(0)=0$ and
\begin{itemize}
\item[(H9)] $J$ is bounded from below and satisfies the Palais-Smale condition;

\item[(H10)] For each $k\in \mathbb{N}$, there exists an $A_k\in \Gamma_k$ 
such that $\sup_{u\in A_k}J(u)<0$.
\end{itemize}
Then, $J$ admits a sequence of critical points $(u_n)_{n\in\mathbb{N}}$ 
such that $J(u_n)\leq 0$, $u_n\neq 0$ and $\lim_{n\to +\infty}u_n=0$.
\end{theorem}

In the following, we prove that Palais-Smale sequences for $I_h$ satisfy the 
properties (H9) and (H10) required in Theorem \ref{09}.

\begin{proposition}
 $I_{h}$ is bounded from below.
\end{proposition}

\begin{proof}
 For any $u\in E$ we introduce the set 
$\Omega_{u}:=\{ x\in\mathbb{R}^3: |u(x)|\leq 1\}$. By the definition of $h$ we have
 \[
 \int_{\mathbb{R}^3}F_{h}(x,u)\,\mathrm{d}x
=\int_{\Omega_{u}}F_{h}(x,u)\,\mathrm{d}x.
 \]
 Hence, in view of (H4), \eqref{nonnegative} and \eqref{growth2} it follows that
 \[
 I_{h}(u) \geq \frac{m_{0}}{2}[u]_{\alpha}^2
+\frac{1}{2}\int_{\mathbb{R}^3}V(x)u^2\,\mathrm{d}x
-\int_{\Omega_{u}}\xi(x)|u|^{\nu}\,\mathrm{d}x.
 \]
 Thus, by using H\"{o}lder inequality and Sobolev embedding we obtain
 \begin{equation}\label{rj44}
 I_{h}(u)\geq \min\big\{ \frac{m_{0}}{2},\frac{1}{2}\big\} 
\|u\|_{\Omega_{u}}^2-C(\mu^{*},\nu)\|\xi\|_{L^{\mu }(\mathbb{R}^3)}
\|u\|_{\Omega_{u}}^{\nu}.
 \end{equation}
 Since $\nu\in(1,2)$ we conclude that $I_{h}$ is bounded from below.
\end{proof}

\begin{lemma}\label{02}
 If $(u_n)_{n\in\mathbb{N}}$ is a Palais-Smale sequence for $I_{h}$,
 then $(u_n)_{n\in\mathbb{N}}$ is bounded in $E$.
\end{lemma}
\begin{proof}
It follows from \eqref{ps} and \eqref{rj44} that
 \[
 C\geq I_{h}(u_n)\geq \min\big\{ \frac{m_{0}}{2},\frac{1}{2}\big\}
\|u_n\|_{\Omega_{u_n}}^2-C(\mu^{*},\nu)
\|\xi\|_{L^{\mu }(\mathbb{R}^3)}\|u_n\|_{\Omega_{u_n}}^{\nu},
 \]
 where $\Omega_{u_n}:=\{ x\in\mathbb{R}^3: |u_n(x)|\leq 1\}$.
Since $\nu\in(1,2)$ we conclude that $\|u_n\|_{\Omega_{u_n}}\leq C$,
where $C$ does not depends on $n\in\mathbb{N}$. Moreover, by using 
\eqref{growth2} we deduce that
 \begin{align*}
&\frac{1}{2}\Big[M([u_n]_{\alpha}^2)+\int_{\mathbb{R}^3}V(x)u_n^2
 \,\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}^3}k(x)\phi_{u_n}u_n^2
 \,\mathrm{d}x\Big]\\
&\leq I_{h}(u_n)+C(\mu^{*},\nu)\|\xi\|_{L^{\mu }(\mathbb{R}^3)}
 \|u_n\|_{\Omega_{u_n}}^{\nu}.
\end{align*}
Since $\|u_n\|_{\Omega_{u_n}}\leq C$ and $I_{h}(u_n)\leq C$ we have
 \[
 \frac{1}{2}\Big[M([u_n]_{\alpha}^2)+\int_{\mathbb{R}^3}V(x)u_n^2
 \,\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}^3}k(x)\phi_{u_n}u_n^2\,\mathrm{d}x\Big]
\leq C,
 \]
 where $C$ does not depends on $n\in\mathbb{N}$. The above boundedness together 
with (H4) implies that
 \[
 C \geq \frac{1}{2}\Big[M([u_n]_{\alpha}^2)+\int_{\mathbb{R}^3}V(x)u_n^2
\,\mathrm{d}x\Big]
 \geq \min\big\{ \frac{m_{0}}{2},\frac{1}{2}\big\}\|u_n\|^2,
 \]
 which implies that $(u_n)_{n\in\mathbb{N}}$ is bounded in $E$.
\end{proof}

In view of Proposition \ref{p_imerscomp} and Lemma \ref{02} we may assume, 
up to a subsequence, that
\begin{gather*}
u_n\rightharpoonup u \quad \text{weakly in }  E; \\
u_n\to u \quad \text{strongly in }  L^{p}(\mathbb{R}^3),  \text{ for } 
 p\in[2,2^{*}_{\alpha}); \\
u_n(x)\to u(x) \quad \text{almost everywhere in }  \mathbb{R}^3. 
\end{gather*}
By using generalized H\"{o}lder inequality we deduce the following convergences:
 \begin{equation*}
 \big|\int_{\mathbb{R}^3}k(x)\phi_{u_n}u_n(u_n-u)\,\mathrm{d}x\big|
\leq \begin{cases} 
\|k\|_{L^{\infty}}\|\phi_{u_n}\|_{L^{2^{*}_{\beta}}}\|u_n\|_{L^{l _\infty}}
 \|u_n-u \|_{L^{l _\infty}}\to 0,\\ 
\|k\|_{L^{r}}\|\phi_{u_n}\|_{L^{2^{*}_{\beta}}}\|u_n\|_{L^{l_r} }
 \|u_n-u\|_{L^{l_r} }\to 0,
 \end{cases}
 \end{equation*}
as $n\to+\infty$. Thus, we conclude that
 \begin{equation}\label{conv1}
 \lim_{n\to+\infty}\int_{\mathbb{R}^3}k(x)(\phi_{u_n}u_n-\phi_{u}u)(u_n-u)
\,\mathrm{d}x=0.
 \end{equation}
Moreover, by using \eqref{growth1} and generalized H\"{o}lder inequality 
we obtain the estimate
\begin{align*}
 &\Big|\int_{\mathbb{R}^3}(f_{h}(x,u_n)-f_{h}(x,u))(u_n-u)\,\mathrm{d}x\Big|\\
 &\leq \nu(\|u_n\|_{L^{\mu^{*}}(\mathbb{R}^3)}^{\nu-1}+\|u\|_{L^{\mu^{*}}
(\mathbb{R}^3)}^{\nu-1})\|\xi\|_{L^{\mu}(\mathbb{R}^3)}\|u_n-u\|_{L^{\mu^{*}}
(\mathbb{R}^3)},
 \end{align*}
which together with the fact that $\mu^{*}\in(2,2^{*}_{\alpha})$ and 
Proposition \ref{p_imerscomp} implies that
 \begin{equation}\label{conv2}
 \lim_{n\to+\infty}\int_{\mathbb{R}^3}(f_{h}(x,u_n)-f_{h}(x,u))(u_n-u)\,\mathrm{d}x=0.
 \end{equation}

\begin{proposition}
 $I_{h}$ satisfies the Palais-Smale condition.
\end{proposition}

\begin{proof}
 It follows from \eqref{ps} and the weak convergence that
 \begin{equation}\label{rj0}
 \langle I'(u_n)-I'(u),u_n-u\rangle=o_n(1),
 \end{equation}
where $o_n$ denotes the standard ``little o notation''. On the other hand we have
\begin{align*}
&\langle I'(u_n)-I'(u),u_n-u\rangle\\
 &=m([u_n]_{\alpha}^2)(u_n,u_n-u)_{\alpha}-m([u]_{\alpha}^2)
(u,u_n-u)_{\alpha}+\int_{\mathbb{R}^3}V(x)(u_n-u)^2\,\mathrm{d}x\\
 &\quad +\int_{\mathbb{R}^3}k(x)(\phi_{u_n}u_n-\phi_{u}u)(u_n-u)\,\mathrm{d}x
 -\int_{\mathbb{R}^3}(f_{h}(x,u_n)-f_{h}(x,u))(u_n-u)\,\mathrm{d}x,
 \end{align*}
which together with \eqref{conv1} and \eqref{conv2} implies that
 \begin{equation}\label{rj1}
 \begin{aligned}
\langle I'(u_n)-I'(u),u_n-u\rangle
&=m([u_n]_{\alpha}^2)(u_n,u_n-u)_{\alpha}
 -m([u]_{\alpha}^2)(u,u_n-u)_{\alpha}\\
&\quad +\int_{\mathbb{R}^3}V(x)(u_n-u)^2\,\mathrm{d}x+o_n(1).
 \end{aligned}
 \end{equation}
Notice that
 \begin{equation}\label{rj2}
 \begin{aligned}
&m([u_n]_{\alpha}^2)(u_n,u_n-u)_{\alpha}-m([u]_{\alpha}^2)(u,u_n-u)_{\alpha}\\
&=m([u_n]_{\alpha}^2)[u_n-u]_{\alpha}^2+\left(m([u_n]_{\alpha}^2)
-m([u]_{\alpha}^2) \right) (u,u_n-u)_{\alpha}.
\end{aligned}
 \end{equation}
Since $(u_n)_{n\in\mathbb{N}}$ is bounded in $E$ and $m$ is a continuous 
function, there exists $A\geq 0$ such that $m([u_n]_{\alpha}^2)\to m(A)$.
In particular, $(m([u_n]_{\alpha}^2))_{n\in\mathbb{N}}$ is bounded.
Thus, by weak convergence one has
 \begin{equation}\label{rj3}
 \left( m([u_n]_{\alpha}^2)-m([u]_{\alpha}^2) \right) (u,u_n-u)_{\alpha}=o_n(1).
 \end{equation}
It follows from (H4), \eqref{rj2} and \eqref{rj3} that
 \begin{equation}\label{rj4}
 m([u_n]_{\alpha}^2)(u_n,u_n-u)_{\alpha}-m([u]_{\alpha}^2)(u,u_n-u)_{\alpha}
\geq m_{0}[u_n-u]_{\alpha}^2+o_n(1).
 \end{equation}
Combining \eqref{rj0}, \eqref{rj1} and \eqref{rj4} we conclude that
 \begin{equation*}
 o_n(1) \geq \min\{m_{0},1\}\|u_n-u\|^2+o_n(1).
 \end{equation*}
Therefore, $u_n\to u$ strongly in $E$ which completes the proof. 
\end{proof}

\begin{lemma}
There exists a sequence of non-trivial critical points $(u_n)_{n\in\mathbb{N}}$ 
for $I_h$.
\end{lemma}

\begin{proof}
The idea is essentially due to \cite[Theorem 2]{kaj} but for the reader's 
convenience we provide the proof here. For simplicity, we assume that 
$x_0=0$ in (H7), that is, there exists a constant $r_{0}>0$ such that
 \[
 \liminf_{s\to0}\Big( \inf_{x\in B_{r_{0}}}\frac{F(x,s)}{s^2}\Big) >-\infty
 \quad  \text{and}  \quad
 \limsup_{s\to0}\Big( \inf_{x\in B_{r_{0}}}\frac{F(x,s)}{s^2}\Big) =+\infty.
 \]
In the following we denote
\[
\mathcal{C}:=\big\{ (x_1,x_2,x_3)\in\mathbb{R}^3
:-\frac{r_0}{2}\leq x_i\leq\frac{r_0}{2}\text{, where }1\leq i\leq 3 \big\}.
\]
By (H7), there exist constants $\vartheta,\epsilon>0$ and two sequences
of positive numbers $\vartheta_n\to 0$ and $M_n\to +\infty$ as $n\to +\infty$ 
such that
\begin{gather}\label{04}
F(x,u)\geq -\epsilon u^2,\quad \text{for all }   x\in\mathcal{C}\text{ and }
|u|\leq \vartheta, \\
\label{07}
\frac{F(x,\delta_n)}{\vartheta_n^2}\geq M_n,\quad \text{for all } 
  x\in\mathcal{C}\text{ and }n\in\mathbb{N}.
\end{gather}
Fix $k\in\mathbb{N}$ arbitrarily and let $p\in\mathbb{N}$ be the smallest 
integer satisfying $p^3\geq k$. We divide $\mathcal{C}$ equally into $p^3$ 
cubes by planes parallel to each face of $\mathcal{C}$ and we denote them 
by $\mathcal{C}_i$, with $1\leq i\leq p^3$. Thus, the edge of each $\mathcal{C}_i$ 
has the length of $a=r_0/p$. For each $1\leq i\leq k$, we make a cube 
$\tilde{\mathcal{C}_i}\subset\mathcal{C}_i$ such that $\tilde{\mathcal{C}_i}$ 
has the same center as that of $\mathcal{C}_i$, the faces of $\tilde{\mathcal{C}_i}$
 and $\mathcal{C}_i$ are parallel and the edge of $\tilde{\mathcal{C}_i}$ 
has the length of $a/2$. Now, we define a continuous function 
$\rho:\mathbb{R}\to\mathbb{R}$ such that
\begin{gather*}
\rho(t)=0 \quad \text{for } t\in\mathbb{R}\setminus [-\frac{a}{2},\frac{a}{2}],\\
\rho(t)=1 \quad \text{for } t\in [-\frac{a}{4},\frac{a}{4}],\\
0\leq\rho(t)\leq 1 \quad \text{for } t\in (-\frac{a}{2},-\frac{a}{4})
\cup(\frac{a}{4},\frac{a}{2}).
\end{gather*}
Define $\eta_{1}:\mathbb{R}^3\to\mathbb{R}$ such that 
$\eta_{1}(x)=\rho(x_1)\rho(x_2)\rho(x_3)$. For each $1\leq i\leq k$, 
let $y_i$ be the center of $\tilde{\mathcal{C}_i}$ and set 
$\eta_{1_i}(x)=\eta_{1}(x-y_i)$ for all $x\in\mathbb{R}^3$. 
It is easy to check that, for each $1\leq i\leq k$, 
$0\leq \eta_{1_i}(x)\leq 1$ for all $x\in\mathbb{R}^3$, 
$\operatorname{supp}\eta_{1_i}\subset\mathcal{C}_i$ and 
$\eta_{1_i}(x)=1$ if $x\in\tilde{\mathcal{C}_i}$. Set
\begin{gather*}
\mathcal{V}_k=\big\{ (t_1,\dots,t_k)\in\mathbb{R}^k:\max_{1\leq i\leq k}|t_i|=1 
\big\}, \\
\mathcal{W}_k=\big\{ \sum_{i=1}^{k}t_i\eta_{1_i}:(t_1,\dots,t_k)\in\mathcal{V}_k 
\big\}.
\end{gather*}
Since $V_k$ is the surface of the k-dimensional cube, it is homeomorphic to
 the sphere $\mathbb{S}^{k-1}$ by an odd mapping. By Proposition \ref{12} 
we have $\gamma(V_k)=k$. If we define the mapping 
$\zeta:\mathcal{V}_k\to\mathcal{W}_k$ by
\[
\zeta(t_1,\dots,t_k)=\sum_{i=1}^{k}t_i\eta_{1_i},
\]
then $\zeta$ is an odd homeomorphism between $\mathcal{V}_k$ and
 $\mathcal{W}_k$, which implies that $\gamma(V_k)=\gamma(W_k)$. 
Since $\mathcal{W}_k$ is compact, there exists a constant $C_k>0$ 
such that $\| u \|\leq C_k$ for all $u\in\mathcal{W}_k$.
Thus, using (H5) and \eqref{phi}, for any 
$\beta\in(0,\min\{\vartheta,t_0/C_k\})$ and
$u=\sum_{i=1}^{k}t_i\eta_{1_i}\in\mathcal{W}_k$ we have
\begin{equation}\label{05}
\begin{aligned}
I_{h}(\beta u)
&=\frac{1}{2}M([\beta u]_{\alpha}^2)
 +\frac{\beta^2}{2}\int_{\mathbb{R}^3}V(x)u^2\,\mathrm{d}x\\
&\quad +\frac{\beta^2}{4}\int_{\mathbb{R}^3}k(x)\phi_uu^2\,\mathrm{d}x
 -\int_{\mathbb{R}^3}F_h\Big(x,\beta\sum_{i=1}^{k}t_i\eta_{1_i}\Big)\,\mathrm{d}x\\
&\leq \beta^2C_1 \| u\|^2+\frac{a_{2}}{2}\beta^{\sigma+2}
 \| u\|^{\sigma+2}\\
&\quad +\beta^2C_2 \|\phi_{u}\|_{\mathcal{D}^{\beta,2}(\mathbb{R}^3)}\|u\|^2
 -\sum_{i=1}^{k}\int_{\mathcal{C}_i}F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x.
\end{aligned}
\end{equation}
On the other hand, by the definition of $\mathcal{V}_k$, there exists some 
integer $1\leq i_u\leq k$ such that $|t_{i_u}|=1$. Then
\begin{align*}
&\sum_{i=1}^{k}\int_{\mathcal{C}_i}F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x \\
&=\int_{\tilde{\mathcal{C}}_{i_u}}F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x
 +\int_{\mathcal{C}_{i_u}\setminus\tilde{\mathcal{C}}_{i_u}}
 F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x
 +\sum_{i\neq i_u}\int_{\mathcal{C}_i}F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x.
\end{align*}
Observe that by \eqref{04},
\begin{equation}\label{06}
\int_{\mathcal{C}_{i_u}\setminus\tilde{\mathcal{C}}_{i_u}}
F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x
+\sum_{i\neq i_u}\int_{\mathcal{C}_i}F_h(x,\beta t_i\eta_{1_i})\,\mathrm{d}x
\geq -\epsilon r_{0}^3\beta^2,
\end{equation}
where we used  that the volume of $\mathcal{C}$ is $r_0^3$.
 We have $|\vartheta_nt_{i_u}\eta_{1_{i_u}}(x)|=\vartheta_n$ for all 
$x\in\tilde{\mathcal{C}_{i_u}}$ and the volume of $\tilde{\mathcal{C}_{i_u}}$ 
is $a^3/8$ . Since $\vartheta_n\to 0$, we assume that there exists
 $n_0\in\mathbb{N}$ such that $\vartheta_n< \min\{\vartheta,t_0/C_k\}$ 
for all $n\geq n_0$. Thus, using \eqref{07}, \eqref{05} and \eqref{06} with 
$\beta=\vartheta_n$ we obtain
\begin{equation}\label{08}
\begin{aligned}
I(u_n)&\leq C_3\vartheta_n^2\left(\| u\|^2
 +\vartheta_n^{\sigma}\| u\|^{\sigma+2}
 +\|\phi_{u}\|_{\mathcal{D}^{\beta,2}(\mathbb{R}^3)}\|u\|^2
 +\epsilon r_{0}^3\vartheta_n^2\right)\\
&\quad -\int_{\tilde{\mathcal{C}}_{i_u}}F_h(x,\vartheta_nt_i\eta_{1_i})\,\mathrm{d}x\\
&\leq C_3\vartheta_n^2\Big(\| u\|^2
 +\vartheta_n^{\sigma}\| u\|^{\sigma+2}
 +\|\phi_{u}\|_{\mathcal{D}^{\beta,2}(\mathbb{R}^3)}\|u\|^2
 +\epsilon r_{0}^3\vartheta_n^2-\frac{a^3M_n}{8}\Big),
\end{aligned}
\end{equation}
where $u_n = \vartheta_n u$. Since $u\in W_k$, one has
\[
\| \phi_u \|_{\mathcal{D}^{\beta,2}(\mathbb{R}^3)}
\leq C_4\| u \|\leq C_4C_k.
\]
Since $\vartheta_n\to 0$ and $M_n\to +\infty$ as $n\to +\infty$,
 we can choose $n\in\mathbb{N}$ large enough such that
\[
C_k^2+\vartheta_n^{\sigma}C_k^{\sigma+2}+C_4C_k^3
+\epsilon r_{0}^3\vartheta_n^2-\frac{a^3M_n}{8}<0.
\]
This implies that the right-hand side of \eqref{08} is negative.
 To complete the proof, we define
\[
A_k=\left\{ \vartheta_{n_0}u:u\in\mathcal{W}_k \right\}.
\]
Thus $\gamma(A_k)=\gamma(\mathcal{W}_k)=k$ and $\sup_{u\in A_k}I_h(u,\phi_u)<0$. 
Thus, all the conditions of Theorem \ref{09} are satisfied. Therefore, 
there exists a sequence of non-trivial critical points $(u_n)_{n\in\mathbb{N}}$ 
for $I_h$.
\end{proof}

\section{Moser iteration method}\label{sec4}

In this section, we focus our analysis for the case $0<\alpha<1$, since the 
local case $\alpha=1$ can be treated similarly as \cite{bao,zhou}. 
For the reader's convenience, before we prove our regularity result, we 
introduce some preliminary concepts about the $\alpha$-harmonic extension 
(see \cite{caf_silv}). We point out that our arguments are local and, 
for this reason, we are able to apply this technique to transform our nonlocal 
problem into a local one.

For $0<\alpha<1$ we define the space $X^\alpha$ as the completion of 
$C^\infty_0(\mathbb{R}^{4}_+)$ with respect to the norm
\begin{equation*}
\| w \|_{X^\alpha} = \Big[ \frac{1}{\kappa _\alpha} 
\int_{\mathbb{R}^{4}_+}y^{1-2\alpha} | \nabla w |^2\,\mathrm{d}x \,\mathrm{d}y
\Big] ^{1/2},
\end{equation*}
where $\kappa_\alpha = (2 ^{1-2\alpha} \Gamma(1 - \alpha)) / \Gamma (\alpha)$ 
and $\Gamma$ is the well known gamma function. By \cite{tracoext}, the space 
$X^\alpha$ is well defined and there is a continuous trace operator 
$\operatorname{Tr} : X^\alpha \to \mathcal{D}^{\alpha,2}(\mathbb{R}^3)$;
 that is, there exists $C>0$ such that 
$\|\operatorname{Tr}(w)\|_{\mathcal{D}^{\alpha,2}(\mathbb{R}^3)} 
\leq C \|w\|_{X^\alpha}$, for all $w \in X^\alpha$.
 When $w\in C(\overline{\mathbb{R}^{4}_+ })$, we have 
 $\operatorname{Tr}(w)(x) = w(x,0 )$, and because of that we also use 
the notation $w(\cdot ,0) = \operatorname{Tr}(w)$. It is also worth to call 
attention that considering the continuous Sobolev embedding 
$ \mathcal{D}^{\alpha,2}(\mathbb{R}^3) \hookrightarrow L ^{2^\ast_\alpha} 
(\mathbb{R}^3)$, we obtain that 
$\|w(\cdot , 0 )\|_{2\alpha ^\ast} \leq C \|w\|_{X^\alpha}$, for all 
$w \in X^\alpha$.

Given $u \in \mathcal{D}^{\alpha,2}(\mathbb{R}^3)$, we call 
$w = E_{\alpha } (u)$ the $\alpha$-harmonic extension of $u$, the unique 
solution of the minimization problem
\begin{equation*}
\min \Big\{ \frac{1}{\kappa_{\alpha}}\int_{\mathbb{R}_+^{4}}y^{1-2\alpha} 
| \nabla w |^2 \,\mathrm{d}x\mathrm{d}y : w \in X^\alpha\text{ and } 
w(\cdot ,0) = u \text{ on }\mathbb{R}^3\Big\} .
\end{equation*}
We have that $E_{\alpha}$ is a well defined operator acting on 
$\mathcal{D}^{\alpha,2}(\mathbb{R}^3)$ into $X^\alpha$. 
Moreover, by \cite[Lemma A.2]{olocomeu}, $E_{\alpha}$ is an isometry, precisely 
$\| E_\alpha (u) \|_{X^\alpha} = \| u \| _{\mathcal{D}^{\alpha,2}(\mathbb{R}^3)}$, 
for all $u \in \mathcal{D}^{\alpha,2}(\mathbb{R}^3)$. We also have that 
$E_{\alpha}$ satisfies
\begin{gather*}
\operatorname{div} (y^{1-2\alpha } \nabla w )  = 0 \quad \text{in } 
 \mathbb{R}^{4} _+, \\
-\frac{1}{\kappa _\alpha}\lim _{y \to 0 ^+} y^{1-2\alpha} w_y (x,y) 
 = (- \Delta) ^\alpha u (x) \quad \text{in }  \mathbb{R}^3,
\end{gather*}
in the weak sense, more precisely
\[
\frac{1}{\kappa _\alpha}\int_{\mathbb{R}^{4}_+} y^{1-2\alpha} 
\langle \nabla E_\alpha (u) , \nabla \psi \rangle \mathrm{d}x\mathrm{d}y 
= \int_{\mathbb{R}^3} (-\Delta) ^{ \alpha /2} u (-\Delta) ^{ \alpha /2} 
\psi(\cdot, 0)\,\mathrm{d}x,
\]
for all $\psi \in X^{\alpha}$. Consequently we see that $u$ is a weak solution 
for \eqref{15} if, and only if, $w = E_\alpha (u)$ is a weak solution for
 the problem
\begin{equation}\label{17}
\begin{gathered}
\operatorname{div} (y^{1-2\alpha} \nabla w ) = 0 \quad \text{in } \mathbb{R}^{4}_+,\\
-\frac{1}{\kappa _\alpha}\lim _{y \to 0 ^+} y^{1-2\alpha} w_y (x,y) 
= g(x,u(x)) \quad \text{in }  \mathbb{R}^3,
\end{gathered} 
\end{equation}
where $g(x,u) = f_{h}(x,u) - V(x) u - k(x) \phi _{u} u$; that is,
\begin{equation*}
\frac{1}{\kappa_{\alpha}}\int_{\mathbb{R}^{4}_+} y^{1-2\alpha} 
\langle \nabla E_\alpha (u) , \nabla \psi \rangle \mathrm{d}x\mathrm{d}y 
= \int_{\mathbb{R}^3} (f_{h}(x,u) - V(x) u - k(x) \phi_{u} u) \psi(\cdot, 0) 
\,\mathrm{d}x,
\end{equation*}
for all $\psi \in X^{\alpha}$. In the following lemma, we show that a
 sequence of critical points of Problem \eqref{17} converges to zero in 
the $L^\infty$-norm. Our proof is based on the Moser iteration method, 
a delicate estimate which take into account the $\alpha$-harmonic extension 
and a suitable interpolation of Lebesgue spaces.

\begin{lemma}\label{it_satan}
Let $(u_n)_{n\in\mathbb{N}}$ be a critical point sequence of $I_h$ satisfying 
$u_n\to 0$ in E, as $n\to +\infty$. Then, $\|u_n\|_{L^{\infty}(\mathbb{R}^3)}\to 0$ 
as $n\to +\infty$.
\end{lemma}

\begin{proof}
We first recall that $w$ is a weak solution to \eqref{17} if $w$ satisfies 
the equality
\begin{equation}\label{18}
\frac{m([ w(\cdot,0) ]_{\alpha}^2)}{\kappa_\alpha}
\int_{\mathbb{R}_{+}^{4}} y^{1-2\alpha}\langle \nabla w , \nabla \psi \rangle 
\,\mathrm{d}x\mathrm{d}y
=\int_{\mathbb{R}^3}g(x,w(\cdot,0))\psi\,\mathrm{d}x,
\end{equation}
for any $\psi\in X^\alpha$. We set $w=E_\alpha (u_n)$, $u = u_n = w(\cdot, 0 )$ 
and $g(x,w(\cdot,0)) = f_{h}(x,u) - V(x) u - k(x) \phi _{u} u$. 
For each $L>0$ we define $w_L:=\min\left\{ w,L \right\}$ and consider 
$\psi:=w_{L}^{2\theta}w\in X^{\alpha}$, where $\theta>0$ will be chosen later. 
By using $\psi$ as test function in \eqref{18} we obtain
\begin{equation}\label{19}
\begin{aligned}
&\frac{m([u]_{\alpha}^2)}{\kappa_\alpha}\Big[\int_{\mathbb{R}_{+}^{4}}
 y^{1-2\alpha}w_{L}^{2\theta}|\nabla w|^2\,\mathrm{d}x\mathrm{d}y
 +\int_{\{ w\leq L\}}2\theta y^{1-2\alpha}w_{L}^{2\theta}|\nabla w|^2
 \,\mathrm{d}x\mathrm{d}y \Big]\\
&=\int_{\mathbb{R}^3}f_{h}(x,u)u_{L}^{2\theta}u\,\mathrm{d}x
-\int_{\mathbb{R}^3}V(x)u^2u_{L}^{2\theta}\,\mathrm{d}x 
- \int_{\mathbb{R}^3}k(x)\phi_u u^2u_{L}^{2\theta}\,\mathrm{d}x.
\end{aligned}
\end{equation}
Taking into account \eqref{growth1}, \eqref{19} and using H\"older inequality 
we deduce that
\begin{equation}\label{20}
\begin{aligned}
 \frac{m_0}{\kappa_\alpha}\int_{\mathbb{R}_{+}^{4}}y^{1-2\alpha}
 w_{L}^{2\theta}|\nabla w|^2\,\mathrm{d}x\mathrm{d}y
 &\leq \nu\int_{\mathbb{R}^3}\xi(x)u^{\nu}u_{L}^{2\theta}\,\mathrm{d}x\\
 &\leq\nu\| \xi \|_{L^{\mu}(\mathbb{R}^3)}\| u^\nu u_L^{2\theta}
 \|_{ L^\frac{\mu}{\mu-1}(\mathbb{R}^3) }^{2\theta+\nu}.
\end{aligned}
\end{equation}
Let us denote $\overline{w}_L=w w_L^{\theta}$. 
Following \cite[Lemma 4.1]{mamamia},  one has
\begin{equation}\label{21}
\| \overline{w}_L(\cdot,0) \|_{L^{2_{\alpha}^{*}}(\mathbb{R}^3)}^2
\leq 4 S_\alpha(\theta+1)^2\int_{\mathbb{R}^{4}}y^{1-2\alpha}w_{L}^{2\theta}
|\nabla w|^2\,\mathrm{d}x\mathrm{d}y.
\end{equation}
 Using \eqref{20} and \eqref{21} we deduce that
 \begin{equation}\label{22}
\| \overline{w}_L(\cdot,0) \|_{L^{2_{\alpha}^{*}}(\mathbb{R}^3)}^2
\leq C(\theta+1)^2\| \xi \|_{L^{\mu}(\mathbb{R}^3)}
\| u^\nu u_L^{2\theta} \|_{ L^\frac{\mu}{\mu-1}(\mathbb{R}^3) }^{2\theta+\nu}.
 \end{equation}
Now, by passing to the limit as $L\to +\infty$ in \eqref{22},  Fatou's Lemma yields
\begin{equation}\label{23}
\| u \|_{L^{(\theta+1)2_{\alpha}^{*}}(\mathbb{R}^3)}
\leq C^\frac{1}{(\theta+1)}(\theta+1)^\frac{1}{(\theta+1)}
\| u \|_{ L^{\alpha_{*}}(\mathbb{R}^3) }^{\frac{2\theta+\nu}{2(\theta+1)}},
\end{equation}
where $\alpha_*=\mu(2\theta+\nu)/(\mu-1)$. For each $n\in\mathbb{N}$, define 
$(\theta_{n-1}+1)2_{\alpha}^{*}=\mu(2\theta_n+\nu)/(\mu-1)$. 
Since $\mu>3/(2\alpha)$, it follows that $\theta_n$ is positive, increasing 
and unbounded. Thus, set
\[
\zeta_n=\sum_{i=0}^{n-1}\frac{\ln(c_0(\beta_i+1))}{\beta_i+1}\quad \text{and}\quad
\sigma_n=\prod_{i=0}^{n-1}\frac{2\beta_i+\nu}{2\beta_i+2}.
\]
Notice that $\zeta_n$ and $\sigma_n$ are convergent sequences 
(see also \cite[Lemma 3.4]{ps}) with $\zeta_n\to \zeta>0$ and 
$\sigma_n\to \sigma \in (0,1)$. We can now iterate \eqref{23} to obtain
\begin{equation}\label{24}
\| u \|_{L^{\mu(2\beta_n+\nu)/(\mu-1)}(\mathbb{R}^3)}
\leq \mathrm{e}^{\zeta_n}\| u \|_{L^{\mu^*}(\mathbb{R}^3)}^{\sigma_n}, 
\quad\text{for all }n\in\mathbb{N}.
\end{equation}
Letting $n\to +\infty$ in \eqref{24} follows 
$\| u \|_{L^{\infty}(\mathbb{R}^3)}
\leq \mathrm{e}^\zeta\| u \|_{L^{\mu^*}(\mathbb{R}^3)}^{\sigma}$.
 Therefore, $u_n\to 0$ strongly in $L^\infty(\mathbb{R}^3)$ as $n\to +\infty$, 
which completes the proof.
\end{proof}

\begin{remark} \rm
Note that for the local case $\alpha=1$, estimate \eqref{22} can be directly
 obtained by the continous Sobolev embedding
 $H^1 (\mathbb{R}^3) \hookrightarrow L^6(\mathbb{R}^3)$.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{B}]
We now look back to the modified problem \eqref{03}. In Section \ref{s_canjica}, 
we  applied Theorem \ref{09} to guarantee the existence of a sequence 
$(u_n)_{n\in\mathbb{N}}$ of critical points for the functional $I_h$. 
Hence, in view of Lemma \ref{it_satan}, there exists $n_{0}\in\mathbb{N}$ 
such that $(u_n,\phi_{u_n})$ is a solution for \eqref{01}, for all $n\geq n_{0}$, 
from which the assertions of Theorem~\ref{B} follows.
\end{proof}

\subsection*{Acknowledgments}
This research was partially supported by the National Institute of Science
and Technology of Mathematics INCT-Mat, CAPES and CNPq.


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\end{document}