\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 76, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/76\hfil 
Schr\"{o}dinger equations involving the fractional Laplacian]
{Existence of standing waves for Schr\"{o}dinger equations involving
the fractional Laplacian}

\author[E. de Medeiros, J. A. Cardoso, M. de Souza \hfil EJDE-2017/76\hfilneg]
{Everaldo S. de Medeiros, Jose Anderson Cardoso, Manass\'{e}s de Souza}

\address{Everaldo S. de Medeiros \newline
Departamento de Matem\'{a}tica,
Universidade Federal da Para\'{\i}ba, 58051-900, 
Jo\~{a}o Pessoa, PB, Brazil}
\email{everaldomedeiros1@gmail.com}

\address{Jose Anderson Cardoso \newline
Departamento de Matem\'{a}tica,
 Universidade Federal de Sergipe, 49000-100, \newline
S\~ao Crist\'ov\~ao, Brazil}
\email{anderson@mat.ufs.br}

\address{Manass\'{e}s de Souza \newline
Departamento de Matem\'{a}tica,
Universidade Federal da Para\'{\i}ba, 58051-900,
Jo\~{a}o Pessoa, PB, Brazil}
\email{manassesxavier@gmail.com}

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted September 16, 2016. Published March 20, 2017.}
\subjclass[2010]{35J20, 35J60, 35R11}
\keywords{Variational methods; critical points; fractional Laplacian}

\begin{abstract}
 We study a class of fractional Schr\"{o}dinger equations of the form
 \[
 \varepsilon^{2\alpha}(-\Delta)^\alpha u+ V(x)u = f(x,u)
 \quad\text{in } \mathbb{R}^N,
 \]
 where $\varepsilon$ is a positive parameter, $0 < \alpha < 1$,
 $2\alpha < N$, $(-\Delta)^\alpha$ is the fractional Laplacian,
 $V:\mathbb{R}^{N}\to \mathbb{R}$ is a potential which may be
 bounded or unbounded and the nonlinearity
 $f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}$ is superlinear and
 behaves like $|u|^{p-2}u$ at infinity for some $2<p< 2^*_\alpha:=2N/(N-2\alpha)$.
 Here we use a variational approach based on the Caffarelli and Silvestre's
 extension developed in \cite{Caffarelli} to obtain a nontrivial solution
 for $\varepsilon$ sufficiently small.
\end{abstract}

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

\section{Introduction}

In this work we are concerned with the existence of standing waves
for a nonlinear differential equation directed by the fractional Laplacian. We
focus on the so-called \emph{fractional Schr\"{o}dinger equation}
\begin{equation}\label{EI1}
i \varepsilon \frac{\partial \Psi}{\partial t} =\varepsilon^{2
\alpha} (-\Delta)^\alpha \Psi + (V(x)+E) \Psi - f(x,\Psi),\,\,\,
(x,t) \in \mathbb{R}^N \times \mathbb{R},
\end{equation}
where $\varepsilon >0$ is a fixed parameter, $E$ is a real constant,
$V$ and $f$ are continuous functions, $0 < \alpha <1$ and
$(-\Delta)^\alpha$ denotes the fractional Laplacian, defined for all
function belongs to the Schwartz space, by
\begin{equation}\label{EI1122}
\widehat{(-\Delta)^\alpha u} (\xi) = |\xi|^{2 \alpha} \widehat{u}(\xi),
\end{equation}
where $\widehat{u}$ denotes the Fourier transform of $u$.
It is worth mentioning that \eqref{EI1} comes from an
expansion of the Feynman path integral from Brownian-like to
L\'{e}vy-like quantum mechanical paths (see \cite{Laskin2} for details).
When $\alpha = 1$ the L\'{e}vy dynamics becomes the Brownian dynamics,
and \eqref{EI1} reduces to the classical Schr\"{o}dinger equation
\[
i \varepsilon \frac{\partial \Psi}{\partial t}
=-\varepsilon^{2 \alpha} \Delta \Psi + (V(x)+E) \Psi - f(x,\Psi),\quad
 (x,t) \in \mathbb{R}^N \times \mathbb{R}.
\]
This equation has been widely investigated by many authors in the last
decades (see, for instance \cite{Rabinowith,Sirakov2002} and references
therein). Standing waves solutions to \eqref{EI1} are
solutions of the form $\Psi(x,t) = u(x) \exp(-i E t)$, where $u$
solves the elliptic equation
\begin{equation}\label{P}
\varepsilon^{2\alpha}(-\Delta)^\alpha u + V(x) u = f(x,u)
\quad \text{in } \mathbb{R}^N.
\end{equation}
Recently, several papers have been performed for classical
elliptic equations involving the fractional Laplacian. In
the sequel, we will list some of them related with the existence
of solutions to \eqref{P} that may be found in the
literature. Using the Nehari variational principle, Cheng \cite{Ming} proved
the existence of a nontrivial solution for the fractional
Schr\"{o}dinger \eqref{P} if $f(x,u)=|u|^{q-2}u$ with
$2<q< 2^*_\alpha$ if $N>2\alpha$ or $2<q < \infty$ if
$N\leq 2\alpha$, where $2_\alpha^*:=2N/(N-2\alpha)$ is the critical
Sobolev exponent. Ground states are found by imposing a coercivity
assumption on $V(x)$,
\begin{equation}\label{HCV}
\lim_{|x| \to +\infty} V(x) = +\infty.
\end{equation}
Applying the method of \cite{Felmer1}, Secchi \cite{Simone} proved
the existence of a ground state under less restrictive
assumptions on $f(x,u)$. It is worthwhile to remark that in \cite{Ming} and
\cite{Simone} the hypothesis \eqref{HCV} is assumed on $V(x)$ in
order to overcome the problem of lack of compactness, typical of
elliptic problems defined in unbounded domains. In
\cite{DiPierro}, Dipierro et al.\ considered the existence
of radially symmetric solutions of \eqref{P} in the situation
where $V(x)$ does not depend explicitly on the space variable $x$.
For the first time, using rearrangement tools and following the
ideas of Berestycki and Lions \cite{BerestyckiLions}, the authors
proved the existence of a nontrivial, radially symmetric solution
to
\[
(-\Delta)^{\alpha}u+u= |u|^{q-2}u \quad \text{in } \mathbb{R}^N,
\]
where $2<q< 2^*_\alpha$ if $N>2\alpha$ or $2<q < \infty$ if $N\leq 2\alpha$.
We would also like to mention that problems involving the existence and
concentration of positive solution to \eqref{P} have been investigated
 by \cite{ALVESMIYAGAKI2016,Chen2014,Davila2015,Davila2014}
when $V$ is positive and $\varepsilon$ is sufficient small.

Here, motivated by the papers
\cite{ALVESMIYAGAKI2016,Ming,Chen2014,Davila2015,Davila2014,DiPierro,
Simone,Sirakov2002},
our main goal is to study the existence of solutions for
 \eqref{P} when $0 < \alpha < 1$, $N > 2\alpha$,
$V:\mathbb{R}^{N}\to \mathbb{R}$ and
$f:\mathbb{R}^{N}\times \mathbb{R}\to \mathbb{R}$ are
continuous functions, $\varepsilon$ is a positive parameter, $V(x)$ is a
nonnegative continuous function satisfying some conditions. More
precisely we assume that $V$ and $f$ satisfy the following assumptions:
\begin{itemize}
\item[(A1)] the set $\mathcal{Z}=\{x\in \mathbb{R}^N: V(x)=0\}$ is nonempty;
\item[(A2)] there exists $A>0$ such that the level set
 $$
 G_A=\{x\in \mathbb{R}^N: V(x)<A\}
 $$
has finite Lesbegue measure;

\item[(A3)] $f(x,s)=o(|s|)$, as $s\to 0$, uniformly in $\mathbb{R}^N$;

\item[(A4)] there exists a constant $C> 0$ such that
$$
| f(x, s)| \leq C(1 + |s|^p),
$$
uniformly in $\mathbb{R}^N$, for all $s \in \mathbb{R}$, for some
$1<p<2_\alpha^*-1$, where $2_\alpha^*:=2N/(N-2\alpha)$;

 \item[(A5)] there exists a constant $\mu \in (2, p+1 ]$ such
 that $$0 < \mu F(x, s) \leq s f(x,s),$$
for all $x �\in \mathbb{R}^N$ and $s \in \mathbb{R} \backslash
\{0\}$; here, as usual, $F(x, s): = \int_0^s f(x, s) ds$.
\end{itemize}
A typical potentials satisfying (A1) and (A2) is
$$
V(x)=\frac{|x-x_0|^\beta}{1+\lambda|x-x_0|}, \quad
\beta\geq1, \; \lambda\geq 0.
$$
Our main result can be summarized as follows.

\begin{theorem} \label{Theo1}
Suppose that {\rm (A1)--(A6)} hold. Then there exists $\varepsilon_0>0$
such that \eqref{P} has a nontrivial and nonnegative solution for
all $\varepsilon\in(0,\varepsilon_0]$.
\end{theorem}

\begin{remark} \label{rmk1} \rm
Our hypotheses on the potential $V$ are inspired in
\cite{Sirakov2002} and the variational approach in the proof of
Theorem~\ref{Theo1} is based on the Caffarelli and Silvestre's
extension developed in \cite{Caffarelli}. We also point
out that the results of this work complement
\cite{ALVESMIYAGAKI2016,Chen2014,Ming,Davila2014,DiPierro,Simone}
in the sense that the potential $V(x)$
belongs to a different class from those treated by them.
To underline the role played by the potential $V(x)$, we suggest
to the reader the papers \cite{Davila2015,Davila2014}.
\end{remark}


This work is organized as follows. In Section \ref{section1}
we gather few notation and definitions. In Section \ref{variatSett}
we make the variational framework to study the geometric properties
and the Palais-Smale sequences of the associated functional. Finally, in Section
\ref{section6} we prove Theorem~\ref{Theo1}.

\section{Notation and definitions}\label{section1}

We recall that the homogeneous Sobolev
$\dot{H}^{\alpha}(\mathbb{R}^{N})$ is defined as the
completion of $C_0^{\infty}(\mathbb{R}^{N})$ with respect to the norm
\[
\|u\|_{\dot{H}^{\alpha}}^2:= \int_{\mathbb{R}^{N}}|2\pi
\xi|^{2\alpha}|\widehat{u}(\xi)|^2 \mathrm{d} \xi =
\int_{\mathbb{R}^{N}}|(-\Delta)^{\alpha/2}u|^2\mathrm{d} x.
\]
For our setting we also consider the space
$X^{2\alpha}(\mathbb{R}_{+}^{N+1})$ defined as the completion of
$C_0^{\infty}(\overline{\mathbb{R}_{+}^{N+1}})$ with respect to the norm
\[
\|w\|_{X^{2\alpha}}^2:=
\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\mathrm{d} x \mathrm{d} y,
\]
where
$\kappa_{\alpha}=2^{1-2\alpha}\Gamma(1-\alpha)/\Gamma(\alpha)$ and
$\mathbb{R}^{N+1}_{+}=\{(x,y)\in \mathbb{R}^N \times \mathbb{R}: y>0\}$.

In \cite{ref13doO}, it is proved that the extension operator
$E_{2\alpha}: \dot{H}^{\alpha}(\mathbb{R}^{N})
\to X^{2\alpha}(\mathbb{R}_{+}^{N+1})$ is well defined. Moreover, for any
$\phi\in X^{2\alpha}(\mathbb{R}_{+}^{N+1})$ if we denote its trace on
$\mathbb{R}^{N}\times \{y=0\}$ as $\phi(x,0)$, there exist
$\mathcal{S}_1, \mathcal{S}_2 >0$ such that
(see \cite[Lemmas 2.2 and 2.3]{ref13doO} for details)
\begin{equation} \label{desdotraco}
\mathcal{S}_1\|\phi(\cdot,0)\|_{2^*_\alpha}\leq
\|\phi(\cdot,0)\|_{\dot{H}^{\alpha}(\mathbb{R}^{N})}
\leq \mathcal{S}_2 \|\phi\|_{X^{2\alpha}(\mathbb{R}_{+}^{N+1})}.
\end{equation}

Given $u\in \dot{H}^{\alpha}(\mathbb{R}^{N})$ we say that
$w=E_{2\alpha}(u)$ is the $\alpha$-harmonic extension of $u$ to
the upper half-space $\mathbb{R}_{+}^{N+1}$, if $w$ is a solution of the
problem
\begin{equation}\label{1.5}
\begin{gathered}
-\operatorname{div}(y^{1-2\alpha}\nabla w)=0 \quad\text{in}\quad
\mathbb{R}_{+}^{N+1},
\\
w=u \quad\text{in } \mathbb{R}^{N}\times\{0\}.
\end{gathered}
\end{equation}
Furthermore, in \cite{Caffarelli} it is proved that
\begin{equation}\label{estrela}
\lim_{y\to 0^{+}}
y^{1-2\alpha}w_{y}(x,y)=-\frac{1}{\kappa_{\alpha}}(-\Delta)^{\alpha}u(x).
\end{equation}
We would like to recall that using the change of
variable $v(x) = u(\varepsilon x)$, Equation \eqref{P} is equivalent
to the problem
\begin{equation}\label{prob1.61}
(-\Delta)^\alpha u+ V(\varepsilon x)u = f(\varepsilon x,u)
\quad\text{in } \mathbb{R}^N.
\end{equation}
Thus it is sufficient to consider \eqref{prob1.61} instead \eqref{P}.
Furthermore, from \eqref{1.5} and \eqref{estrela}, we may
consider the problem
\begin{equation}\label{prob1.6}
\begin{gathered}
-\operatorname{div}(y^{1-2\alpha}\nabla w)=0 \quad \text{in } \mathbb{R}_{+}^{N+1},\\
-\kappa_{\alpha}\frac{\partial w}{\partial \nu}=-V(\varepsilon x)u
+f(\varepsilon x,u) \quad \text{in } \mathbb{R}^{N}\times \{0\},
\end{gathered}
\end{equation}
where $\frac{\partial w}{\partial \nu}= \underset{y\to 0^{+}}{\lim }
y^{1-2\alpha}w_{y}(x,y)$. To obtain a weak solution to \eqref{prob1.6}, by using
variational methods, we will consider the following subspace of
$X^{2\alpha}(\mathbb{R}_{+}^{N+1})$:
\[
X_\varepsilon:=\big\{w\in
X^{2\alpha}(\mathbb{R}_{+}^{N+1}):\int_{\mathbb{R}^{N}}V(\varepsilon
x)w(x,0)^2\mathrm{d} x < \infty \big\},
\]
endowed with the inner product
\[
\langle w,v\rangle_\varepsilon
=\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}\nabla
w\nabla v\,\mathrm{d} x\mathrm{d} y+ \int_{\mathbb{R}^{N}}V(\varepsilon x)w(x,0)v(x,0)\,\mathrm{d} x
\]
and the induced norm $\|w\|_\varepsilon=\langle w,w\rangle^{1/2}$
(see Lemma \ref{lem11}).


Throughout this work we say that $w \in X_\varepsilon$ is a weak
solution to \eqref{prob1.6} if for any $\varphi \in X_\varepsilon$
\begin{align*}
&\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}\nabla
w\nabla \varphi\,\mathrm{d} x\mathrm{d} y+ \int_{\mathbb{R}^{N}}V(\varepsilon
x)w(x,0)\varphi(x,0)\,\mathrm{d} x \\
&-\int_{\mathbb{R}^{N}}f(\varepsilon
x,w(x,0))\varphi(x,0)\mathrm{d} x=0,
\end{align*}
and consequently $u = w(x,0) \in H^\alpha(\mathbb{R}^N)$ is a weak
solution to \eqref{prob1.61} (see \cite{Caffarelli}). Here
$H^\alpha(\mathbb{R}^N)$ stands to the fractional Sobolev space
\[
H^\alpha(\mathbb{R}^N):=\{u\in L^2(\mathbb{R}^N): \|(-\Delta)^\alpha
u\|^2_2 + \|u\|_2^2<\infty\},
\]
endowed with the norm $\|u\|_{H^\alpha}=(\|(-\Delta)^\alpha
u\|^2_2+\|u\|_2^2)^{1/2}$. We also recall that the imbedding
$H^\alpha(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)$ is
continuous for any $q\in[2,2_\alpha^*]$ (see Proposition 3.6 in
\cite{Di Nezza}).

\begin{remark} \label{rmk2} \rm
We point out that if $u = w(x,0) \in H^\alpha(\mathbb{R}^N)$ is a weak
solution to \eqref{prob1.61}, then it is possible
to get the pointwise expression of the fractional Laplacian as in \eqref{EI1122}.
 For details, the reader may see the paper \cite{ServadeiValdinoci2014},
which addresses regularity results of weak solutions and viscosity
solutions of the fractional Laplace equation.
\end{remark}

\section{Variational Setting} \label{variatSett}

Our first lemma enables us to settle the
variational setting.

\begin{lemma}\label{lem11}
Suppose condition {\rm (A2)} holds. Then, for each $0< \varepsilon < 1$
there exists $C=C(\varepsilon) >0$ such that
\begin{equation}\label{imersaoXemLq}
\|w(x,0)\|_{q} \leq C \|w\|_\varepsilon,\quad
\text{for all $w \in X_\varepsilon$ and }2\leq q \leq 2_{\alpha}^{*}.
\end{equation}
\end{lemma}


\begin{proof}
 First we show that there exists $\tau_1 >0$ such that
\begin{equation}
\begin{aligned}\label{Eita10}
&\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\,\mathrm{d} x\mathrm{d} y + \int_{\mathbb{R}^{N}}V(x)|w(x,0)|^2\,\mathrm{d} x \\
&\geq \tau_1 \Big(\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\,\mathrm{d} x\mathrm{d} y + \|w(x,0)\|_2^2 \Big),
\end{aligned}
\end{equation}
for every $w$ for which the quantity in the left-hand side of \eqref{Eita10}
is finite. Indeed, since $G_A$ has finite measure, there exists
$C_2=C_2(|G_A|,\alpha, N)>0$ such that
\begin{align*}
&\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\,\mathrm{d} x\mathrm{d} y + \int_{\mathbb{R}^{N}}V(
x)|w(x,0)|^2\,\mathrm{d} x \\
& \geq \frac{1}{2}\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}
 y^{1-2\alpha}|\nabla w|^2\,\mathrm{d} x\mathrm{d} y + C_2 \int_{G_{A}}|w(x,0)|^2 \mathrm{d} x
+ A \int_{\mathbb{R}^N \setminus G_{A}} |w(x,0)|^2\, \mathrm{d} x\\
&\geq \min\{1/2,C_2,A\}\Big(\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}
y^{1-2\alpha}|\nabla w|^2\,\mathrm{d} x\mathrm{d} y + \|w(x,0)\|_2^2 \Big),
\end{align*}
where we used the H\"{o}lder and Sobolev inequalities, which give
\[
\int_{G_{A}}|w(x,0)|^2 \mathrm{d} x \leq
|G_{A}|^{2\alpha/N}\|w(x,0)\|^2_{2_\alpha^*}\leq
C_{3}
\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\,\mathrm{d} x\mathrm{d} y.
\]
Take $w \in X_\varepsilon$ and put $v(x,y) = w(x/\varepsilon,y)$.
Then from \eqref{Eita10},
\begin{align*}
\|w\|_\varepsilon
& = \varepsilon^{-n} \Big(\int_{\mathbb{R}_{+}^{N+1}}
\varepsilon^2\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
v|^2\,\mathrm{d} x\mathrm{d} y + \int_{\mathbb{R}^{N}}V(
x)|v(x,0)|^2\,\mathrm{d} x \Big)\\
& \geq \varepsilon^{-n +2} \Big(\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
v|^2\,\mathrm{d} x\mathrm{d} y + \int_{\mathbb{R}^{N}}V(
x)|v(x,0)|^2\,\mathrm{d} x \Big)\\
& \geq \varepsilon^{-n +2}\tau_1\Big(\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
v|^2\,\mathrm{d} x\mathrm{d} y + \|v(x,0)\|_2^2 \Big)\\
& = \varepsilon^2\tau_1 \Big(\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}|\nabla
w|^2\,\mathrm{d} x\mathrm{d} y + \|w(x,0)\|_2^2 \Big).
\end{align*}
Thus, from \eqref{desdotraco}
\[
\|w\|_\varepsilon \geq C_1(\varepsilon) \|w(x,0)\|_{H^\alpha}^2.
\]
This together with the Sobolev imbedding
$H^\alpha(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)$ imply
the desired result.
\end{proof}

It follows by Lemma \ref{lem11}, (A2)--(A4), that the functional
$$
I_\varepsilon(w)=\frac{1}{2}\|w\|_\varepsilon^2-\int_{\mathbb{R}^{N}}F(\varepsilon
x,w(x,0))\mathrm{d} x
$$
is well defined in $X_\varepsilon$ and belongs to
$C^1(X_\varepsilon,\mathbb{R})$, with G\^{a}teaux derivative given by
\begin{align*} % \label{derivadaP}
I_\varepsilon'(w)\cdot \varphi
&=\int_{\mathbb{R}_{+}^{N+1}}\kappa_{\alpha}\,y^{1-2\alpha}\nabla
w\nabla \varphi\,\mathrm{d} x\mathrm{d} y+ \int_{\mathbb{R}^{N}}V(\varepsilon
x)w(x,0)\varphi(x,0)\,\mathrm{d} x \\
&\quad -\int_{\mathbb{R}^{N}}f(\varepsilon x,w(x,0))\varphi(x,0)\mathrm{d} x,
\end{align*}
for any $\varphi\in C_0^{\infty}(\overline{\mathbb{R}_{+}^{N+1}})$.
Thus critical points of $I_\varepsilon$ are weak solutions to
\eqref{prob1.6} and reciprocally.
The functional $I_\varepsilon$ satisfies the following geometric
properties.


\begin{lemma}\label{Positivelevel-a}
Suppose that {\rm (A2)--(A5)} hold. Then
\begin{itemize}
 \item[(i)] there exist
$\delta,\rho>0$ such that $I_\varepsilon(w)\geq\delta$ if
 $\|w\|_{\varepsilon}=\rho$;
\item[(ii)] there exists $e \in X_\varepsilon$ such that
$\|e\|_{\varepsilon} > \rho$ and $I_\varepsilon(e) < 0$.
\end{itemize}
\end{lemma}

\begin{proof} By (A3) and (A4), given $\epsilon>0$, there exists
$C_{\epsilon}>0$ such that
\begin{equation} \label{estimativaG}
\int_{\mathbb{R}^{N}}|F(\varepsilon x,w(x,0))|\,\mathrm{d} x\leq
\frac{\epsilon}{2}\int_{\mathbb{R}^{N}}|w(x,0)|^2\mathrm{d}
x+\frac{C_{\epsilon}}{p+1}\int_{\mathbb{R}^{N}}|w(x,0)|^{p+1}\mathrm{d}
x,
\end{equation}
for all $ w \in X_\varepsilon$.
This and the imbedding \eqref{imersaoXemLq} imply that
there exist positive constants $C_1$ and $C_2$ such that
\[
I_\varepsilon(w) \geq \Big(\frac{1}{2}-\frac{\epsilon C_1
}{2}\Big)\rho^2- C_2\rho^{p+1}\quad\text{if }\\|w\|_\varepsilon=\rho.
\]
Since $p+1 >2$, we  choose $\epsilon, \rho >0$ sufficiently
small such that
$\delta:= \inf_{\|w\|_\varepsilon= \rho} I_\varepsilon(w) >0$,
which proves (i). To prove (ii), we consider
 $\varphi\in C_0^{\infty}(\mathbb{R}^{N+1}_{+},\mathbb{R}_{+})$ such that
$\varphi(x,0)\not \equiv 0$. Then by (A4) and (A5) there exists a positive
function $d(x) \in L^\infty(\mathbb{R}^N)$ such that
\begin{equation}\label{AR}
F(x,s) \geq d(x) |s|^\mu, \quad \text{for all } ( x,s)\in \mathbb{R}^{N}
\times \mathbb{R}.
\end{equation}
Thus for $t>0$, we get that $I_\varepsilon(t\varphi) \to - \infty$, as
$t \to +\infty$.
Setting $e=t\varphi$ with $t$ large enough, the condition (ii) is satisfied.
\end{proof}

From Lemma \ref{Positivelevel-a}, for each $\varepsilon>0$
the minimax level
$$
c_\varepsilon:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}
I_\varepsilon(\gamma(t))
$$
is positive, where $\Gamma:=\{\gamma\in C([0,1], X_\varepsilon);
\gamma(0)=0\text{ and } \gamma(1) = e\}$.

\begin{lemma}\label{SmallLevel}
 If $1 < p < 2^*_\alpha-1$, it holds
$\lim_{\varepsilon \to 0^+} c_\varepsilon= 0$.
 \end{lemma}

\begin{proof} Without loss of generality we may assume that $0 \in \mathcal{Z}$.
 Note that by \eqref{AR}, for all $w\in X_\varepsilon$ there exists $C_0 >0$
such that
 \begin{equation*}
 c_\varepsilon\leq\max_{t\geq0} I_\varepsilon(t w)
\leq C_0\bigg(\frac{ \int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla w|^2\mathrm{d} x\mathrm{d} y
+\int_{\mathbb{R}^N} V(\varepsilon x)w(x,0)^2 \, \mathrm{d} x}{(\int_{\mathbb{R}^N}
d(\varepsilon x) |w(x,0)|^\mu\, \mathrm{d} x)^
{2/\mu}}\bigg)^{\mu/(\mu-2)}.
 \end{equation*}
Defining
${\mathcal{M}_\varepsilon}:=\big\{w \in
X_\varepsilon:\int_{\mathbb{R}^N}d(\varepsilon x) |w(x,0)|^\mu\,
\mathrm{d} x=1\big\}$
and
\[
\bar{c}_\varepsilon:=\inf_{w\in \mathcal{M}_\varepsilon}
\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla w|^2\mathrm{d} x\mathrm{d} y
+\int_{\mathbb{R}^N} V(\varepsilon x)w(x,0)^2 \, \mathrm{d} x.
\]
We obtain that
\begin{equation} \label{prime}
c_\varepsilon\leq C_0 \bar{c}_\varepsilon^{\mu/(\mu-2)}.
\end{equation}
We claim that $\bar{c}_\varepsilon\to 0$ as
$\varepsilon\to 0^+$. Suppose by contradiction that for
some sequence $\varepsilon_n\to 0^+$, 
$ \bar{c}_{\varepsilon_n}\geq c_0>0$ for all $n\in \mathbb{N}$.
 Since $\mu \leq p+1< 2^*_\alpha$, it is known that (see
\cite[Theorem 4]{Zhuo2014} and \cite{Caffarelli})
\begin{equation}\label{Sirakov Idea}
\inf_{\substack{
w\in C^{\infty}_0(\mathbb{R}_{+}^{N+1})\\
\int_{\mathbb{R}^N}|w(x,0)|^{\mu}\, \mathrm{d} x=1}}
\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla w|^2\,\mathrm{d}\,x\mathrm{d} y=0.
\end{equation}
Thus we can take a sequence $(w_n)\subset C^{\infty}_0(\mathbb{R}_{+}^{N+1})$ 
such that
\begin{equation}\label{CaffZero}
\lim_{n\to \infty}
\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla w_n|^2\mathrm{d} x\mathrm{d}
y=0 \quad \text{and} \quad 
\int_{\mathbb{R}^N}|w_n(x,0)|^{\mu}\, \mathrm{d} x=1.
\end{equation}
Defining
\[
v_n = \frac{w_n}{ (\int_{\mathbb{R}^N}d(\varepsilon
x)|w_n(x,0)|^{\mu}\, \mathrm{d} x )^{1/\mu}},
\]
we have that $v_n \in \mathcal{M}_{\varepsilon}$. For simplicity we
suppose that $d(0)=1$. By using \eqref{CaffZero} we see that for
each $n \in \mathbb{N}$, we can find $\varepsilon_n > 0$ such that
\begin{equation}\label{eqt0}
\int_{\mathbb{R}^N}d(\varepsilon x)|w_n(x,0)|^{\mu}\, \mathrm{d} x > \frac{1}{2},
\end{equation}
for $\varepsilon < \varepsilon_n$. Thus, for every $n$ and every
$$
\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla v_n|^2\mathrm{d} x\mathrm{d} y
\leq 2^{2/\mu}\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla
w_n|^2\mathrm{d} x\mathrm{d} y.
$$
From \eqref{CaffZero}, we can find $n_0$ such that for every
$\varepsilon < \varepsilon_{n_0}$,
\begin{equation}\label{eqT1}
\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla v_{n_0}|^2\mathrm{d}
x\mathrm{d} y < \frac{c_0}{2}.
\end{equation}
On the other hand, for all $n \in \mathbb{N}$ and $\varepsilon>0$
we have
\begin{equation}\label{eqt2}
c_0 \leq \int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla
v_{n}|^2\mathrm{d} x\mathrm{d} y + \int_{\mathbb{R}^N} V(\varepsilon x)v_n(x,0)^2\, \mathrm{d} x.
\end{equation}
Hence for $\varepsilon < \varepsilon_{n_0}$, from \eqref{eqt0},
\eqref{eqT1} and \eqref{eqt2} we have
\begin{align*}
\int_{\mathbb{R}^N} V(\varepsilon x)|w_{n_0}(x,0)|^2\, \mathrm{d} x 
&\geq \frac{\int_{\mathbb{R}^N} V(\varepsilon x) |w_{n_0}(x,0)|^2\, \mathrm{d}
x}{2^{2/\mu}(\int_{\mathbb{R}^N} d(\varepsilon
x)|w_{n_0}(x,0)|^{\mu}\, \mathrm{d} x )^{2/\mu}}\\ 
&= \frac{1}{2^{2/\mu}} \int_{\mathbb{R}^N} V(\varepsilon
x)|v_{n_0}(x,0)|^2\, \mathrm{d} x 
\geq \frac{c_0}{2^{\frac{2}{\mu}+1}},
\end{align*}
which contradicts hypothesis (A1). Therefore,
$\bar{c}_\varepsilon \to 0$ as $\varepsilon \to
0^+$,
 which combined with \eqref{prime} completes the proof.
\end{proof}

Combining Lemma \ref{Positivelevel-a} and the Ekeland's variational principle,
 we obtain a sequence $(w_n)\subset X_\varepsilon$ such that
\begin{equation}\label{SequenPS}
I_\varepsilon(w_n)\to c_\varepsilon\quad\text{and}\quad
\|I_\varepsilon'(w_n)\|_{*}\to 0 \quad\text{as } n \to
+\infty.
\end{equation}

\begin{lemma}\label{limita} 
If $(w_n)\subset X_\varepsilon$ is a sequence satisfying \eqref{SequenPS}, 
then there exists a positive constant $\nu>0$ such that
\[
\limsup_{n\to \infty} \|w_n\|^2_{\varepsilon}\leq \nu
c_{\varepsilon}\quad \text{for all }\varepsilon>0.
\]
\end{lemma}

\begin{proof} 
Because of \eqref{SequenPS} and (A5) we have
$$
\big(\frac{\mu}{2}-1\big) \|w_n\|^2_{\varepsilon} 
\leq \mu I_\varepsilon(w_n) - I_\varepsilon'(w_n)(w_n)\\
 \leq \mu c_\varepsilon + o_n(1) + o_n(1) \|w_n\|_{\varepsilon},
$$
which implies that $(w_n)$ is bounded in $X_\varepsilon$ and the
lemma follows easily. In the last inequality we used $o_n(1)$ to
denotes a quantity that tends to zero as $n\to\infty$.
\end{proof}

By Lemma \ref{lem11}, we may assume that, up to a subsequence,
$w_{n}\rightharpoonup w_0$ weakly in $X_\varepsilon$,
$w_{n}(x,0)\to w_0(x,0)$ in $L_{loc}^{q}(\mathbb{R}^N)$
for all $2\leq q < 2^*_\alpha,$ and 
$w_{n}(x,0)\to w_0(x,0)$ almost everywhere in $\mathbb{R}^N$. 
Taking the limit in \eqref{SequenPS} we obtain that $w_0$ is a weak solution of
\eqref{prob1.6}. To prove that $w_0$ is not trivial we need the
following result.

\begin{lemma}\label{limitaOut} 
Let $\varepsilon>0 $ and $(w_n)$ a sequence satisfying \eqref{SequenPS}. 
Then there exist positive constants $\eta=\eta(A, \mathcal{S}_2, \nu)>0$ 
and $R=R(\varepsilon)>0$ such that
$$
\limsup_{n\to \infty} \|w_n(x,0)\|^2_{H^\alpha(\mathbb{R}^N \setminus B_R)} \leq
\eta\,c_{\varepsilon}.
$$
\end{lemma}

\begin{proof} 
From \eqref{desdotraco}, we have 
\[
\|w_n(\cdot,0)\|_{\dot{H}^{\alpha}(\mathbb{R}^N \setminus B_R)}
\leq \mathcal{S}_2 \|w_n\|_{X^{2\alpha}(\mathbb{R}_{+}^{N+1})}.
\]
Thus, by Lemma \ref{limita}, it is sufficient to show that for each 
$\varepsilon>0$ there exist
positive constants $C=C(A)$ and $R=R(\varepsilon)$ such that
\begin{equation}\label{CrucEst}
\int_{|x|>R}|w_n(x,0)|^2\mathrm{d} x\leq C\|w_n\|^2_\varepsilon.
\end{equation}
We denote by $B_R=\{x\in \mathbb{R}^N: |x|\leq R\}$ the closed
ball of radius $R$ centered at the origin, $B^c_R$ the complement
of $B_R$, $G_{A,\varepsilon}=\{ x\in \mathbb{R}^N: V(\varepsilon
x)<A\}$ and $v_n(x,y)=w_n(x/\varepsilon ,y)$.
Since $G_{A}$ has finite measure, by using the
H\"{o}lder and Sobolev inequalities, we obtain 
$C_1= C(\mathcal{S}_1,\mathcal{ S}_2) >0$ such that
\begin{align*}
\int_{B_R^c\cap G_{A,\varepsilon}}|w_n(x,0)|^2\mathrm{d} x
&=\varepsilon^{-N}\int_{B_{\varepsilon R}^c\cap G_{A}}|v_n(x,0)|^2\mathrm{d} x \\
&\leq\varepsilon^{-N} |B_{\varepsilon R}^c\cap G_{A}|^{2\alpha/N}
\Big(\int_{\mathbb{R}^N}|v_n(x,0)|^{2^*_\alpha}\mathrm{d} x\Big)^{{2}/{2^*_\alpha}}\\
&\leq C_1\varepsilon^{-N} |B_{\varepsilon R}^c\cap G_{A}|^{2\alpha/N}
 \int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla v_n|^2\mathrm{d} x\mathrm{d} y\\
&= C_1 \varepsilon^{-2} |B_{\varepsilon R}^c\cap
G_{A}|^{2\alpha/N}\int_{\mathbb{R}_{+}^{N+1}}\,y^{1-2\alpha}|\nabla
w_n|^2\mathrm{d} x\mathrm{d} y,
\end{align*}
and we have that, for each $\varepsilon>0$, we can find a radius
$R=R(\varepsilon)>0$ such that
\[
|B_{\varepsilon R}^c\cap G_{A}|^{2\alpha/N}< \varepsilon^2 C_1^{-1} A^{-1}.
\] 
Thus
\begin{equation}\label{EK1}
\int_{B_R^c\cap G_{A,\varepsilon}}|w_n(x,0)|^2\mathrm{d} x\leq \frac{1}{A}
\|w_n\|^2_{\varepsilon}.
\end{equation}
Moreover,
\begin{equation}\label{KE2}
\int_{B_R^c\setminus G_{A,\varepsilon}}|w_n(x,0)|^2\mathrm{d} x\leq
\frac{1}{A}\int_{B_R^c\setminus G_{A,\varepsilon}}V(\varepsilon x)w_n^2(x,0)\mathrm{d}
x\leq \frac{1}{A}\|w_n\|^2_{\varepsilon}.
\end{equation}
Combining \eqref{EK1} and \eqref{KE2}, we obtain that
\eqref{CrucEst} holds.
\end{proof}

\section{Proof of Theorem \ref{Theo1}} \label{section6}

To obtain a nonnegative solution of \eqref{prob1.61}, we replace
$f(x,s)$ by $f^+(x,s)$ where $f^+(x,s)=f(x,s)$ if $s \geq 0$ and
$0$ if $s < 0$. Let $\eta>0$ given in Lemma \ref{limitaOut}. From
(A3) and (A4) there exists $C>0$ such that
\[
|f(x,s)| \leq \frac{1}{\eta}|s| + C |s|^p\quad \text{for all } 
 ( x,s)\in \mathbb{R}^{N}\times \mathbb{R}.
\]
Since $F(x,s)\geq0$, from \eqref{SequenPS}
\begin{align*}
c_\varepsilon
&= \lim_{n\to \infty}\big[ I_\varepsilon( w_n)-\frac{1}{2}I'_\varepsilon 
( w_n)( w_n)\big]\\
&=\lim_{n\to \infty}\int_{\mathbb{R}^N} 
\big[\frac{1}{2}f(\varepsilon x, w_n(x,0))w_n(x,0) -F(\varepsilon x, w_n(x,0))\big]
 \mathrm{d} x\\
&\leq  \liminf_{n\to \infty}
\int_{\mathbb{R}^N}\big[\frac{1}{2 \eta} |w_n(x,0)|^2+ C
|w_n(x,0)|^{p+1}\big] \, \mathrm{d} x.
\end{align*}
Thus
\begin{align*}
c_\varepsilon 
&\leq \frac{1}{2 \eta}\|w_0(x,0)\|^2_{L^2(B_R)}
 + C \|w_0(x,0)\|^{p+1}_{L^{p+1}(B_R)} \\
&\quad + \frac{1}{2 \eta}\limsup_{n \to +\infty}\|w_n(x,0)\|^2_{L^2(\mathbb{R}^N 
\setminus B_R)} 
+ C \limsup_{n \to +\infty}\|w_n(x,0)\|^{p+1}_{L^{p+1}(\mathbb{R}^N
\setminus B_R)}.
\end{align*}
Invoking Lemma \ref{limitaOut} we get
\[
c_\varepsilon \leq \frac{1}{2 \eta}\|w_0(x,0)\|^2_{L^2(B_R)}+ C
\|w_0(x,0)\|^{p+1}_{L^{p+1}(B_R)} + \frac{c_\varepsilon}{2} + C
\eta^{\frac{p+1}{2}} c_\varepsilon^{\frac{p+1}{2}}.
\]
Therefore,
\begin{equation}\label{EEETF}
\frac{1}{2 \eta}\|w_0(x,0)\|^2_{L^2(B_R)}+ C
\|w_0(x,0)\|^{p+1}_{L^{p+1}(B_R)} 
\geq c_\varepsilon \Big(\frac{1}{2}- C \eta^{\frac{p+1}{2}}
c_\varepsilon^{\frac{p-1}{2}}\big).
\end{equation}
From Lemma \ref{SmallLevel}, we can find
$\varepsilon_0 > 0$ such that
\[
c_\varepsilon < \Big(\frac{1}{4 C \eta^{\frac{p+1}{2}}}\Big)^{\frac{2}{p-1}}
\quad \text{for all } \varepsilon \in (0,\varepsilon_0] .
\]
The last inequality and \eqref{EEETF} imply that
\[
\frac{1}{2 \eta}\|w_0(x,0)\|^2_{L^2(B_R)}+ C
\|w_0(x,0)\|^{p+1}_{L^{p+1}(B_R)} \geq \frac{c_\varepsilon}{4} >0.
\]
Consequently, $w_0 \not \equiv 0$ for all
$\varepsilon \in (0, \varepsilon_0 ]$. This completes the proof of
Theorem \ref{Theo1}.


\subsection*{Acknowledgments}
The authors were partially supported by CNPq/Brazil,  grant
306498/2016-2. The authors would like to thank the anonymous referees for their
 useful suggestions.

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\end{document}