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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 133, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/133\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for $p(x)$-Laplacian equations
in $\mathbb{R}^N$}

\author[B. Ge, Q. Zhou \hfil EJDE-2014/133\hfilneg]
{Bin Ge, Qingmei Zhou}  % in alphabetical order

\address{Bin Ge \newline
Department of Applied Mathematics, Harbin Engineering  University,
Harbin 150001, China}
\email{gebin04523080261@163.com}

\address{Qingmei Zhou \newline
 Library, Northeast Forestry University, Harbin 150040, China}
\email{zhouqingmei2008@163.com}

\thanks{Submitted March 4, 2014. Published June 10, 2014.}
\subjclass[2000]{35J60, 35J20, 58E30}
\keywords{$p(x)$-Laplacian; variational method; radial solution;
\hfill\break\indent Ambrosetti-Rabinowitz condition}

\begin{abstract}
 This article concerns the existence and multiplicity of solutions
 to a class of $p(x)$-Laplacian equations. We introduce a revised
 Ambrosetti-Rabinowitz condition, and show that the problem has a
 nontrivial solution and infinitely many solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}


The study of various mathematical problems with variable exponent
growth condition has received considerable attention in recent years;
see e.g. \cite{a1,k1,d3,g1,g2,g3}.
For background information, we refer the reader to \cite{r2,z1}.
The aim of this paper is to discuss the existence and
multiplicity of solutions of the following $p(x)$-Laplacian equation
in $\mathbb{R}^N$:
\begin{equation}
  \begin{gathered}
-\Delta_{p(x)}u+|u|^{p(x)-2}u= K(x)f(u),\quad\text{in }\mathbb{R}^N, \\
   u\in W^{1,p(x)}(\mathbb{R}^N),
 \end{gathered} \label{eP}
\end{equation}
 where $p(x)=p(|x|)\in C((\mathbb{R}^N)) $ with
$2\leq N<p^-:=\inf_{\mathbb{R}^N}p(x)\leq p^+:=\sup_{\mathbb{R}^N}p(x)<+\infty$,
$K:\mathbb{R}^N\to \mathbb{R} $ is a measurable function and
$f\in C(\mathbb{R},\mathbb{R})$.

Problem \eqref{eP} has been widely studied.  The following equation also has been
studied very well
\begin{equation}
  \begin{gathered}
-\Delta_{p(x)}u+|u|^{p(x)-2}u=f(x,u),\quad\text{in }\mathbb{R}^N, \\
   u\in W^{1,p(x)}(\mathbb{R}^N).
 \end{gathered}
\label{eP1}
\end{equation}
When $p(x)=p(|x|)\in C(\mathbb{R}^N)$ with
$2\leq N<p^-\leq p^+<+\infty$, the authors in \cite{d1} proved the existence
of infinitely many distinct homoclinic radially symmetric solutions
for \eqref{eP1}, under adequate hypotheses about the nonlinearity at zero
(and at infinity).

The case of $p$  Lipschitz continuous with $1<p^-\leq p^+<N$ was
discussed by \cite{f1,f6}.
Fu-Zhang \cite{f6} uses a nonlinearity on the right-hand side
of the form $h(x)|u|^{\beta(x)-1}$ where
$h\in L_+^\infty(\mathbb{R}^N)\cap L^{q(x)}(\mathbb{R}^N)$,
$1<\beta(x)<p(x)$, $q(x)=\frac{p^*(x)}{p^*{x}-\beta(x)}$,
$p^*(x)=\frac{Np(x)}{N-p(x)}$, and they prove the existence of at
least two nontrivial solutions to problem \eqref{eP1}.
In \cite{f1}, through the critical point theory, three main results on the
existence of solutions of problem \eqref{eP1} obtained, treating
separately the three cases; i.e., when the nonlinear term $f(x,u)$
is sublinear, superlinear and concave-convex nonlinearity.

Fan and Han  \cite{f1} established the existence of nontrivial
solutions for problem \eqref{eP} under the case of superlinear, by
assuming the following key condition:
\begin{itemize}
\item[(F1')] there exist $\theta>p^+$ and $M>0$ such that
$$
0<\theta F(t):=\theta\int_0^tf(s)ds\leq f(t)t,\quad \forall |t|\geq M.
$$
\end{itemize}
This condition is originally due to Ambrosetti and
Rabinowitz \cite{a2} in the case $p(x)\equiv 2$, and then was used
in \cite{b1,d2,f2,f3} for $p(x)$-Laplacian equations. Actually,
condition (F1')  is quite natural and important not only to
ensure that the Euler-Lagrange functional associated to problem
\eqref{eP1} has a mountain pass geometry, but also to guarantee that
Palais-Smale sequence of the Euler-Lagrange functional is bounded.
But this condition is very restrictive eliminating many
nonlinearities. In this paper, we introduce a new condition (F1),
below, which is different from the Ambrosetti-Rabinowitz-type
condition (F1').
\begin{itemize}
\item[(F1)] there exist a constant $M\geq 0$ and a decreasing function
$\tau$ in the space $C(\mathbb{R}\setminus(-M,M),\mathbb{R})$, such that
$$
0<(p^++\tau(t) )F(t):=(p^++\tau(t) )\int_0^tf(s)ds\leq f(t)t ,\quad  |t|\geq M,
$$
where $\tau(t)>0$,
$\lim_{|t|\to+\infty}|t|\tau(t)=+\infty$ and
$\lim_{|t|\to+\infty}\int_M^{|t|}\frac{\tau(s)}{s}ds=+\infty$.
\end{itemize}

\begin{remark} \label{rmk1.1}\rm
 Obviously, when $\inf_{|t|\geq M}\tau(t)>0$, condition (F1) and (F1')
are equivalent.
However, condition (F1) is weaker than (F1') when
$\inf_{|t|\geq M}\tau(t)=0$. For example, let $|t|\geq M=2$,
and assume that $F(t)=|t|^{p^+}{\rm ln}|t|$. Then
  $f(t)=(p^++\tau(t)){\rm sgn}(t)|t|^{p^+-1}{\rm ln}|t|$
satisfies condition (F1) not (F1'), where
$\tau(t)=\frac{1}{{\rm lnt}}\in C(\mathbb{R}\setminus(-M,M),\mathbb{R})$.
\end{remark}

The aim of this paper is twofold. First, we want to handle the case
when $p^->N$ and the unbounded area $\mathbb{R}^N$. Although
important problems can be treated within this framework, only a few
works are available in this direction, see \cite{d1}. The main
difficulty in studying problem \eqref{eP} lies in the fact that no
compact embedding is available for
$W^{1,p(x)}(\mathbb{R}^N)\hookrightarrow L^{\infty}(\mathbb{R}^N)$.
However, the subspace of radially symmetric functions of
$W^{1,p(x)}(\mathbb{R}^N)$, denoted  further by
$W_r^{1,p(x)}(\mathbb{R}^N)$, can be embedded compactly into
$L^{\infty}(\mathbb{R}^N)$ whenever $N<p^-\leq p^+<+\infty$
(cf. \cite[Theorem 2.1]{d1}). Second, instead of some usual assumption on the
nonlinear term $f$, we assume that it satisfies a modified
Ambrosetti-Rabinowitz-type condition (F1).

To state our results, we first introduce the following assumptions:
\begin{itemize}
\item[(H1)]  $K\in L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$
is radial, nonnegative, $K(x)\geq 0$ for any $x\in \mathbb{R}^N$
and $\sup_{d>0}{\rm ess}\inf_{|x|\leq d}K(x)>0$.

\item[(H2)] $f(t)=o(t^{p^+-1})$ for $t$ near 0.
\end{itemize}

Now, we are ready to state the main result of this paper.

\begin{theorem} \label{thm1.1}
Suppose that {\rm (H1), (H2), (F1)} hold.  Then
problem \eqref{eP} has a nontrivial radially symmetric solution.
Furthermore, if $f(t)=f(-t)$,  then problem \eqref{eP} has infinitely
many pairs of radially symmetric solutions.
\end{theorem}

In the remainder of this section, we recall some definitions and
basic properties of variable spaces $L^{p(x)}(\mathbb{R}^N)$ and
$W^{1,p(x)}(\mathbb{R}^N)$.  For a deeper treatment on these spaces,
we refer to \cite{f4,f5}.

Let $p\in L^\infty(\mathbb{R}^N)$, $p^->1$. The variable exponent
Lebesgue space $L^{p(x)}(\mathbb{R}^N)$ is defined by
$$
L^{p(x)}(\mathbb{R}^N)=\{
u:\mathbb{R}^N\to\mathbb{R}: u\text{ is measurable and }
\int_{\mathbb{R}^N}|u|^{p(x)}dx<+\infty\}
$$
 endowed with the norm
$|u|_{p(x)}=\{\lambda>0:\int_{\mathbb{R}^N}|\frac{u}{\lambda}|^{p(x)}dx\leq
1\}$. Then we define the variable exponent Sobolev space
\[
W^{1,p(x)}(\mathbb{R}^N)=\{u\in L^{p(x)}(\mathbb{R}^N): |\nabla
u|\in L^{p(x)}(\mathbb{R}^N)\}
\]
 with the norm
$\|u\|=|u|_{p(x)}+|\nabla u|_{p(x)}$.

\begin{proposition}[\cite{f1}] \label{prop1.1}
Set $\psi(u)=\int_{\mathbb{R}^N}(|\nabla u(x)|^{p(x)}+|u(x)|^{p(x)})dx$.
If $u, u_{k}\in W^{1, p(x)}(\mathbb{R}^N)$, then
\begin{itemize}
\item[(1)] $\| u\|<1(=1; >1)\Leftrightarrow I(u)<1(=1; >1);$

\item[(2)] If $\| u\|>1$, then $\|u\|^{p^{-}}\leq \psi(u)\leq
\|u\|^{p^{+}};$

\item[(3)] If $\|u\|<1$, then $\|u\|^{p^{+}}\leq \psi(u)\leq
\|u\|^{p^{-}};$

\item[(4)] $\lim_{k\to +\infty}\|u_{k}\|=0\Leftrightarrow
\lim_{k\to +\infty}\psi(u_{k})=0;$
\end{itemize}
\end{proposition}

 \section{Proof of Theorem \ref{thm1.1}}


In this section we prove Theorem \ref{thm1.1}  when
$\inf_{|t|\geq M}\tau(t)=0$. If $\inf_{|t|\geq M}\tau(t)>0$, then conditions
(F1') and  (F1) are equivalent, and
the proof is rather standard. We may assume that $M\geq 1$, and that
there is constant $N_0>0$ such that $|\tau(t)|\leq N_0$ for all
$t\in\mathbb{R}\backslash(-M,M)$.

We introduce the energy function $\varphi$ associated to problem
\eqref{eP}  defined by
$$
\varphi(u)=\int_{\mathbb{R}^N}\frac{1}{p(x)}(|\nabla
u(x)|^{p(x)}+|u(x)|^{p(x)})dx-\int_{\mathbb{R}^N}K(x)F(u)dx,\quad
u\in W_r^{1, p(x)}(\mathbb{R}^N)
$$
Due to the principle of symmetric
criticality of Palais (see \cite{w1}), the critical points of
$\varphi|_{W_r^{1, p(x)}(\mathbb{R}^N)}$ are critical points of
$\varphi$ as well, so radially symmetric, weak solutions of problem
\eqref{eP}.


\begin{claim} \label{claim2.1}
Let $W=\{w\in W_r^{1, p(x)}(\mathbb{R}^N):\|w\|=1\}$. Then,
 for any $w\in W$, there exist $\delta_w>0$ and $\lambda_w>0$, such that
$$
\varphi(\lambda v)<0,\quad \forall v\in W\cap B(w,\delta_w),
\forall |\lambda|\geq\lambda_w,
$$
 where $B(w,\delta_w)=\{v\in W_r^{1,p(x)}(\mathbb{R}^N):\|v-w\|<\delta_w\}$.
\end{claim}

\begin{proof}
 Since the embedding $W_r^{1,p(x)}(\mathbb{R}^N)\hookrightarrow
L^\infty(\mathbb{R}^N)$ is
compact, there is constant $C>0$ such that $|u|_\infty\leq C\|u\|$.
Thus, for all $w\in W$ and a.e. $x\in\mathbb{R}^N$, we have
$|w(x)|\leq C$. By  the definition of $\tau(t)$, we deduce that
there exists $t_\lambda\in\{t\in\mathbb{R}:M\leq|t|\leq|\lambda|C\}$
such that
$\tau(t_\lambda)=\min_{M\leq|t|\leq|\lambda|C}\tau(t)$. Then
$|\lambda|\geq\frac{t_\lambda}{C}$ and
$\lim_{|\lambda|\to+\infty}|t_\lambda|\to+\infty$.
From condition (F1), we conclude that $ F(t)\geq C_1|t|^{p^+}H(|t|)$
for all $|t|\geq M$, where $H(t)=\exp(\int_M^{|t|}\frac{\tau(s)}{s}ds)$.
 Hence, using $\lim_{|t|\to+\infty}\int_M^{|t|}\frac{\tau(s)}{s}ds=+\infty$,
it follows that $H(|t|)$ increases when $|t|$ increases, and
$\lim_{|t|\to+\infty}H(|t|)=+\infty$.

Fix $w\in W$. By $\|w\|=1$, we deduce that
$\mu(\{x\in\mathbb{R}^N:w(x)\neq 0\})>0$, and that there exists a
$\overline{t}_w>M$ such that $\mu(\{x\in\mathbb{R}^N:|\overline{t}_w
w(x)|\geq M\})>0$, where $\mu$ is the Lebesgue measure.

Set $\Omega_1:=\{x\in\mathbb{R}^N:|\overline{t}_w w(x)|\geq M\}$ and
$\Omega_2:=\mathbb{R}^N\backslash\Omega_1$. Then $\mu(\Omega_1)>0$.
Therefore, for any $x\in\Omega_1$, we have that
$|w(x)|\geq\frac{M}{\overline{t}_w}$. Now take
$\delta_w=\frac{M}{2C\overline{t}_w}$. Then, for any
$v\in W\cap B(w,\delta_w)$,
$|v-w|_\infty\leq C\|v-w\|<\frac{M}{2\overline{t}_w}$. Hence, for all
$x\in\Omega_1$,
we deduce that $|v(x)|\geq \frac{M}{2\overline{t}_w}$ and $|\lambda
v(x)|\geq M$ for any $x\in \Omega_1$ and $\lambda\in\mathbb{R}$ with
$|\lambda|\geq2\overline{t}_w$. Thus, for
$|\lambda|\geq2\overline{t}_w$, by the above estimates and $H(|t|)$
increases when $|t|$ increases, we have
\begin{equation} \label{2.1}
\begin{aligned}
\int_{\Omega_1}K(x)F(\lambda v(x))dx
&\geq C_1|\lambda|^{p^+}\int_{\Omega_1}K(x)|v(x)|^{p^+}H(|\lambda
v(x)|)dx\\
&\geq C_1|\lambda|^{p^+}(\frac{M}{2\overline{t}_w})^{p^+}
H(|\lambda|\frac{M}{2\overline{t}_w})\int_{\Omega_1}K(x)dx.
\end{aligned}
\end{equation}
On the other hand, by continuity, we deduce that there exists a
$C_2>0$ such that $F(t)\geq -C_2$ when $|t|\leq M$. Note that
$F(t)>0$ if $|t|\geq M$. Hence,
\begin{equation} \label{2.2}
\begin{aligned}
\int_{\Omega_2}K(x)F(\lambda v(x))dx
&=\int_{\Omega_2\cup\{x\in\mathbb{R}^N:|\lambda v(x)|\geq M\}}K(x)F(\lambda v(x))dx\\
&\quad +\int_{\Omega_2\cup\{x\in\mathbb{R}^N:|\lambda v(x)|
 \leq M\}}K(x)F(\lambda v(x))dx\\
&\geq \int_{\Omega_2\cup\{x\in\mathbb{R}^N:|\lambda v(x)|\leq M\}}K(x)F(\lambda v(x))dx\\
&\geq -C_2|K|_1.
\end{aligned}
\end{equation}
Hence, for $v\in W\cap B(w,\delta_w)$ and $|\lambda|>1$,  from
\eqref{2.1} and \eqref{2.2}, we have
\begin{align*}
\varphi(\lambda v)
&=\int_{\mathbb{R}^N}\frac{|\lambda|^{p(x)}}{p(x)}(|\nabla v|^{p(x)}+|v|^{p(x)})dx
-\int_{\mathbb{R}^N}K(x)F(\lambda v(x))dx\\
&\leq |\lambda|^{p^+}-C_1|\lambda|^{p^+}(\frac{M}{2\overline{t}_w})^{p^+}
H(|\lambda|\frac{M}{2\overline{t}_w})\int_{\Omega_1}K(x)dx
+C_2|K|_1\\
&=|\lambda|^{p^+}\Big[1-C_1(\frac{M}{2\overline{t}_w})^{p^+}
H(|\lambda|\frac{M}{2\overline{t}_w})\int_{\Omega_1}K(x)dx\Big]+C_2|K|_1\\
&\to-\infty,
\end{align*}
as $|\lambda|\to+\infty$, because
$\lim_{|t|\to+\infty}H(|t|) =+\infty$.
\end{proof}

\begin{claim} \label{claim2.2} 
There exist $\nu>0$ and $\rho>0$ such that
$\inf_{\|u\|=\nu}\varphi(u)\geq\rho>0$.
\end{claim}

\begin{proof}
Note that $|u|_\infty\to 0$ if
$\|u\|\to 0$. Then, by  hypothesis (H2), we have
\[
\int_{\mathbb{R}^N}K(x)F(u)dx=|K|_1o(|u|_\infty^{p^+})=|K|_1o(\|u\|^{p^+}),
\]
which implies 
\begin{align*}
\varphi(u)&= \int_{\mathbb{R}^N}\frac{1}{p(x)}(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-\int_{\mathbb{R}^N}K(x)F(u)dx\\
& \geq \frac{1}{p^+}\|u\|^{p^+}-|K|_1o(\|u\|^{p^+}).
\end{align*}
Therefore, there exist $1>\nu>0$ and $\rho>0$ such that
$\inf_{\|u\|=\nu}\varphi(u)\geq\rho>0$.
\end{proof}

\begin{claim} \label{claim2.3}
The functional $\varphi$ satisfies the (PS)
condition.
\end{claim}

\begin{proof}
Let $\{u_n\}\subset W_r^{1,p(x)}(\mathbb{R}^N)$ be a
(PS) sequence of the functional $\varphi$; that is,
$|\varphi(u_n)|\leq c$ and $ |\langle \varphi'(u_n),h\rangle | \leq
\varepsilon_n \|h\|$ with $\varepsilon_n\to 0$, for all
$h\in W_r^{1,p(x)}(\mathbb{R}^N)$. We will prove that the sequence
$\{u_n\}$ is bounded in $ W_r^{1,p(x)}(\mathbb{R}^N)$. Indeed, if
$\{u_n\}$ is unbounded in $ W_r^{1,p(x)}(\mathbb{R}^N)$, we may
assume that $\|u_n\|\to\infty$ as $n\to\infty$. Let
$u_n=\lambda_nw_n$, where $\lambda_n\in\mathbb{R}$, $w_n\in W$. It
follows that $|\lambda_n|\to\infty$.

Let $\Omega_1^n:=\{x\in\mathbb{R}^N:|\lambda_n w_n(x)|\geq M\}$ and
$\Omega_2^n:=\mathbb{R}^N\backslash\Omega_1^n$. Then
\begin{align*}
-\varepsilon_n  |\lambda_n|&= -\varepsilon_n \|u_n\|\\
&\leq  \langle \varphi'(u_n),u_n\rangle \\
&= \int_{\mathbb{R}^N}\left(|\nabla u_n|^{p(x)}+|u_n|^{p(x)}\right)dx
-\int_{\mathbb{R}^N}K(x)f(u_n)u_ndx\\
&\leq  \int_{\mathbb{R}^N}|\lambda_n|^{p(x)}\left(|\nabla w_n|^{p(x)}
+|w_n|^{p(x)}\right)dx-\int_{\Omega_1^n}K(x)f(\lambda_nw_n)\lambda_nw_ndx\\
&\quad -\int_{\Omega_2^n}K(x)f(\lambda_nw_n)\lambda_nw_ndx,
\end{align*}
which implies that
\begin{align*}
\int_{\Omega_1^n}K(x)f(\lambda_nw_n)\lambda_nw_n\,dx
&\leq \int_{\mathbb{R}^N}|\lambda_n|^{p(x)}\left(|\nabla
w_n|^{p(x)}+|w_n|^{p(x)}\right)dx  \\
&\quad +\varepsilon_n
|\lambda_n|-\int_{\Omega_2^n}K(x)f(\lambda_nw_n)\lambda_nw_ndx.
\end{align*}
Note that $0<(p^++\tau(t_{\lambda_n}))F(\lambda_nw_n)\leq
f(\lambda_nw_n)\lambda_nw_n$ in $\Omega_1^n$. So,
\[
\int_{\Omega_1^n}K(x)F(\lambda_nw_n)dx
\leq\frac{1}{p^++\tau(t_{\lambda_n})}\int_{\Omega_1^n}K(x)
f(\lambda_nw_n)\lambda_nw_ndx.
\]
Then it follows that
\begin{align*}
 \varphi(u_n)
&= \varphi(\lambda_nw_n)\\&= \int_{\mathbb{R}^N}\frac{|\lambda_n|^{p(x)}}{p(x)}(|\nabla w_n|^{p(x)}+|w_n|^{p(x)})dx-\int_{\mathbb{R}^N}K(x)F(\lambda_n w_n)dx\\
&= \int_{\mathbb{R}^N}\frac{|\lambda_n|^{p(x)}}{p(x)}
 \left(|\nabla w_n|^{p(x)}+|w_n|^{p(x)}\right)dx
 -\int_{\Omega_1^n}K(x)F(\lambda_n w_n)dx\\
&\quad -\int_{\Omega_2^n}K(x)F(\lambda_n w_n)dx\\
&\geq \frac{1}{p^+}\int_{\mathbb{R}^N} |\lambda_n|^{p(x)}
 \left(|\nabla w_n|^{p(x)}+|w_n|^{p(x)}\right)dx\\
&\quad -\frac{1}{p^++\tau(t_{\lambda_n})}\int_{\Omega_1^n}K(x)
 f(\lambda_nw_n)\lambda_nw_ndx
 -\int_{\Omega_2^n}K(x)F(\lambda_n w_n)dx\\
&\geq \frac{1}{p^+}\int_{\mathbb{R}^N} |\lambda_n|^{p(x)}
 \left(|\nabla w_n|^{p(x)}+|w_n|^{p(x)}\right)dx\\
&\quad -\frac{1}{p^+
 +\tau(t_{\lambda_n})} \Big[\int_{\mathbb{R}^N}|\lambda_n|^{p(x)}
\left(|\nabla w_n|^{p(x)}+|w_n|^{p(x)}\right)dx
 +\varepsilon_n  |\lambda_n|\Big]\\
&\quad   +\frac{1}{p^++\tau(t_{\lambda_n})}\int_{\Omega_2^n}K(x)
 f(\lambda_nw_n)\lambda_nw_ndx-\int_{\Omega_2^n}K(x)F(\lambda_n w_n)dx\\
&= \frac{\tau(t_{\lambda_n})}{p^+(p^++\tau(t_{\lambda_n}))}
 \int_{\mathbb{R}^N} |\lambda_n|^{p(x)} \left(|\nabla w_n|^{p(x)}+|w_n|^{p(x)}
 \right)dx\\
&\quad -\frac{1}{p^++\tau(t_{\lambda_n})}\varepsilon_n  |\lambda_n|+T(\lambda_nw_n)\\
&\geq \frac{\tau(t_{\lambda_n})}{p^+(p^++N_0)}|\lambda_n|^{p^-}
  -\frac{1}{p^+}\varepsilon_n  |\lambda_n|+T(\lambda_nw_n)\\
&= |\lambda_n|\Big[ \frac{|\lambda_n|^{p^--1}\tau(t_{\lambda_n})}{p^+(p^++N_0)}
 -\frac{\varepsilon_n}{p^+}\Big]+T(\lambda_nw_n)\\
&\geq |\lambda_n|\Big[ \frac{|\lambda_n|^{p^--1}\tau(t_{\lambda_n})}{p^+(p^++N_0)}
-\frac{\varepsilon_n}{p^+}\Big]-C_2,
\end{align*}
where
$$
T(\lambda_nw_n)=\frac{1}{p^++\tau(t_{\lambda_n})}
\int_{\Omega_2^n}K(x)f(\lambda_nw_n)\lambda_nw_n\,dx
-\int_{\Omega_2^n}K(x)F(\lambda_n w_n)\,dx
$$
is bounded from below. We know that
$|\lambda_n|\to+\infty$, and so
$|t_{\lambda_n}|\to+\infty$, as $n\to+\infty$. It
follows from (F1) and $p^->N\geq 2$ that
$$
\lim_{n\to+\infty}|\lambda_n|^{p^--1}\tau(t_{\lambda_n})
\geq\lim_{n\to+\infty}\frac{|t_{\lambda_n}|\tau(t_{\lambda_n})}{M}=+\infty.
$$
This means that
$\lim_{n\to+\infty}\varphi(u_n)\to+\infty$.
This is a contradiction. So, the sequence $\{u_n\}$ is bounded in
$W_r^{1,p(x)}(\mathbb{R}^N)$. Note that the embedding
$W_r^{1,p(x)}(\mathbb{R}^N)\hookrightarrow L^\infty(\mathbb{R}^N)$ is
compact, there exists a $u\in W_r^{1,p(x)}(\mathbb{R}^N)$ such that
passing to subsequence, still denoted by  $\{u_n\}$, it converges
strongly to $u$ in $L^\infty(\mathbb{R}^N)$, and in the same way as
the proof of \cite[Proposition 3.1]{k2} we can conclude that $u_n$
converges strongly also in $W_r^{1,p(x)}(\mathbb{R}^N)$. Thus,
$\varphi$ satisfies the (PS) condition.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Due to Claims \ref{claim2.1}, \ref{claim2.2} and \ref{claim2.3},
we know that $\varphi$ satisfies the conditions of
the classical mountain pass theorem due to Ambrosetti and Rabinowitz
\cite{a2}. Hence, we obtain a nontrivial critical point, which gives
rise to a nontrivial radially symmetric solution to problem \eqref{eP}.

Furthermore, if $f(t)=f(-t)$, then $\varphi$ is even. We will use
the following $\mathbb{Z}_2$ version of the mountain pass theorem in
\cite{r1}.
\end{proof}

\begin{theorem} \label{thm2.1}
Let $E$ be an infinite-dimensional Banach space,
and  $\varphi\in C(E,\mathbb{R})$ be even, satisfying the (PS)
condition, and having $\varphi(0)=0$. Assume that $E=V\oplus X$, where
$V$ is finite dimensional. Suppose that the following hold.
\begin{itemize}
\item[(a)] there are constants $\nu, \rho>0$  such that
$\inf_{\partial B_\nu\cup X}\varphi\geq\rho$.

\item[(b)] for each finite-dimensional subspace
$\overline{E}\subset E$, there is an $\sigma=\sigma(\overline{E})$
such that $\varphi\leq 0$ on $\overline{E}\backslash B_{\sigma}$.
\end{itemize}
Then $\varphi$ possesses an unbounded sequence of critical values.
\end{theorem}

From Claims \ref{claim2.1} and  \ref{claim2.2}, $\varphi$ satisfies
(a) and the (PS) condition. For any finite-dimensional subspace
$\overline{E}\subset E$, $S\cap \overline{E}=\{w\in \overline{E}:\|w\|=1\}$
 is compact.
By Claim \ref{claim2.1} and the finite covering theorem, it is easy to verify
that $\varphi$ satisfies condition (b). Hence, by the $\mathbb{Z}_2$
version of the mountain pass theorem, $\varphi$ has a sequence of
critical points $\{u_n\}_{n=1}^{\infty}$. That is, problem \eqref{eP} has
infinitely many pairs of radially symmetric solutions.

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of
China (No. 11126286, No. 11201095), the Fundamental Research Funds
for the Central Universities (No. 2014), China Postdoctoral Science
Foundation funded project (No. 20110491032), and China Postdoctoral
Science (Special) Foundation (No. 2012T50325).

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