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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 184, pp. 1--12.\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-2017/184\hfil Existence of solutions]
{Existence of solutions for Kirchhoff type equations with unbounded
 potential}

\author[Y. Duan, Y. Zhou \hfil EJDE-2017/184\hfilneg]
{Yueliang Duan, Yinggao Zhou}

\address{Yueliang Duan \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083,  China}
\email{duanyl@csu.edu.cn}

\address{Yinggao Zhou (corollaryresponding author) \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083,   China}
\email{ygzhou@mail.csu.edu.cn}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted March 7, 2017. Published July 19, 2017.}
\subjclass[2010]{35A15, 35A20, 35J20}
\keywords{Cut-off functional; Kirchhoff type problem; unbounded potential;
\hfill\break\indent variational method}

\begin{abstract}
 In this article, we study the Kirchhoff type  equation
 $$
 \Big(a+\lambda\int_{\mathbb{R}^3}|\nabla u|^2
 +\lambda b\int_{\mathbb{R}^3}u^2\Big)[-\Delta u+b u]
 =K(x)|u|^{p-1}u,\quad \text{in } \mathbb{R}^3,
 $$
 where  $a,b>0$, $p\in(2,5)$, $\lambda\geq0$ is a parameter, and
 $K$ may be an unbounded  potential function. By using  variational methods,
 we prove  the existence of nontrivial solutions for the above equation.
 A cut-off functional and some estimates are used to obtain the bounded
 Palais-Smale sequences.
\end{abstract}

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

\section{Introduction}

In this article, we consider the Kirchhoff type problem
\begin{equation}\label{1.1}
\begin{gathered}
\Big(a+\lambda\int_{\mathbb{R}^3}|\nabla u|^2
 +\lambda b\int_{\mathbb{R}^3}u^2\Big)[-\Delta u+b u]
 =K(x)|u|^{p-1}u,\quad \text{in } \mathbb{R}^3,\\
 u\in H^1_r(\mathbb{R}^3),
 \end{gathered}
\end{equation}
 where $a>0$, $b>0$, $p\in(2,5)$, $\lambda\geq0$ is a parameter and $K(x)$ 
is a given potential satisfying  the following conditions:
\begin{itemize}
\item[(H1)] $K:\mathbb{R}^{3}\to \mathbb{R}$ is a nonnegative continuous function, 
$K$ is radial (that is $k(x)=k(|x|)$) and $K\not\equiv 0$;

\item[(H2)] there exists $C_0>0$ and  $0\leq l< p-2$ such that
 $K(x)\leq C_0(1+|x|^{l})$ for all $x\in \mathbb{R}^3$.
 \end{itemize}

When $\Omega$ is a bounded domain in $\mathbb{R}^N$, the Kirchhoff type problem
\begin{equation}\label{1.2}
 \begin{gathered}
-\Big(a+b\int_{\Omega}|\nabla u|^2\Big)\Delta u=f(x,u), \quad  \text{in } \Omega,\\
u=0,\quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
has attracted a lot of attention in recent years.
Equation \eqref{1.2} is a nonlocal problem because  of the 
appearance of the term $\int_{\Omega}|\nabla u|^2$.  This causes some 
mathematical difficulties but makes the study of \eqref{1.2} very interesting.
 One can refer to 
\cite{aco,cbwx,cbwx1,liusun,leiliao,suntang,xiewu,pkzz,yyzj,hxzw,hezouw,ld} 
and references therein  for the existence and multiplicity of positive 
solutions to \eqref{1.2}.  Mao and Luan \cite{mazz}, Mao and Luan \cite{maoluan},
 Zhang and Perera \cite{zzpk} have investigated the existence of sign-changing 
solutions of \eqref{1.2}.


When $\Omega=\mathbb{R}^N$, the existence and multiplicity of solutions to 
\eqref{1.2} have been treated in \cite{liuliao,jinwu,liuhe,wux,zouwhe,zouwhe1}. 
In particular, Nie and Wu \cite{niewu}
have studied the  Schr\"odinger-Kirchhoff type problem
\begin{equation}\label{1.2.1}
 \begin{gathered}
-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^2\Big)\Delta u+V(|x|)u=Q(|x|)f(u), \\
u(x)\to 0,\quad \text{as } |x|\to\infty,
\end{gathered}
\end{equation}
where $N\geq2$, $a,b>0$, $f\in C(\mathbb{R},\mathbb{R})$, $V$ and $Q$ are 
both radial functions in $\mathbb{R}^N$.
Under some conditions on $V$ and $Q$, the existence  of nontrivial solutions 
and a sequence of high energy solutions for problem \eqref{1.2.1} has been 
 proved by the Mountain Pass theorem and symmetric Mountain Pass theorem. 
From \cite[Remarks 1 and 2]{niewu}, we see that the potential  $Q$ is
 bounded in $\mathbb{R}^N$. 


In \cite{lylfs}, Li, Li and Shi  showed the existence of nontrivial solutions 
of the following problem with zero mass
\begin{gather*}
-\Big(a+b\int_{\mathbb{R}^{N}}|\nabla u|^2\Big)\Delta u=K(x)f(u),\quad \text{in }
\mathbb{R}^{N},\\
u\in \mathcal{D}^{1,2}(\mathbb{R}^{N}),
\end{gather*}
where the  potential function $K(x)$ is a nonnegative continuous function, 
$K\in[L^{s}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})]\backslash\{0\}$ 
for some $s\geq2N/(N+2)$ and
$|x\cdot\nabla K(x)|\leq\alpha K(x)$ for a.e. 
$x\in \mathbb{R}^{N}$ and some $\alpha\in(0,2)$.

To be best of our knowledge, there is no result about the Kirchhoff type 
problems with unbounded potential at infinity.  Motivated by the above work,
 in this article, we are lead to study problem \eqref{1.1} with unbounded potential 
$K$. Our main results are as follows:

\begin{theorem}\label{thm1.3} 
Assume  {\rm (H1)} and {\rm (H2)} hold. Then  there exists $\lambda_0>0$
such that for any $\lambda\in[0,\lambda_0)$, problem \eqref{1.1} has at least
 one nontrivial  solution in $H_r^1(\mathbb{R}^3)$.
\end{theorem}

\begin{corollary}
Assume  {\rm (H1)} and {\rm (H2)} hold. Then the local  problem
\begin{equation} 
-\Delta u+b u=K(x)|u|^{p-1}u,\quad \text{in } \mathbb{R}^3
\end{equation}
 has at least one nontrivial solution in  $H_r^1(\mathbb{R}^3)$.
\end{corollary}

We remark that in \cite{lylfs},  a cut-off functional and Pohozaev type 
identity are utilized to obtain the bounded Palais-Smale sequences.  
However, we find   that when $f(u)=|u|^{p-1}u$ can be expressed explicitly, 
it is not necessary to establish the Pohozaev type identity. 
In the case that $K$ is unbounded,   we need the Radial Lemma 
(see Lemma~\ref{l1}) to overcome the loss of compactness, which is
 different from \cite{lylfs}.

The remaining of this paper is organized as follows. 
In Section 2, we give some notations and elementary lemmas which will be 
used in the paper. 
In Section 3, we are devoted to the proof of our main results.

\section{Preliminaries}

In this paper, we shall use the following notation:\\
$\bullet$  $C$ stands for different positive constants.\\
$\bullet$ For $r>0$,  $B_r(x)$ is an open ball in $\mathbb{R}^3$ with radius 
$r$ centered at $x$.\\
$\bullet$  $H^{1}(\mathbb{R}^{3})$ denotes the usual Sobolev space equipped 
with the inner product 
$$
(u,v)=\int_{\mathbb{R}^{3}}(\nabla u\cdot\nabla v+buv)dx,
$$
and  the corresponding  norm  $\|u\|=(u,u)^{1/2}$.\\
$\bullet$ $H^{1}_r(\mathbb{R}^{3})$ is the set of all radial functions in 
$H^1(\mathbb{R}^3)$. \\
$\bullet$ For $\Omega\subset \mathbb{R}^3$ and $1\leq q<\infty$, $L^{q}(\Omega)$ 
denotes the Lebesgue space with the norm
 $$
|u|_{L^q(\Omega)}=(\int_{\Omega}|u|^qdx)^{\frac{1}{q}}.
$$
 When $\Omega=\mathbb{R}^3$, we write $|u|_q=|u|_{L^q(\mathbb{R}^3)}$ 
for simplicity of notations.
Furthermore, when  $q=\infty$, we write
 $$
|u|_{\infty}=\operatorname{ess\,sup}_{x\in\mathbb{R}^3}|u(x)|.
$$
$\bullet$  $H^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$ 
continuously for $q\in[2,6]$, while
 $H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$ 
compactly for $q\in(2,6)$. \\
 $\bullet$ The dual space of $H_r^1(\mathbb{R}^3)$ is denoted by
 $H_r^{-1}(\mathbb{R}^3)$. \\
 $\bullet$  Let $\langle \cdot,\cdot\rangle$ be the duality pairing between 
$H_r^1(\mathbb{R}^3)$ and $H_r^{-1}(\mathbb{R}^3)$.


Before establishing the variational setting for \eqref{1.1}, we need the 
following lemma.

\begin{lemma}[\cite{bhpl}] \label{l1}
Let $N\geq2$. Then for any radial function $u\in H^{1}_{r}(\mathbb{R}^{N})$,
$$
|u(r)|\leq C_1\|u\|r^{\frac{1-N}{2}},  \quad \text{for } r\geq 1,
$$
where $C_1$ is a positive constant and  only depends on $N$.
\end{lemma}

\begin{remark}\label{r1} \rm
Since $H^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$
continuously for $q\in[2,6]$, for any $u\in H_{r}^{1}(\mathbb{R}^{3})$, 
by (H1) and  (H2),  we have
\begin{align*}
0&\leq\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x \\
&\leq C_0\int_{\mathbb{R}^{3}}(1+|x|^{l})|u|^{p+1}\mathrm{d}x\\
&= C_0\int_{B_{1}(0)}{(1+|x|^{l})|u|^{p+1}\mathrm{d}x}
 +C_0\int_{\mathbb{R}^{3}\setminus{B_{1}(0)}}{(1+|x|^{l})|u|^{p+1}\mathrm{d}x}\\
&\leq C(\int_{B_{1}(0)}{|u|^{p+1}\mathrm{d}x}
 +\int_{1}^{\infty}{r^{l+2}|u(r)|^{p+1}\mathrm{d}r})\\
&\leq C\|u\|^{p+1}+C\int_{1}^{\infty}{r^{l+2}|u(r)|^{p+1}\mathrm{d}r}.
\end{align*}
Note that $0\leq l<p-2$. Then by Lemma~\ref{l1}, we deduce
$$
\int_{1}^{\infty}{r^{l+2}|u(r)|^{p+1}\mathrm{d}r}
\leq C\|u\|^{p+1}\int_{1}^{\infty}{r^{l-p+1}\mathrm{d}r}
\leq \frac{C}{p-2-l}\|u\|^{p+1}.
$$
Thus,
$$
0\leq\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x<C\|u\|^{p+1}.
$$
\end{remark}

In view of  Remark~\ref{r1}, we consider the Sobolev space 
$H_r^1(\mathbb{R}^3)$. As usual, the energy functional 
$J_{\lambda}:H_{r}^{1}(\mathbb{R}^{3})\to\mathbb{R}$ associated with \eqref{1.1} 
is well defined by
$$
J_{\lambda}(u)=\frac{1}{2} a \|u\|^{2}
+\frac{1}{4}\lambda\|u\|^{4}
-\frac{1}{p+1}\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x.
$$
It is easy to check that $J_{\lambda}$ is $C^1$-functional, whose Gateaux 
derivative is given by
$$
\langle J'_{\lambda}(u),v\rangle=a(u,v)+\lambda\|u\|^{2}(u,v)
-\int_{\mathbb{R}^{3}}K(x)|u|^{p-1}uv\mathrm{d}x 
\quad u,v\in H_{r}^{1}(\mathbb{R}^{3}).
$$
In the weak sense, solutions to \eqref{1.1} correspond to the critical points 
of the functional $J_{\lambda}$.

\section{Proof of our main results}

In this section, we  prove the  existence of nontrivial solutions to \eqref{1.1}. 
We first have the following lemma.

\begin{lemma}\label{l3}
If $u$ is a nontrivial weak solution of \eqref{1.1}, then $\|u\|\geq r$ 
for some $r>0$.
\end{lemma}

\begin{proof}
Since $u$ is a nontrivial  weak solution of \eqref{1.1},  by Remark \ref{r1}, 
we have
$$
a\|u\|^2\leq(a+\lambda\|u\|^2)\|u\|^2=\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}
\mathrm{d}x\leq C\|u\|^{p+1}.
$$
Because $p+1>2$, there exists $r>0$ such that  $\|u\|>r$.
\end{proof}

To  find a  bounded Palais-Smale sequence for the energy  
functional $J_{\lambda}$, by following \cite{kh,jjlcsl}
 (also see \cite{lylfs,lylfs1}), we introduce  a cut-off 
function $\phi\in C^{\infty}(\mathbb{R}_{+},\mathbb{R})$ such that
\begin{gather*}
\phi(t)=1, \quad   t\in[0,1],\\
0\leq\phi(t)\leq1, \quad t\in(1,2),\\
\phi(t)=0, \quad   t\in[2,\infty),\\
|\phi'|_{\infty}\leq2.
\end{gather*}
 So we can define the functional 
$J_{\lambda}^{T}:H_{r}^{1}(\mathbb{R}^{3})\to \mathbb{R}$ by
$$
J_{\lambda}^{T}(u)=\frac{1}{2}a\|u\|^2+\frac{1}{4}\lambda h_{T}(u)\|u\|^{4}
-\frac{1}{p+1}\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x,\quad
 u\in H_{r}^{1}(\mathbb{R}^{3}),
$$
where for every $T>0$,
$$
h_{T}(u)=\phi(\frac{\|u\|^{2}}{T^{2}}).
$$
Moreover,  for every $u,v\in H^1_r(\mathbb{R}^3)$, we have
\begin{equation}
\begin{aligned}
\langle (J_{\lambda}^T)'(u),v\rangle
&=a(u,v)+\lambda h_{T}(u)\|u\|^{2}(u,v)+\frac{\lambda}{2T^{2}}
\phi'(\frac{\|u\|^{2}}{T^{2}})\|u\|^{4}(u,v) \\
&\quad -\int_{\mathbb{R}^{3}}K(x)|u|^{p-1}uv\mathrm{d}x.
\end{aligned}
\end{equation}
It is easy to see that if  $u$ is a critical point of $J_{\lambda}^{T}$ 
such that $\|u\|\leq T$, then 
$\langle (J_{\lambda}^T)'(u),v\rangle=\langle J_{\lambda}'(u),v\rangle$. 
So the arbitrary of $v$ yields 
$J_{\lambda}'(u)=0$ and thus $u$ is also a critical point of $J_{\lambda}$.

Next we recall a theorem, for which a corollary  was proved
 by Struwe \cite{ms}.

\begin{theorem}[\cite{jl}]\label{t2}
Let $(X,\|\cdot\|)$ be a Banach space and $I\subset \mathbb{R}_{+}$ an interval. 
Consider the family of $C^{1}-$ functional on $X$
$$
J_{\mu}(u)=A(u)-\mu B(u), \quad\mu\in I,
$$
with $B$ nonnegative and either $A(u)\to\infty$ or $B(u)\to\infty$ as 
$\|u\|\to\infty$ and such that $J_{\mu}(0)=0$.

For every $\mu\in I$, we set
 $$
\Gamma_{\mu}=\{\gamma\in C([0,1],X):\gamma(0)=0,J_{\mu}(\gamma(1))<0\}.
$$
If for every $\mu\in I$, the set $\Gamma_{\mu}$ is nonempty and 
$$
c_{\mu}=\inf_{\gamma\in\Gamma_{\mu}}\max_{t\in[0,1]}J_{\mu}(\gamma(t))>0,
$$
then for almost every $\mu\in I$, there is a sequence $\{u_n\}\subset X$ such that
\begin{itemize}
\item[(i)]  $ \{u_n\}$ is bounded;
\item[(ii)] $J_{\mu}(u_n)\to c_{\mu}$;
 \item[(iii)]  $J'_{\mu}(u_n)\to 0$ in the dual  space $X^{-1}$ of $X$.
\end{itemize}
\end{theorem}

In our case, $X=H^{1}_{r}(\mathbb{R}^{3})$,
\begin{gather*}
 A(u)=\frac{1}{2}a\|u\|^{2}+\frac{1}{4}\lambda h_{T}(u)\|u\|^{4}, \\
 B(u)=\frac{1}{p+1}\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x,
\end{gather*}
and the associated  perturbed functional we study is 
$$
J_{\lambda,\mu}^{T}(u)=\frac{1}{2}a\|u\|^2
 +\frac{1}{4}\lambda h_{T}(u)\|u\|^{4}
 -\frac{\mu}{p+1}\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x,\quad
 u\in H_{r}^{1}(\mathbb{R}^{3}).
$$
It is clear that this functional is $C^1$-functional and for every 
$u,v\in H^1_r(\mathbb{R}^3)$,
\begin{equation}\label{3.1}
\begin{aligned}
\langle (J_{\lambda,\mu}^T)'(u),v\rangle
&=a(u,v)+\lambda h_{T}(u)\|u\|^{2}(u,v)+\frac{\lambda}{2T^{2}}
\phi'(\frac{\|u\|^{2}}{T^{2}})\|u\|^{4}(u,v) \\
&\quad -\mu\int_{\mathbb{R}^{3}}K(x)|u|^{p-1}uv\mathrm{d}x.
\end{aligned}
\end{equation}
Notice that  $J_{\lambda,\mu}^{T}(0)=0$, $B$ is nonnegative in 
$H^1_r(\mathbb{R}^3)$  and $A(u)\to+\infty$ as $\|u\|\to\infty$.
Next we shall  prove the following two lemmas which show that the functional
 $J_{\lambda,\mu}^{T}$ satisfies the  other conditions of
Theorem~\ref{t2}.

\begin{lemma}\label{l4}
We have $\Gamma_{\mu}\neq\emptyset$ for all $\lambda\geq 0$ and  
$\mu\in I:=[\frac{1}{2},1]$.
\end{lemma}

\begin{proof}
By (H1), there exist  $R,C_{2}>0$ and $\varphi\in C_0^{\infty}(\mathbb{R}^{3})$ 
such that
\begin{equation}\label{20}
\begin{gathered}
\varphi\geq 0,\quad \|\varphi\|=1,\\
\operatorname{supp}(\varphi)\subset B_{R}(0),\\
\int_{B_{R}(0)}K(x)\varphi^{p+1}\mathrm{d}x\geq C_{2}.
\end{gathered}
\end{equation}
Then for $t^{2}\geq2T^{2}$,
\begin{align*}
J_{\lambda,\mu}^{T}(t\varphi)
&=\frac{1}{2}a t^{2}\|\varphi\|^2
 +\frac{1}{4}\lambda \phi(\frac{t^{2}\|\varphi\|^{2}}{T^{2}})t^{4}\|\varphi\|^{4}
-\frac{\mu}{p+1}|t|^{p+1}\int_{\mathbb{R}^{3}}K(x)\varphi^{p+1}\mathrm{d}x\\
&=\frac{1}{2}a t^{2}-\frac{\mu}{p+1}|t|^{p+1}
 \int_{\mathbb{R}^{3}}K(x)\varphi^{p+1}\mathrm{d}x\\
&\leq\frac{1}{2}a t^{2}-\frac{1}{2(p+1)}|t|^{p+1}\int_{B_{R}(0)}K(x)
\varphi^{p+1}\mathrm{d}x\\
&\leq\frac{1}{2}a t^{2}-\frac{ C_{2}}{2(p+1)}|t|^{p+1}.
\end{align*}
Since $p+1>2$, we can choose $t_0>0$ large such that 
$J_{\lambda,\mu}^{T}(t_0\varphi)<0$. We set
$$
\gamma(t)=tt_0\varphi,\ t\in[0,1].
$$
So $\gamma\in \Gamma_{\mu}$ and $\Gamma_{\mu}\neq \emptyset$. 
The proof is complete.
\end{proof}

\begin{lemma}\label{l5}
There exists a positive constant $\alpha$ such that $c_{\mu}\geq \alpha$ 
for all $\mu\in I$.
\end{lemma}

\begin{proof}
For any  $\mu\in I$ and  $u\in H_{r}^{1}(\mathbb{R}^{3})$, by Remark~\ref{r1}, 
we have that 
\begin{align*}
J_{\lambda,\mu}^{T}(u)
&=\frac{1}{2}a\|u\|^2+\frac{1}{4}\lambda h_{T}(u)\|u\|^{4}
 -\frac{\mu}{p+1}\int_{\mathbb{R}^{3}}K(x)|u|^{p+1}\mathrm{d}x\\
&\geq \frac{1}{2}a\|u\|^{2}-\frac{\mu}{p+1}C\|u\|^{p+1}\\
&\geq\frac{1}{2}a\|u\|^{2}-\frac{C}{p+1}\|u\|^{p+1}.
\end{align*}
Take $\rho:=\big{(}\frac{ap+a}{4C}\big{)}^{\frac{1}{p-1}}$  and 
$\alpha:=\frac{1}{2}a\rho^{2}-\frac{C}{p+1}\rho^{p+1}>0$. 
Then $J_{\lambda,\mu}^{T}(u)>0$
for any $\mu\in I$ and  $0<\|u\|\leq \rho$. Moreover, 
$J_{\lambda,\mu}^{T}(u)\geq \alpha$ if $\|u\|=\rho$.
 By the definition of $\Gamma_{\mu}$, for each $\gamma\in \Gamma_{\mu}$, 
there is  $t^*\in(0,1)$ such that $\|\gamma(t^*)\|=\rho$. 
Therefore, the arbitrary of $\mu\in I$ yields that
$$
c_{\mu}\geq\inf_{\gamma\in\Gamma_{\mu}
}J_{\lambda,\mu}^{T}(\gamma(t^*))\geq \alpha.
$$
The proof is complete.
\end{proof}

Next we need some compactness on  Palais-Smale sequences of the functional 
$J_{\lambda,\mu}^{T}$ in order to prove  our main results.

\begin{lemma}\label{l6}
Assume  $4\lambda T^{2}<a$. For any $\mu\in I$, any bounded Palais-Smale 
sequence of $J_{\lambda,\mu}^{T}$ contains a convergent subsequence.
\end{lemma}

\begin{proof}
Fix $\mu\in I$. We assume  $\{u_n\}$ be a bounded Palais-Smale sequence of
 $J_{\lambda,\mu}^{T}$, namely
\begin{gather*}
\text{$\{u_n\}$ and $\{J_{\lambda,\mu}^{T}(u_n)\}$ are bounded}, \\
(J_{\lambda,\mu}^{T})'(u_n)\to 0 \quad \text{in } H_{r}^{-1}(\mathbb{R}^{3}).
\end{gather*}
Up to a subsequence,  there exists $u\in H_{r}^{1}(\mathbb{R}^{3})$  such that
\begin{gather*}
u_n\rightharpoonup u,\quad \text{in } H_{r}^{1}(\mathbb{R}^{3}),\\
u_n\to u,\quad \text{in  } L^{s}(\mathbb{R}^{3}),\; s\in(2,6),\\
u_n\to u,\quad \text{in } L^{s}_{\rm loc}(\mathbb{R}^{3}),\; s\in[1,6),\\
u_n(x)\to u(x),\quad \text{a.e.  } x\in \mathbb{R}^{3}.
\end{gather*}
For $R>1$,  from (H2) we obtain
\begin{equation}\label{3.5}
\begin{aligned}
\big|\int_{\mathbb{R}^{3}}K(x)|u_n|^{p-1}u_n(u_n-u)\mathrm{d}x\big| 
&\leq \int_{\mathbb{R}^{3}}K(x)|u_n|^{p}|u_n-u|\mathrm{d}x\\
&\leq C_0\int_{\mathbb{R}^{3}}(1+|x|^{l})|u_n|^{p}|u_n-u|\mathrm{d}x\\
&=C_0\int_{B_{R}(0)}(1+|x|^{l})|u_n|^{p}|u_n-u|\mathrm{d}x \\
&\quad +C_0\int_{\mathbb{R}^{3}\backslash B_{R}(0)}(1+|x|^{l})|u_n|^{p}
 |u_n-u|\mathrm{d}x\\
&=:I_{1}+I_{2}.
\end{aligned}
\end{equation}
By H\"older's inequality, we have
\begin{equation}\label{3.6}
\begin{aligned}
I_{1}&=C_0\int_{B_{R}(0)}(1+|x|^{l})|u_n|^{p}|u_n-u|\mathrm{d}x\\
&\leq C(R)(\int_{B_{R}(0)}|u_n|^{p+1}\mathrm{d}x)^{\frac{p}{p+1}}
\Big(\int_{B_{R}(0)}|u_n-u|^{p+1}\mathrm{d}x\Big)^{\frac{1}{p+1}}\\
&=C(R)|u_n|_{L^{p+1}(B_R(0))}^{p}|u_n-u|_{L^{p+1}(B_R(0))}\\
&\leq C(R)\|u_n\|^{p}|u_n-u|_{L^{p+1}(B_R(0))}.
\end{aligned}
\end{equation}
Moreover,  from Lemma~\ref{l1} we deduce that
\begin{equation}\label{3.7}
\begin{aligned}
I_{2}&=C_0\int_{\mathbb{R}^{3}\backslash B_{R}(0)}
 (1+|x|^{l})|u_n|^{p}|u_n-u|\mathrm{d}x\\
&\leq 2C_0\int_{\mathbb{R}^{3}\backslash B_{R}(0)}|x|^{l}(|u_n|^{p+1}
 +|u_n|^{p}|u|)\mathrm{d}x\\
&\leq C(\|u_n\|^{p+1}+\|u_n\|^p\|u\|)\int_{R}^\infty r^{l-p+1}\mathrm{d}r\\
&= C\frac{R^{l-p+2}}{p-2-l}(\|u_n\|^{p+1}+\|u_n\|^p\|u\|).
\end{aligned}
\end{equation}
Since $3< p+1<6$, $u_n\to u$ in $L^{p+1}(B_R(0))$. Therefore, by 
\eqref{3.5}-\eqref{3.7},  we can find $R>1$ large enough such that
\begin{equation}\label{3.8}
\int_{\mathbb{R}^{3}}K(x)|u_n|^{p-1}u_n(u_n-u)\mathrm{d}x\to 0\quad 
\text{as }n\to\infty.
\end{equation}
Note that $(J_{\lambda,\mu}^{T})'(u_n)\to 0$. Then by \eqref{3.8}, we deduce that
\begin{align*}
0&\leftarrow\langle(J_{\lambda,\mu}^{T})'(u_n),u_n-u\rangle\\
&=a(u_n,u_n-u)+\lambda h_{T}(u_n)\|u_n\|^{2}(u_n,u_n-u)\\
&\quad +\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}(u_n,u_n-u)
-\mu\int_{\mathbb{R}^{3}}K(x)|u_n|^{p-1}u_n(u_n-u)\mathrm{d}x\\
&=[a+\lambda h_{T}(u_n)\|u_n\|^{2}+
\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}](u_n,u_n-u)+o(1).
\end{align*}
So
\begin{equation}\label{3.3}
(a+\lambda h_{T}(u_n)\|u_n\|^{2}
+\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4})(u_n,u_n-u)\to 0.
\end{equation}
Next, we claim that
\begin{equation}\label{3.2}
(u_n,u_n-u)\to 0\quad \text{as }n\to\infty.
\end{equation}
Note that  if $\|u_n\|^{2}>2T^{2}$, then $h_{T}(u_n)=0$ and 
$\phi'(\frac{\|u_n\|^{2}}{T^{2}})=0$.
If $\|u_n\|^{2}\leq2T^{2}$, then 
$$
|\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}|\leq 8T^{4}.
$$
Since $4\lambda T^{2}<a$, we have
\begin{align*}
a+\lambda h_{T}(u_n)\|u_n\|^{2}
+\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}
&\geq a+\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}\\
&\geq a+\frac{\lambda}{2T^{2}}(-8T^{4})\\
&= a-4\lambda T^{2}>0
\end{align*}
and
\begin{align*}
a+\lambda h_{T}(u_n)\|u_n\|^{2}+\frac{\lambda}{2T^{2}}
\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}
&\leq a+\lambda \|u_n\|^{2}+\frac{\lambda}{2T^{2}}
 \phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}\\
&\leq a+2\lambda T^{2}+\frac{\lambda}{2T^{2}}(8T^{4})\\
&= a+6\lambda T^{2}\leq\frac{5 a}{2}.
\end{align*}
This combined with \eqref{3.3} yields  that \eqref{3.2} holds.

Since $u_n\rightharpoonup u$ in $H_{r}^{1}(\mathbb{R}^{3})$,
 we infer from the above claim  that up to a subsequence,
 $$
u_n\to u\quad\text{in }H_{r}^{1}(\mathbb{R}^{3}).
$$
The proof is complete.
\end{proof}

\begin{lemma}\label{l7}
Let $4\lambda T^{2}<a$. For almost every $\mu\in I$,
there exists $u^{\mu}\in H_{r}^{1}(\mathbb{R}^{3})\backslash\{0\}$ such that
\begin{equation}\label{3.9}
(J_{\lambda,\mu}^{T})'(u^{\mu})=0\quad\text{and}\quad
J_{\lambda,\mu}^{T}(u^{\mu})=c_{\mu}.
\end{equation}
\end{lemma}

\begin{proof}
By Lemma~\ref{l4}, Lemma~\ref{l5} and Theorem~\ref{t2}, for almost every 
$\mu\in I$, we can find  a bounded sequence 
$\{u_n^{\mu}\}\subset H_{r}^{1}(\mathbb{R}^{3})$ satisfying
\begin{gather*}
J_{\lambda,\mu}^{T}(u_n^{\mu})\to c_{\mu},\\
(J_{\lambda,\mu}^{T})'(u_n^{\mu})\to 0.
\end{gather*}
Furthermore, by using Lemma~\ref{l6}, we conclude  that there is 
 $u^{\mu}\in H_{r}^{1}(\mathbb{R}^{3})$ such that $u_n^{\mu}\to u^{\mu}$ 
in $H_{r}^{1}(\mathbb{R}^{3})$. So the continuity of $J_{\lambda,\mu}^{T}$ and
$(J_{\lambda,\mu}^{T})'$ imply that \eqref{3.9} holds.
This  completes the proof.
\end{proof}


According to Lemma~\ref{l7}, we obtain that there exist  sequences
$\{\mu_n\}\subset I$ with  $\mu_n\to 1^{-}$
and  $\{u_n\}\subset H_{r}^{1}(\mathbb{R}^{3})$
such that
\begin{equation}\label{19}
J_{\lambda,\mu_n}^{T}(u_n)=c_{\mu_n},\quad (J_{\lambda,\mu_n}^{T})'(u_n)=0.
\end{equation}
In what follows, we shall show $\|u_n\|\leq T$, which is a critical key 
in the proof of  existence of solutions to \eqref{1.1}.

\begin{lemma}\label{l8}
Assume $\{\mu_n\}\subset I$ with $\mu_n\to 1^{-}$ and $\{u_n\}$ satisfies \eqref{19}. 
Then for $T>0$ sufficiently large, there exists $\lambda_0=\lambda_0({T})$  
with $4\lambda_0T^{2}<a$ such that for any $\lambda\in[0,\lambda_0)$, up to 
a subsequence,
$$
\|u_n\|\leq T\quad\forall n.
$$
\end{lemma}

\begin{proof}
The proof consists of three steps.
\smallskip

\noindent \textbf{Step 1}. By the  definition of $c_\mu$ and Lemma~\ref{l5},
 we conclude  that
\begin{equation}\label{21}
\begin{aligned}
\alpha
&\leq c_{\mu_n}
\leq \sup_{t\in[0,\infty)}J_{\lambda,\mu_n}^{T}(t\varphi)\\
&= \sup_{t\in[0,\infty)}\big{[}\frac{1}{2}a t^{2}
 +\frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4}-\frac{\mu_n}{p+1}|t|^{p+1}
\int_{\mathbb{R}^{3}}K(x)\varphi^{p+1}\mathrm{d}x\big{]}\\
&\leq \max_{t\in[0,\infty)}\big{[}\frac{1}{2}a t^{2}
 -\frac{1}{2(p+1)}|t|^{p+1}\int_{B_{R}(0)}K(x)\varphi^{p+1}\mathrm{d}x\big{]}\\
&\quad +\sup_{t\in[0,\infty)}\frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4}\\
&= \frac{a(p-1)}{2(p+1)}(\frac{2a}{ \int_{B_{R}(0)}K(x)\varphi^{p+1}
 \mathrm{d}x})^{\frac{2}{p-1}}+\sup_{t\in[0,\infty)}
 \frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4}\\
&=:A+\sup_{t\in[0,\infty)}\frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4},
\end{aligned}
\end{equation}
where $\varphi$ is defined in Lemma~\ref{l4}.
\smallskip

\noindent\textbf{Step 2}. It is easy to see that  if $t^2\geq2T^{2}$, 
then $\phi(\frac{t^{2}}{T^{2}})=0$ and so 
$$
\sup_{t\in[\sqrt{2}T,\infty)}\frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4}=0,
$$
while if  $t^2<2T^{2}$, then
$$
\sup_{t\in[0,\sqrt{2}T)}\frac{1}{4}\lambda \phi(\frac{t^{2}}{T^{2}})t^{4}
<\lambda T^{4}.
$$
This, combined with \eqref{21}, yields 
 $$
0<\alpha\leq c_{\mu_n}\leq A+\lambda T^{4}.
$$
\smallskip

\noindent\textbf{Step 3}. Note that
$$
c_{\mu_n}=J_{\lambda,\mu_n}^{T}(u_n)
=\frac{1}{2}a\|u_n\|^2+\frac{1}{4}\lambda h_{T}(u_n)\|u_n\|^{4}
-\frac{\mu_n}{p+1}\int_{\mathbb{R}^{3}}K(x)|u_n|^{p+1}\mathrm{d}x
$$
and
\begin{align*}
0&=\langle(J_{\lambda,\mu_n}^{T})'(u_n),u_n\rangle \\
&=a(u_n,u_n)+\lambda h_{T}(u_n)\|u_n\|^{2}(u_n,u_n)\\
&\quad +\frac{\lambda}{2T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{4}
(u_n,u_n)-\mu_n\int_{\mathbb{R}^{3}}K(x)|u_n|^{p+1}\mathrm{d}x.
\end{align*}
Thus, we have that
\begin{equation}\label{22}
\begin{aligned}
c_{\mu_n}
&= J_{\lambda,\mu_n}^{T}(u_n)-\frac{1}{p+1}\langle(J_{\lambda,\mu_n}^{T})'(u_n),
 u_n\rangle\\
&=a(\frac{1}{2}-\frac{1}{p+1})\|u_n\|^{2}+(\frac{1}{4}
 -\frac{1}{p+1})\lambda h_{T}(u_n)\|u_n\|^{4} \\
&\quad -\frac{\lambda}{2(p+1)T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{6}.
\end{aligned}
\end{equation}
We claim: there exists a subsequence of $\{u_n\}$ which is uniformly bounded by $T$.
 By way of contradiction, we distinguish  two cases to prove the claim.
\smallskip

\noindent\textbf{Case 1.} Up to a subsequence, $\|u_n\|^{2}>2T^{2}$ for all $n$.
It is easy to check that  $h_{T}(u_n)=0$ and $\phi'(\frac{\|u_n\|^{2}}{T^{2}})=0$. 
Then we deduce from \eqref{22} and Step 2 that
$$
\Big(\frac{1}{2}-\frac{1}{p+1}\Big)a\|u_n\|^{2}
=c_{\mu_n}\leq A+\lambda T^{4}.
$$
Since  $4\lambda T^{4}<a$,
$$
2T^{2}<\|u_n\|^{2}\leq\Big[\Big(\frac{1}{2}-\frac{1}{p+1}\Big)a\Big]^{-1}
(A+\lambda T^{4})
\leq\Big[\Big(\frac{1}{2}-\frac{1}{p+1}\Big)a\Big]^{-1}(A+\frac{a}{4}),
$$
which contradicts with the assumption that  $T$ is sufficiently large.
\smallskip

\noindent\textbf{Case 2.}
 Up to a subsequence, $T^{2}<\|u_n\|^{2}\leq2T^{2}$ for all $n$.
By \eqref{22} and Step 2, we have
$c_{\mu_n} \leq A+\lambda T^{4}$.
Furthermore, since $p\in(2,5)$, we obtain
\begin{align*}
&(\frac{1}{2}-\frac{1}{p+1})aT^{2} \\
&<(\frac{1}{2}-\frac{1}{p+1})a\|u_n\|^{2}\\
&= c_{\mu_n}+\frac{\lambda}{2(p+1)T^{2}}\phi'(\frac{\|u_n\|^{2}}{T^{2}})\|u_n\|^{6}
-(\frac{1}{4}-\frac{1}{p+1})\lambda h_{T}(u_n)\|u_n\|^{4}\\
&\leq A+\lambda T^{4}+\frac{\lambda}{2(p+1)T^{2}}\big{|}
 \phi'(\frac{\|u_n\|^{2}}{T^{2}})\big{|}\|u_n\|^{6}
+(\frac{1}{4}+\frac{1}{p+1})\lambda h_{T}(u_n)\|u_n\|^{4}\\
&\leq A+\lambda T^{4}+\frac{8}{p+1}\lambda T^{4}
 +(\frac{1}{4}+\frac{1}{p+1})\times 4\lambda T^4\\
&\leq A+\frac{a(p+14)}{4(p+1)},
\end{align*}
which contradicts  the assumption that  $T$ is  sufficiently large.
Then the  claim holds and we complete the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm1.3}]
  We assume  $T,\lambda_0$ are defined as in Lemma~\ref{l8} and for each 
$\mu_n\in I$, $u_n$ is a  critical point for $J_{\lambda,\mu_n}^{T}$ 
at level $c_{\mu_n}$. By using  Lemma~\ref{l8},  up to a subsequence, we 
can also assume  $\|u_n\|\leq T$.
Hence, $h_T(u_n)=1$ and
 $$
J_{\lambda,\mu_n}^{T}(u_n)=\frac{1}{2}a\|u_n\|^2+\frac{1}{4}
\lambda \|u_n\|^{4}-\frac{\mu_n}{p+1}\int_{\mathbb{R}^{3}}K(x)|u_n|^{p+1}\mathrm{d}x.
$$


Next, we claim that $\{u_n\}$ is a Palais-Smale sequence of $J_{\lambda}$.
Indeed, since  $\|u_n\|$ is  bounded for all $n$, we conclude that  
$|J_{\lambda}(u_n)|$ is  bounded for all $n$. Moreover, we have
\[
\langle J'_{\lambda}(u_n),v\rangle=\langle(J_{\lambda,\mu_n}^{T})'(u_n),
v\rangle+(\mu_n-1)\int_{\mathbb{R}^{3}}K(x)|u_n|^{p-1}u_nv\mathrm{d}x,
\]
for any $v\in H_{r}^{1}(\mathbb{R}^{3})$.
Note that $(J_{\lambda,\mu_n}^{T})'(u_n)=0$ and $\mu_n\to 1^-$. 
Then we obtain $J'_{\lambda}(u_n)\to0$ and the claim holds. 
By Lemma~\ref{l6}, $\{u_n\}$ has a convergent subsequence. We may assume that 
$u_n\to \bar{u}$ in $H^1_r(\mathbb{R}^3)$. Therefore, $J'_{\lambda}(\bar{u})=0$ 
and by Lemma \ref{l5},   
$$
J_{\lambda}(\bar{u})=\lim_{n\to\infty
}J_{\lambda}(u_n)=\lim_{n\to\infty
}J_{\lambda,\mu_n}^{T}(u_n)=c_{\mu_n}\geq \alpha>0.
$$ 
Consequently,  $\bar{u}$ is nontrivial solution of \eqref{1.1}. 
The proof is complete. 
\end{proof}

\subsection*{Acknowledgments}
This research was supported partially by the
National Natural Science Foundation of  China (Grant No.
11571371).


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\end{document}