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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 102, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/102\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for nonhomogeneous Klein-Gordon-Maxwell
equations}

\author[L. Xu, H. Chen \hfil EJDE-2015/102\hfilneg]
{Liping Xu, Haibo Chen}

\address{Liping Xu \newline
School of Mathematics and Statistics, Central South University,
Changsha 410075,  China.\newline
Department of Mathematics and Statistics,
Henan University of Science and Technology, Luoyang 471003, China}
\email{x.liping@126.com}

\address{Haibo Chen (corresponding author)\newline
School of Mathematics and Statistics, Central South University,
Changsha 410075,  China}
\email{math\_chb@csu.edu.cn}

\thanks{Submitted January 21, 2015. Published April 16, 2015.}
\makeatletter
\@namedef{subjclassname@2010}{\textup{2010} Mathematics Subject Classification}
\makeatother
\subjclass[2010]{35J20, 35J65, 35J60}
\keywords{Nonhomogeneous Klein-Gordon-Maxwell equations;
\hfill\break\indent  multiple solutions;  Poho\u{z}aev identity; variational method}

\begin{abstract}
 This article concerns the  nonhomogeneous Klein-Gordon-Maxwell equation
 \begin{gather*}
 -\Delta u+u-(2\omega +\phi)\phi u=  |u|^{p-1}u +h(x),\quad\text{in }\mathbb{R}^3,\\
 \Delta \phi=(\omega +\phi)u^2,\quad\text{in }\mathbb{R}^3,
 \end{gather*}
 where $\omega>0$ is constant, $p\in(1,5)$.  Under appropriate assumptions
 on $h(x)$, the existence of at least two solutions is obtained by applying
 the Ekeland's variational principle and the Mountain Pass Theorem in
 critical point theory.

\end{abstract}

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

\section{Introduction}

In this article, we consider the existence of multiple solutions for the 
nonhomogeneous Klein-Gordon-Maxwell equation
\begin{equation}
\begin{gathered}
-\Delta u+u-(2\omega +\phi)\phi u=  |u|^{p-1}u +h(x),\quad\text{in }\mathbb{R}^3,\\
\Delta \phi=(\omega +\phi)u^2, \quad\text{in }\mathbb{R}^3,
\end{gathered}\label{e1.1}
\end{equation}
where $\omega>0$ is constant, $1<p<5$. We assume that the  function $h(x)$
satisfies the following hypotheses.
\begin{itemize}
\item[(H1)] $0 \leq h(x)\in L^2(\mathbb{R}^3)\bigcap C^1(\mathbb{R}^3)$ and
 $h(x)=h(|x|)\not\equiv 0$.
\item[(H2)] $\|h(x)\|_{L^2}<m_p$, where
$m_p=\frac{p-1}{2p}(\frac{p+1}{2p \eta_p^{p+1}})^{\frac{1}{p-1}}$,
$\eta_p>0$ is the  Sobolev embedding constant.
\item[(H3)] $\langle\nabla h(x),x\rangle \in L^2(\mathbb{R}^3)$.
\end{itemize}
Such  system was first introduced in \cite{b1} as a model which describes
 the nonlinear Klein-Gordon field interacting with the electromagnetic
field in the electrostatic case. The unknowns of the system are the field
$u$ associated to the particle and the electric potential $\phi$, while
 $\omega$ denotes the phase. The presence of the nonlinear term simulates
the interaction between many particles or external nonlinear perturbations.

When $h(x)=0$, the homogeneous case, a several 
works have been devoted to the Klein-Gordon-Maxwell:
\begin{equation}
 \begin{gathered}
-\Delta u+[m^2-(\omega+\phi)^2]u=|u|^{p-1}u ,\quad\text{in } \mathbb{R}^3,\\
\Delta \phi=( \omega+\phi)u^2,\quad\text{in }\mathbb{R}^3.
\end{gathered} \label{e1.2}
\end{equation}
 The first result is due to Benci and Fortunato. In \cite{b1}, they proved the
existence of infinitely many radially symmetric solutions for \eqref{e1.2}
under the assumption $3<p<5$. D'Aprile and Mugnai \cite{d1} covered the case $1<p<3$
and the case $p=3$.  Under the assumption $1<p<5$,
Azzollini and Pomponio proved the existence of a ground state solution
for \eqref{e1.2} in \cite{a1}.
In \cite{d2}, some nonexistence results of nontrivial  solutions for \eqref{e1.2}
were obtained when $p\geq5$ or $p\leq 1$.

Recently, by combining the minimization of the corresponding Euler-Lagrange 
functional on the Nehari manifold with the Brezis and Nirenberg technique, 
Carri\~ao, Cunha and Miyagaki proved  the existence of positive ground 
state solutions of system \eqref{e1.1} with $h(x)=0$ when the nonlinearity 
exhibits critical growth, see \cite{c1}.

The nonhomogeneous case, that is $h(x)\neq 0$. The authors \cite{c2} considered the 
following nonhomogeneous Klein-Gordon-Maxwell equations:
 \begin{equation}
\begin{gathered}
-\Delta u+[m^2-(\omega+\phi)^2] u=  |u|^{p-2}u +h(x)\quad\text{in }\mathbb{R}^3,\\
\Delta \phi=( \omega+\phi)u^2,\quad\text{in }\mathbb{R}^3,
\end{gathered} \label{e1.3}
\end{equation}
where $m>\omega>0$ and $2<p<6$. This is the first paper dealing with the
nonhomogeneous Klein-Gordon-Maxwell equations. However,
since \cite[equality (9)]{c2} is in error, the authors could not obtain
the boundedness of $\{u_n\} $ under the assumption $2<p<6$.
Then \cite[Lemma 3.6 and Theorem 1.3]{c2} could not be obtained.

Motivated by the works described above, in the present paper, we establish 
the existence of multiple solution results for system \eqref{e1.1}. 
The method is inspired by \cite{j3}.

 By Ekeland's variational principle, it is not difficult to get a solution 
$u_0$ of \eqref{e1.1} for all $\omega >0,\,1<p<5$ and $\|h\|_{L^2}$ suitably small. 
Moreover, $u_0$ is a local minimizer of $I_\omega$ and $I_\omega(u_0)<0$, where 
$I_\omega$ is defined by \eqref{e2.2}. However, under our assumptions it seems 
difficult to get a second solution(different from $u_0$) of \eqref{e1.1} 
by applying the Mountain Pass Theorem. So we have to study problem \eqref{e1.1} 
in the following two cases: $p\in(1,2]$ and $p\in(2,5)$, respectively.

For $p\in[3,5)$, we can  directly prove  the boundedness of $\{u_n\}$ and the 
$(PS)_c$ condition. But for $p\in(2,3)$, it is difficult to show if the $(PS)_c$ 
condition satisfies. To overcome the difficulty, by introducing a suitable 
approximation problem, we use an indirect method to obtain the boundedness 
of $\{u_n\}$ sequence for $I_\omega$ based on the weak solutions of the 
approximation problem, and then show that this special (PS) sequence
converges to a solution of problem \eqref{e1.1}.
However, when $p\in(1,2]$, it is more delicate. For this case, we note 
that \eqref{e1.1} has no positive energy solution for $\omega>0$ large 
enough (see Theorem \ref{thm5.1}). Based on this observation, by using the cut-off 
technique as in \cite{j1},  we finally get a positive energy solution for 
problem \eqref{e1.1} with $\omega>0$ small enough.

Our main results read as follows.

\begin{theorem} \label{thm1.1}  
Let $p\in (2,5)$ and {\rm (H1)--(H3)} hold. Then, for all $\omega >0$,  
problem \eqref{e1.1} has at least two nontrivial solutions  $u_0$ and $u_1$ 
such that $I_\omega(u_0)<0<I_\omega(u_1)$.
\end{theorem}

\begin{theorem} \label{thm1.2} 
Assume that $p\in (1,2]$ and {\rm (H1)--(H2)} hold. Then, 
if $\omega >0$ small, problem \eqref{e1.1} possesses two nontrivial 
solutions $u_0$ and $\tilde{u_1}$ such that $I_\omega(u_0)<0<I_\omega(\tilde{u_1})$. 
However, if $\omega >0$ large enough, problem \eqref{e1.1} has no solution
 with positive energy.
\end{theorem}

\begin{remark} \label{rmk1.3}\rm
 According to our results, for any $\omega>0$,  problem \eqref{e1.1}
 has always a solution with negative energy.
\end{remark}

Throughout this article m$C$ denotes various positive constants.

\section{Variational setting}

In this section, we introduce  some preliminary results concerning the 
variational structure for \eqref{e1.1}. Our working space is 
$E:=H^1(\mathbb{R}^3)$ equipped with the inner product and norm
$$
\langle u,v\rangle :=\int_{\mathbb{R}^3}{(\nabla u\cdot \nabla v+uv)dx},\quad 
\| u\|:=\langle u,u\rangle^{1/2}.
$$
Let $D^{1,2}(\mathbb{R}^3)$ be the completion of  $C_0^\infty(\mathbb{R}^3,R)$ 
with respect to the norm 
$$
\|u\|_{D^{1,2}}=(\int_{\mathbb{R}^3}{|\nabla u|^2dx})^{\frac{1}{2}}.
$$
And for any $1\leq s<\infty$, 
$\|u\|_{L^s}:=(\int_{\mathbb{R}^3}|u|^s dx)^{\frac{1}{s}}$ 
denotes  the usual norm of the Lebesgue space $L^s(\mathbb{R}^3)$.

Due to the variational nature of problem \eqref{e1.1}, its weak solutions 
$(u,\phi )\in E\times D^{1,2}(\mathbb{R}^3)$ are critical points of the functional 
$J:E\times D^{1,2}(\mathbb{R}^3)\to R$ defined by
\begin{align*}
J(u,\phi)&=\frac{1}{2}\|u\|^2-\frac{1}{2}\int_{\mathbb{R}^3}{|\nabla \phi|^2dx}
-\frac{1}{2}\int_{\mathbb{R}^3}(2\omega +\phi)\phi u^2dx \\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^3}{|u|^{p+1}dx}-\int_{\mathbb{R}^3}{h(x)u\,dx}.
\end{align*}
Obviously, the action functional $J$ belongs to 
$C^1(E\times D^{1,2}(\mathbb{R}^3),R)$ and  exhibits a strong indefiniteness.
To avoid the indefiniteness we apply a reduction method, as has been done 
by the aforementioned authors.

\begin{lemma}[\cite{d1,d2}] \label{lem2.1}
 For every $u\in E$ there exists a unique $\phi=\phi_u \in D^{1,2}(\mathbb{R}^3)$
 which solves $\Delta \phi=(w+\phi)u^2$. Furthermore 
\begin{itemize}
\item[(i)] in the set $\{x:u(x)\neq 0\}$ we have $-\omega \leq \phi_u\leq 0$ 
 for $\omega >0$;
\item[(ii)] if $u$ is radially symmetric, $\phi_u$ is radial too.
\end{itemize}
 According to Lemma \ref{lem2.1},  we can consider the functional $I_\omega:E\to R$ 
defined by $I_\omega(u)=J(u,\phi_u)$. After multiplying both members of the 
second equation in  equations \eqref{e1.1} by  $\phi_u$ and integrating 
by parts, we obtain
\begin{equation}
\int_{\mathbb{R}^3}|\nabla \phi_u|^2dx
=-\int_{\mathbb{R}^3}\omega\phi_uu^2dx-\int_{\mathbb{R}^3}\phi_u^2u^2dx.
\label{e2.1}
\end{equation}
Then, the reduced functional takes the form
\begin{equation}
I_\omega(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+u^2-\omega\phi_uu^2)dx
-\frac{1}{p+1}\int_{\mathbb{R}^3}{|u|^{p+1}dx}-\int_{\mathbb{R}^3}{h(x)u\,dx}.
\label{e2.2}
\end{equation}
Furthermore $I$ is $C^1$ and we have for any $u,v\in E$,
\begin{equation}
\begin{aligned}
\langle I'_\omega(u),v\rangle
&=\int_{\mathbb{R}^3}(\nabla u\cdot\nabla v+uv-(2\omega+\phi_u)\phi_uuv)dx\\
&\quad -\int_{\mathbb{R}^3}{|u|^{p-1}uv dx}-\int_{\mathbb{R}^3}{h(x)v\,dx}.
\end{aligned}
\label{e2.3}
\end{equation}
\end{lemma}

\begin{remark} \label{rmk2.2} \rm
By \eqref{e2.1}, we can note that
$$
\|\phi_u\|_{D^{1,2}(\mathbb{R}^3)}^2\leq\int_{\mathbb{R}^3}\omega|\phi_u|u^2dx
\leq\omega \|\phi_u\|_{L^6}\|u\|^2_{L^{12/5}},
$$
then
$$
\|\phi_u\|_{D^{1,2}(\mathbb{R}^3)}\leq C_1\omega\|u\|^2_{L^{12/5}}, \quad
\int_{\mathbb{R}^3}\omega|\phi_u|u^2dx\leq\omega C_1 \|u\|^4.
$$
\end{remark}

Now, we can apply \cite[Lemma 2.2]{d2} to our functional $I_\omega$ and obtain
the following result.

\begin{lemma} \label{lem2.3} 
The following statements are equivalent:
\begin{itemize}
\item[(1)] $(u,\phi)\in E\times D^{1,2}(\mathbb{R}^3)$ is a critical point of 
$J$ (i.e.$(u,\phi)$ is a solution of \eqref{e1.1}.
\item[(2)] $u$ is a critical point of $I_\omega$ and $\phi =\phi_u$.
\end{itemize}
\end{lemma}

Set 
$$
H^1_r(\mathbb{R}^3):=\{u\in H^1(\mathbb{R}^3):u=u(r),\; r=|x|\}.
$$
We shall consider the functional $I_\omega$ on $H^1_r(\mathbb{R}^3)$. 
Then any critical point $u\in H^1_r(\mathbb{R}^3)$ of 
$I_\omega|_{H^1_r(\mathbb{R}^3)}$ is also a critical point of 
$I_\omega$ since $H^1_r(\mathbb{R}^3)$ is a natural constraint for $I_\omega$. 
Thus we are reduced to look for critical points of $I_\omega|_{H^1_r(\mathbb{R}^3)}$. 
In the following, we still denote $I_\omega|_{H^1_r(\mathbb{R}^3)}$ by 
$I_\omega$. It follows from \cite{b1} that for $2<s<6$, $H^1_r(\mathbb{R}^3)$ 
is compactly embedded into $L^s(\mathbb{R}^3)$. Therefore, there exists a 
positive constant $\eta_s>0$ such that
$$
\|u\|_{L^s}\leq \eta_s\|u\|,\quad \forall u\in H^1_r(\mathbb{R}^3).
$$
To obtain our results, the following theorem will be needed in our argument.

\begin{theorem}[\cite{j2}] \label{thm2.4}
 $(X,\|\cdot\|)$  is a Banach space and $S\subset R_+$ an interval. 
Let us consider the family of $C^1$ functionals on $X$
$$
I_\lambda (u)=A(u)-\lambda B(u),\quad \lambda \in S ,
$$
with $B$ nonnegative and either $A(u)\to +\infty $ or $B(u)\to+\infty$ 
as $\|u\|\to \infty$ and such that $I_\lambda(0)=0$. Set
$$
\Gamma _\lambda=\{ \gamma \in C([0,1],X):\gamma (0)=0,I_\lambda(\gamma(1))<0\},
\quad\text{for any }\lambda \in S.
$$
If for every $\lambda\in S$ the set $\Gamma_\lambda $ is nonempty and
$ c_\lambda=\inf_{\gamma \in \Gamma _\lambda}\max _{t\in[0,1]}I_\lambda(\gamma(t))>0$,
then for almost every $\lambda\in S$, there exists a sequence 
$\{u_n\}\subset X$ satisfying
\begin{itemize}
\item[(i)] $\{u_n\}$ is bounded;
\item[(ii)] $I_\lambda (u_n)\to c_\lambda$;
\item[(iii)] $I'_\lambda (u_n)\to 0$ in the dual  $X^{-1}$ of $X$.
\end{itemize}
\end{theorem}

\section{A weak solution with negative energy}

In this section, we prove that \eqref{e1.1} has a weak solution with negative 
energy for any $\omega >0$ and $p\in (1,5)$.
With the aid of Ekeland's variational principle, this weak solution is obtained 
by seeking a local minimum of the energy functional $I_\omega$.

\begin{lemma} \label{lem3.1} 
Suppose that $p\in(1,5)$ and {\rm (H1)--(H2)} hold. Then there 
exist $\rho$, $\alpha$, and $m_p$ positive  such that $I_\omega(u)|_{\|u\|=\rho}\geq \alpha >0$
 for all $h$ satisfying $\|h\|_{L^2}<m_p$, where 
$m_p=\frac{p-1}{2p}(\frac{p+1}{2p\eta_P^{p+1}})^{\frac{1}{p-1}}$.
\end{lemma}

\begin{proof} 
For all $\omega>0$ and $u\in H^1(\mathbb{R}^3)$, by Lemma \ref{lem2.1}, the H\"older 
inequality and  Sobolev's embedding theorem, we have
\begin{equation}
\begin{aligned}
I_\omega(u)
&\geq\frac{1}{2}\|u\|^2-\frac{1}{p+1}\|u\|_{L^p}^{p+1}-\|h\|_{L^2}\|u\|\\
&\geq\frac{1}{2}\|u\|^2-\frac{\eta_p^{p+1}}{p+1}\|u\|^{p+1}-\|h\|_{L^2}\|u\|\\
&=\|u\|(\frac{1}{2}\|u\|-\frac{\eta_p^{p+1}}{p+1}\|u\|^{p}-\|h\|_{L^2}).
\end{aligned} \label{e3.1}
\end{equation}
Set
\[
g(t)=\frac{1}{2}t-\frac{\eta_p^{p+1}}{p+1}t^p\quad\text{for }t\geq 0.
\]
 By direct calculations, we see that
$\max_{t\geq 0}g(t)=g(\rho)=\frac{p-1}{2p}(\frac{p+1}{2p \eta_p^{p+1}})
^{\frac{1}{p-1}}:=m_p $, where $\rho=(\frac{p+1}{2p \eta_p^{p+1}})^{\frac{1}{p-1}}$.
 Then it follows from \eqref{e3.1} that, if $\|h\|_{L^2}<m_p$, there exists
$\alpha=\rho(g(\rho)-\|h\|_{L^2})>0$ such that
$I_\omega(u)|_{\|u\|=\rho}\geq \alpha >0$ for all $\omega>0$.
\end{proof}

\begin{lemma} \label{lem3.2}
 If $p\in (1,5)$ and {\rm (H1)--(H2)} hold. Then, for any $\omega>0$, 
there exists $u_0\in H^1_r(\mathbb{R}^3)$ such that
$$
I_\omega(u_0)=\inf\{I_\omega(u):u\in H^1_r(\mathbb{R}^3)\text{ and }
\|u\|\leq \rho\}<0.
$$
where $\rho$ is given by Lemma \ref{lem3.1}. Moreover, $u_0$ is a solution of problem
 \eqref{e1.1}.
\end{lemma}

\begin{proof} 
 By (H1), we can choose a function $\varphi \in H^1_r(\mathbb{R}^3)$ 
such that $\int_{\mathbb{R}^3}h(x)\varphi dx>0$.
Hence, for $t>0$ small enough, we obtain
\begin{align*}
I_\omega(t\varphi)
&=\frac{t^2}{2}\int_{\mathbb{R}^3}(|\nabla \varphi|^2+\varphi^2)dx
 -\frac{1}{2}\int_{\mathbb{R}^3}\omega\phi_{t\varphi}(t\varphi)^2dx\\
&\quad -\frac{t^{p+1}}{p+1}\int_{\mathbb{R}^3}{|\varphi|^{p+1}dx}
 -t\int_{\mathbb{R}^3}{h(x)\varphi dx}\\
&\leq \frac{t^2}{2}\|\varphi\|^2+\frac{t^4C_1\omega}{2}\|\varphi\|^4
 -\frac{t^{p+1}}{p+1}\int_{\mathbb{R}^3}{|\varphi|^{p+1}dx}
 -t\int_{\mathbb{R}^3}{h(x)\varphi dx}<0,
\end{align*}
which shows that $c_0=\inf\{I_\omega(u):u\in \bar{B}_\rho\}<0$, 
where 
\[
\bar{B}_\rho=\{u\in H^1_r(\mathbb{R}^3)\text{ and }\|u\|\leq \rho\}.
\]
 By the Ekeland's variational principle, there exists a sequence 
$\{u_n\}\subset \bar{B}_\rho$ such that
$$
c_0\leq I_\omega(u_n)\leq c_0+\frac{1}{n},\quad
I_\omega(\vartheta)\geq I_\omega(u_n)-\frac{1}{n}\|\vartheta-u_n\|\quad
\forall \vartheta\in \bar{B}_\rho.
$$
By a standard procedure, see, for example \cite{z1}, we can show that 
$\{u_n\}$ is bounded (PS) sequence of $I_\omega$. Then, by the compactness 
of the embedding $H^1_r(\mathbb{R}^3)\hookrightarrow L^s(\mathbb{R}^3)(2<s<6)$, 
there exists $u_0\in H^1_r(\mathbb{R}^3)$ such that $\{u_n\}\to u_0$ strongly 
in $H^1_r(\mathbb{R}^3)$. Hence $I_\omega(u_0)=c_0<0, ~I'_\omega(u_0)=0$.
\end{proof}

\section{Positive energy solution for $p\in(2,5)$}

  In this section, we aim to prove that problem \eqref{e1.1} has a positive 
energy solution for any $\omega>0,~p\in (2,5)$. It is well-known that,
 for $p\in[3,5)$, we can directly prove  the boundedness of $\{u_n\}$ 
of the functional $I_\omega$. But for  $p\in(1,3)$, it is not easy to do this. 
Particularly, $p\in(1,2)$ is the hardest case. To show the boundedness of 
a (PS) sequence of $I_\omega$ when $p\in(2,5)$ is also nontrivial.
 Here we have to use Theorem \ref{thm2.4}. Consider the  approximation problem
\begin{equation}
\begin{gathered}
-\Delta u+u-(2\omega +\phi)\phi u=  \lambda|u|^{p-1}u +h(x),\quad\text{in }
\mathbb{R}^3,\\
\Delta \phi=(\omega +\phi)u^2,\quad\text{in }\mathbb{R}^3,
\end{gathered} \label{e*}
\end{equation}
where $p\in(2,5)$ and $\lambda\in[1/2,1]$. Set $X=H_r^1(\mathbb{R}^3)$,
\[
A(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}\int_{\mathbb{R}^3}\omega\phi_uu^2dx
-\int_{\mathbb{R}^3}h(x)u\,dx
\]
 and $ B(u)=\frac{1}{p+1}\int_{\mathbb{R}^3}{|u|^{p+1}dx}$.
Thus we study the perturbed functional
$$
I_{\omega,\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2
+u^2-\omega\phi_{u}u^2)dx-\int_{\mathbb{R}^3}h(x)u\,dx
-\frac{\lambda}{p+1}\int_{\mathbb{R}^3}|u|^{p+1}dx.
$$
Then, $I_{\omega,\lambda}$ is a family of $C^1$-functionals on $X$,
$B(u)\geq 0$ and $A(u)\geq \frac{1}{2}\|u\|^2 -\|h\|_{L^2}\|u\|\to+\infty$
as $\|u\|\to \infty$.

\begin{lemma} \label{lem4.1}
 Assume $p\in (1,5)$ and {\rm (H1)--(H2)} satisfy. Then, the following hold.
\begin{itemize}
\item[(i)] $\Gamma _\lambda \neq \emptyset$, for any 
$\lambda \in [1/2,1]$;
\item[(ii)]
 There exists a constant $\tilde{c}$ such that $c_\lambda \geq \tilde{c}>0$ 
for all $\lambda \in [1/2,1]$.
\end{itemize}
\end{lemma}

\begin{proof} (i) For any $\lambda\in[1/2,1]$, we choose a function
$\psi\in X\geq (\not\equiv0)$. Then, by Lemma \ref{lem2.1}, we obtain
$$
I_{\omega,\lambda}(t\psi)\leq \frac{t^2}{2}\|\psi\|^2
+\frac{t^2}{2}\omega ^2\int_{\mathbb{R}^3}\psi^2dx
-\frac{t^{p+1}}{p+1}\int_{\mathbb{R}^3}{|\psi|^{p+1}dx}.
$$
Since $p\in (1,5)$, there exists $t_0$ large enough such that 
$I_{\omega,\lambda}(t_0\psi)<0$. Hence $(i)$ holds.

(ii) By Lemma \ref{lem2.1}, for any $u\in X$ and $\lambda \in [1/2,1]$, we have
$$
I_{\omega,\lambda}(u) \geq \frac{1}{2}\|u\|^2
-\frac{1}{p+1}\int_{\mathbb{R}^3}{|u|^{p+1}dx}
$$
Since $p>1$, we conclude that there exists $\rho >0$ such that 
$I_{\omega,\lambda} (u)>0$ for any $u\in X$ and 
$\lambda \in [1/2,1]$ with $\|u\|\leq \rho$.
 In particular, for any $\|u\|=\rho$, we have 
$I_{_\omega,\lambda} (u)>\tilde{c}>0$. Now fix $\lambda \in [1/2,1]$
and $\gamma \in \Gamma _\lambda$, by the definition of $ \Gamma _\lambda$, 
certainly $\|\gamma (1)\|>\rho$. By continuity, we deduce that there exists
$t_\gamma \in (0,1)$ such that $\|\gamma (t_\gamma)\|=\rho$. 
Therefore, for any $\lambda \in [1/2,1]$,
 we have
$$ 
c_\lambda \geq \inf_{\gamma\in \Gamma_\lambda}I_{\omega,\lambda} 
(\gamma (t_\gamma))\geq \tilde{c}>0.
$$
Thus, (ii) holds.
\end{proof}

Since $I_{\omega,\lambda}(0)=0$, then by Lemma \ref{lem4.1} and Theorem \ref{thm2.4}, there 
exist  (i) $\{\lambda_j\}\subset [1/2,1]$ such that 
$\lambda_j\to 1$ as $j\to \infty$ and (ii)  a bounded sequence
 $\{v_n^{j}\}$ of the functional $I_{\omega,\lambda_j}$. 
By the compactness of  the embedding 
$H^1_r(\mathbb{R}^3)\hookrightarrow L^s(\mathbb{R}^3)(2<s<6)$ 
and \cite[Lemma 2.1]{r1}, we can show that for each $j\in \text N$ there exists 
$v_j\in H^1_r(\mathbb{R}^3)$ such that $v_n^j\to v_j$ strongly in 
$H^1_r(\mathbb{R}^3)$. Moreover, for all $j\in \text N$, we have
\begin{equation}
0<\tilde{c}\leq I_{\omega,\lambda_j}(v_j)
=c_{\omega,\lambda_j}\leq c_{\omega,\frac{1}{2}},\quad
I'_{\omega,\lambda_j}(v_j)=0.
\label{e4.1}
\end{equation}


\begin{lemma} \label{lem4.2} 
 If $v_j\in X$ solves the problem $(*),$  then the following Poho\u{z}aev type 
identity
\begin{equation}
\begin{aligned}
\frac{1}{2}\int_{\mathbb{R}^3}|\nabla v_j|^2dx
+\frac{3}{2}\int_{\mathbb{R}^3}v_j^2dx
-\int_{\mathbb{R}^3}(\frac{5}{2}\omega
+\phi_{v_j})\phi_{v_j}v_j^2dx\\= \int_{\mathbb{R}^3}[\frac{3\lambda }{p+1}|v_j|^{p+1}
+(3h(x)+\langle x,\nabla h(x) \rangle) v_j]dx.
\end{aligned}\label{e4.2}
\end{equation}
holds.
\end{lemma}

The proof can be done as in \cite[Lemma 3.1]{d2} and details are omitted here. 
In what follows, we turn to showing that $\{v_j\}$ converges to a solution 
of problem \eqref{e1.1}. For this purpose, we have to prove $\{v_j\}$ 
is the bounded in $H^1_r(\mathbb{R}^3)$.

\begin{lemma} \label{lem4.3} 
Under the conditions of Theorem \ref{thm1.1}, if $p\in (2,5)$, then  $\{v_j\}$ 
is bounded in $H^1_r(\mathbb{R}^3)$.
\end{lemma}

\begin{proof}
 The proof of this theorem is divided into two steps.
\smallskip

\noindent\textbf{Step 1:} $\{\|v_j\|_{L^2}\}$ is bounded.
By contradiction, we assume that $\|v_j\|_{L^2}\to\infty$ as 
$j\to\infty$. Set $u_j=\frac{v_j}{\|v_j\|_{L^2}}$,
$X_j=\int_{\mathbb{R}^3}|\nabla u_j|^2dx$, 
$Y_j=\int_{\mathbb{R}^3}\omega\phi_{v_j} u_j^2dx$,
$Z_j=\int_{\mathbb{R}^3}\phi_{v_j}^2 u^2_jdx$, and
 $T_j=\lambda_j\|u_j\|^{p+1}_{L^{p+1}}\|v_j\|^{p-1}_{L^{2}}$. 
By \eqref{e4.1}, we have
\begin{equation}
\begin{gathered}
\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla v_j|^2+v_j^2-\omega\phi_{v_j}v_j^2)dx
-\int_{\mathbb{R}^3}h(x)v_jdx-\frac{\lambda_j}{p+1}\int_{\mathbb{R}^3}|v_j|^{p+1}dx
=c_{\omega,\lambda_j},\\
\int_{\mathbb{R}^3}|\nabla v_j|^2+v_j^2-(2\omega+\phi_{v_j})\phi_{v_j}v_j^2)dx
-\int_{\mathbb{R}^3}{h(x)v_jdx}=\lambda_j\int_{\mathbb{R}^3}{|v_j|^{p+1}dx},
\end{gathered} \label{e4.3}
\end{equation}
and $\{c_{\omega,\lambda_j}\}$ is bounded. Note that
$h(x), \langle x,h(x)\rangle\in L^2(\mathbb{R}^3)$. Multiplying \eqref{e4.2}
and \eqref{e4.3} by $\frac{1}{\|v_j\|_{L^2}}$, we obtain
\begin{equation} \label{e4.4}
\begin{gathered}
\frac{1}{2}X_j-\frac{5}{2}Y_j-Z_j-\frac{3}{p+1}T_j=o(1)-\frac{3}{2},\\
\frac{1}{2}X_j-\frac{1}{2}Y_j-\frac{1}{p+1}T_j=o(1)-\frac{1}{2},\\
X_j-2Y_j-Z_j-T_j=o(1)-1,
\end{gathered}
\end{equation}
where $o(1)$ denotes that the quantity tends to zero as $j\to\infty$.
Solving \eqref{e4.4}, we have
$$
X_j=\frac{(1-p)(1+Z_j)}{2(p-2)}+o(1),\quad\text{for } p\in(2,5).
$$
Since $Z_j\geq 0$ and $X_j\geq0$ for all $j\in \text N$, \eqref{e4.4}
is a contradiction for $j$ large enough.  Thus, $\{\|v_j\|_{L^2}\}$
is bounded for $p\in(2,5)$.
\smallskip

\noindent\textbf{Step 2:} $\|\nabla v_j\|_{L^2}$ is bounded.
Similarly, by contradiction, we can assume that
 $\|\nabla v_j\|_{L^2}\to\infty$ as $j\to\infty$. 
Set $w_j=\frac{v_j}{\|\nabla v_j\|_{L^2}}$, 
$M_j=\int_{\mathbb{R}^3}\omega\phi_{v_j} w_j^2dx$,
$N_j=\int_{\mathbb{R}^3}\phi_{v_j}^2 w^2_jdx$,~
$S_j=\lambda_j\|w_j\|^{p+1}_{L^{p+1}}\|\nabla v_j\|^{p-1}_{L^{2}}$. 
Then, multiplying \eqref{e4.2} and \eqref{e4.3} by 
$\frac{1}{\|\nabla v_j\|^2_{L^2}}$, and noting that  $\|v_j\|_{L^2}$  
is bounded, we obtain
\begin{equation}
\begin{gathered}
-\frac{5}{2}M_j-N_j-\frac{3}{p+1}S_j=o(1)-\frac{1}{2},\\
-\frac{1}{2}M_j-\frac{1}{p+1}S_j=o(1)-\frac{1}{2},\\
-2M_j-N_j-S_j=o(1)-1.
\end{gathered} \label{e4.5}
\end{equation}
For $p\in (2,5)$, solving \eqref{e4.5}, we obtain
$$
N_j=\frac{2(2-p)}{(p-1)}+o(1),\quad\text{for }p\in(2,5),
$$
which implies a contradiction for $j$ large enough since $N_j\geq 0$
for all $j\in \text N$. Thus, $\{\|\nabla v_j\|_{L^2}\}$ is bounded for
$p\in(2,5)$. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
  Lemma \ref{lem4.3} implies that $\{v_j\}$ is a bounded sequence of $I_\omega$. 
Then, by the compactness of the embedding 
$H^1_r(\mathbb{R}^3)\hookrightarrow L^s(\mathbb{R}^3)(2<s<5)$, 
for any $\omega>0$, we show that problem \eqref{e1.1} has a solution $u_1$ 
satisfying $I_\omega(u_1)>0$. Combining with  Lemma \ref{lem3.2},  
we complete the proof.
\end{proof}

\section{Positive energy solution for $p\in (1,2]$}

In this section, we first prove that  \eqref{e1.1} with $1<p\leq 2$ 
has  no solution with positive energy for $\omega>0$ large enough.

\begin{theorem} \label{thm5.1} 
Assume that $p\in(1,2]$ and {\rm (H1)--(H2)} hold (in fact, $h(x)$
 may not be radially symmetric). Then  \eqref{e1.1} has no
 solution with positive energy if $\omega>0$ is large enough.
\end{theorem}

\begin{proof} 
Let $u\in H^1(\mathbb{R}^3)$ be a solution of  \eqref{e1.1}. 
Then $\langle I'_\omega(u),u\rangle=0$. By \eqref{e2.2} and \eqref{e2.3}, we have
\begin{equation}
\begin{aligned}
I_\omega(u)&=-(\frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2dx
 -\frac{3}{2}\int_{\mathbb{R}^3}\omega\phi_uu^2dx
 -\int_{\mathbb{R}^3}\phi_u^2u^2dx)\\
&\quad -\frac{1}{2}\int_{\mathbb{R}^3}u^2dx
 +\frac{p}{p+1}\int_{\mathbb{R}^3}|u|^{p+1}dx.
\end{aligned}
\label{e5.1}
\end{equation}
Similar to \cite[(20)]{r1}, we  obtain
\begin{equation}
\sqrt{\frac{3}{4}}\int_{\mathbb{R}^3}(\omega+\phi_u)|u|^3
\leq \frac{1}{4}\int_{\mathbb{R}^3}|\nabla u|^2dx
+\frac{3}{4}\int_{\mathbb{R}^3}|\nabla \phi|^2dx.
\label{e5.2}
\end{equation}
Then, by Lemma \ref{lem2.1}, one has
\begin{equation}
\begin{aligned}
\sqrt{3}\int_{\mathbb{R}^3}(\omega+\phi_u)|u|^3
&\leq \frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2dx
 +\frac{3}{2}\int_{\mathbb{R}^3}|\nabla \phi|^2dx\\
&=\frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2dx
 -\frac{3}{2}\int_{\mathbb{R}^3}\omega\phi_uu^2dx
 -\frac{3}{2}\int_{\mathbb{R}^3} \phi_u^2u^2dx\\
&\leq\frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2dx
 -\frac{3}{2}\int_{\mathbb{R}^3}\omega\phi_uu^2dx
 -\int_{\mathbb{R}^3}\phi_u^2u^2dx.
\end{aligned} \label{e5.3}
\end{equation}
For $p\in (1,2]$ and $\omega>0$ large enough such that $\omega+\phi_u>0$,
it follows from \eqref{e5.1} and \eqref{e5.3} that
$$
I_\omega(u)\leq-\{\sqrt{3}\int_{\mathbb{R}^3}[(\omega+\phi_u)|u|^3
+\frac{1}{2}u^2-\frac{ p}{p+1}|u|^{p+1}]dx\}<0.
 $$
Hence, problem \eqref{e1.1} must have no solution with positive energy if
$\omega>0$  is large enough.
\end{proof}

Obviously, when $p\in (1,2]$, Theorem \ref{thm5.1} implies that we may find a solution 
with positive energy to problem \eqref{e1.1} only for $\omega>0$ small.
To overcome the difficulty in finding bounded $(PS)_c(c>0)$ sequence for 
the associated functional $I_\omega$, following \cite{k1}, we introduce the
 cut-off function
$\eta \in C^\infty (\mathbb{R}^+,\mathbb{R}^+)$ satisfying
\begin{gather*}
\eta (t)=1,\quad\text{for } t\in [0,1],\\
0\leq \eta (t)\leq 1,\quad\text{for }t\in (1,2),\\
\eta (t)=0,\quad\text{for } t\in [2,+\infty),\\
|\eta'|_\infty \leq 2,
\end{gather*}
and consider the  modified functional 
\begin{equation}
\begin{aligned}
I_{\omega,T}(u)&=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)dx
-\frac{\omega}{2}\int_{\mathbb{R}^3}K_T(u)\phi_uu^2dx\\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^3}{|u|^{p+1}dx}-\int_{\mathbb{R}^3}{h(x)u\,dx}.
\end{aligned}\label{e5.4}
\end{equation}
where, for $T>0$, $K_T(u)=\eta (\frac{\|u\|^2}{T^2})$.
If $h(x)=h(|x|)\in L^2(\mathbb{R}^3)$ and $p\in (1,5]$, then
$I_{\omega,T}$ is a $C^1$ functional, and
\begin{equation}
\begin{aligned}
\langle I_{\omega,T}'(u),v\rangle
&=\int_{\mathbb{R}^3}(\nabla u\nabla v+uv)dx
 -\int_{\mathbb{R}^3}K_T(u)(2\omega+\phi_u)\phi_uuv\,dx\\
&\quad -\frac{\omega}{T^2}\eta' (\frac{\|u\|^2}{T^2})\int_{\mathbb{R}^3}\phi_uu^2dx
\int_{\mathbb{R}^3}(\nabla u\nabla v+uv)dx\\
&\quad -\int_{\mathbb{R}^3}{|u|^{p-1}uv\,dx}-\int_{\mathbb{R}^3}{h(x)v\,dx},
\end{aligned}\label{e5.5}
\end{equation}
for every $u,v\in E$.

\begin{lemma} \label{lem5.2} 
Assume that $p\in (1,5)$ and {\rm (H1)--(H2)}. Then the functional
$I_{\omega,T}$ satisfies the following:
\begin{itemize}
\item[(i)] $I_{\omega,T}|_{\|u\|=\rho}>\alpha>0$ for all $\omega,T>0$.
\item[(ii)]  For each $T>0$, there exists  a function $e_T\in H^1_r(\mathbb{R}^3)$ 
 with
$\|e_T\|> \rho$ such that $I_{\omega,T}(e_T)<0$, where 
$\rho,\alpha$ is given by Lemma \ref{lem3.1}.
\end{itemize}
\end{lemma}

\begin{proof}   The proof of (i) is similar to that of Lemma \ref{lem3.1}.

(ii) we choose $\varphi \in E$ with $\varphi \geq 0$, $\|\varphi\|=1$. 
By \eqref{e5.4}  and the definition of $\eta$,  there exists $t_T\geq 2T>0$ 
large enough such that $K_T(t_T\varphi)=0$ and $I_{\omega,T}(t_T\varphi)<0$.
 Hence, $(ii)$ holds by taking $e_T=t_T\varphi$.
Set
$$
c_{\omega,T}=\inf_{\gamma\in\Gamma_{\omega,T}}\max_{t\in[0,1]}
I_{\omega,T}(\gamma(t)),
$$
where $\Gamma_{\omega,T}:=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)=e_T\}$. 
Then, by Lemma \ref{lem5.2}, we have
\begin{equation}
c_{\omega,T}\geq \alpha>0,\quad\text{for  all }\omega,\; T>0.
\label{e5.6}
\end{equation}
Applying the Mountain Pass Theorem, there exists
$\{u^n_{\omega,T}\}\in H_r^1(\mathbb{R}^3)$ (denoted by $\{u_n\}$
for simplicity) such that
\begin{equation}
I_{\omega,T}(u_n)\to c_{\omega,T},~(1+\|u_n\|)\|I'_{\omega,T}(u_n)\|_{H_r^{-1}}\to 0
\label{e5.7}
\end{equation}
as $n\to\infty$, where $H_r^{-1}$ denotes the dual space of $H_r^1(\mathbb{R}^3)$.
\end{proof}

\begin{lemma} \label{lem5.3} 
Suppose that $p\in(1,5)$ and {\rm (H1)--(H2)} hold. Let $\{u_n\}$ be given 
by \eqref{e5.7}. Then there exists $T_0>0$ such that 
$$
\lim_{n\to\infty}\sup\|u_n\|\leq \frac{T_0}{2},\quad
\forall 0<\omega<T_0^{-3},
$$
which implies $\{u_n\}$ being a bounded $(PS)$ sequence of
 $I_\omega$ in $H^1_r(\mathbb{R}^3)$.
\end{lemma}

\begin{proof}  
Motivated by \cite{k1}, we will argue by contradiction. Assume that, 
for every $T>0$ there exists $0<\omega_T<T^{-3}$ such that
$\lim_{n\to\infty}\sup\|u_n\|> \frac{T}{2}$.
So, up to a subsequence, we obtain $\|u_n\|\geq \frac{T}{2}$ for all
 $n\in \text N$. On the one hand, by $\eqref{e5.4}$, $\eqref{e5.5}$ and 
Lemma \ref{lem2.1}, we have
\begin{align*}
&(p+1)I_{\omega,T}(u_n)-\langle I'_{\omega,T}(u_n),u_n\rangle\\
&=\frac{p-1}{2}\|u_n\|^2 -\frac{\omega(p-3)}{2}\int_{\mathbb{R}^3}
 K_T(u_n)\phi_{u_n}u_n^2dx\\
&\quad +\int_{\mathbb{R}^3}K_T(u_n)\phi^2_{u_n}u_n^2dx
+\frac{\omega}{T^2}\eta' (\frac{\|u_n\|^2}{T^2})\|u_n\|^2
 \int_{\mathbb{R}^3}\phi_{u_n}u_n^2dx
 -p\int_{\mathbb{R}^3}h(x)u_ndx.
\end{align*}
Consequently,
\begin{equation}
\begin{aligned}
&\frac{p-1}{2}\|u_n\|^2 -\|I'_{\omega,T}(u_n)\|\|u_n\|\\
&\leq\frac{p-1}{2}\|u_n\|^2 +\langle I'_{\omega,T}(u_n),u_n\rangle\\
&\leq(p+1)I_{\omega,T}(u_n)+\frac{\omega(p-3)}{2}
 \int_{\mathbb{R}^3}K_T(u_n)\phi_{u_n}u_n^2dx\\
&\quad +\int_{\mathbb{R}^3}K_T(u_n)\phi^2_{u_n}u_n^2dx
-\frac{\omega}{T^2}\eta' (\frac{\|u_n\|^2}{T^2})\|u_n\|^2
 \int_{\mathbb{R}^3}\phi_{u_n}u_n^2dx+p\int_{\mathbb{R}^3}h(x)u\,dx\\
&\leq (p+1)I_{\omega,T}(u_n)+\frac{\omega(p-3)}{2}
 \int_{\mathbb{R}^3}K_T(u_n)\phi_{u_n}u_n^2dx \\
& -\omega\int_{\mathbb{R}^3}K_T(u_n)\phi_{u_n}u_n^2dx
-\frac{\omega}{T^2}\eta' (\frac{\|u_n\|^2}{T^2})\|u_n\|^2
 \int_{\mathbb{R}^3}\phi_{u_n}u_n^2dx
 +p\int_{\mathbb{R}^3}h(x)u\,dx\\
&=(p+1)I_{\omega,T}(u_n)+\frac{\omega(5-p)}{2}
 \int_{\mathbb{R}^3}K_T(u_n)(-\phi_{u_n})u_n^2dx\\
&\quad +\frac{\omega}{T^2}\eta' (\frac{\|u_n\|^2}{T^2})
 \|u_n\|^2\int_{\mathbb{R}^3}(-\phi_{u_n})u_n^2dx
 +p\int_{\mathbb{R}^3}h(x)u\,dx.
\end{aligned}\label{e5.8}
\end{equation}
On the other hand, we claim that there exist $T_1, C, M_1>0$ such that
\begin{equation}
c_{\omega,T}\leq C\omega T^4+M_1,\quad\forall T\geq T_1. \label{e5.9}
\end{equation}
Let $\varphi$ be the function taken in the proof of (ii) of Lemma \ref{lem5.2}.
By $\eqref{e5.4}$, we have
\begin{equation}
I_{\omega,T}(2T\varphi)\leq2T^2
-\frac{ 2^{p+1}}{p+1}T^{p+1}\|\varphi\|^{p+1}_{L^{p+1}}.\label{e5.10}
\end{equation}
Then there exists $T_1>0$ such that $I_{\omega,T}(2T\varphi)<0$ for all $T>T_1$.
 Thus
\begin{equation}
c_{\omega,T}\leq \max_{t\in [0,1]}I_{\omega,T}(2tT\varphi),
\quad\forall T\geq T_1. \label{e5.11}
\end{equation}
By \eqref{e5.4} and Remark \ref{rmk2.2}, we have
\begin{equation}
\begin{aligned}
&\max_{t\in [0,1]}I_{\omega,T}(2tT\varphi)\\
&\leq \max_{t\in [0,1]}\{2(tT)^2-\frac{ 2^{p+1}}{p+1}(tT)^{p+1}
 \|\varphi\|^{p+1}_{L^{p+1}}\}
 +\max_{t\in[0,1]}\{-\frac{\omega}{2}\int_{\mathbb{R}^3}\phi_{2tT\varphi}
 (2tT\varphi)^2dx\}\\
&\leq\max_{m\geq 0}\{2(m)^2-\frac{2^{p+1}}{p+1}(m)^{p+1}
 \|\varphi\|^{p+1}_{L^{p+1}}\}+C\omega T^4\\
&=M_1+C\omega T^4.
\end{aligned}\label{e5.12}
\end{equation}
It follows from \eqref{e5.11} and \eqref{e5.12} that \eqref{e5.9} holds.
By Remark \ref{rmk2.2},  and noting that $K_T(u_n)=0$ for $\|u_n\|^2\geq 2T^2$,
we obtain
\begin{gather}
\int_{\mathbb{R}^3}K_T(u_n)(-\phi_{u_n})u_n^2dx\leq CT^4, \label{e5.13}\\
\eta' (\frac{\|u_n\|^2}{T^2})\frac{\|u_n\|^2}{T^2}
 \int_{\mathbb{R}^3}(-\phi_{u_n})u_n^2dx\leq CT^4. \label{e5.14}
\end{gather}
Combining \eqref{e5.7}, \eqref{e5.8}, \eqref{e5.9}, \eqref{e5.13} with \eqref{e5.14},
 one has, for all $T>T_1$,
\begin{equation}
\frac{p-1}{2}\|u_n\|^2 \leq C_2\omega T^4 +M_2+p\int_{\mathbb{R}^3}h(x)u\,dx,
\label{e5.15}
\end{equation}
where $C_2, M_2>0$ independent of $T$. Then, for any $\varepsilon>0$,
by the inequality
$\int_{\mathbb{R}^3}h(x)u_n\leq \varepsilon\|u_n\|^2+C(\varepsilon,\|h\|_{L^2})$
and \eqref{e5.15},  there exist $C,M>0$ independent of $T$ such that, for all
$T>T_1$,
\begin{equation}
\|u_n\|^2 \leq C\omega T^4 +M. \label{e5.16}
\end{equation}
Since $0<\omega<T_0^{-3} $ and $\|u_n\|\geq \frac{T}{2}$, \eqref{e5.16}
is impossible for $T>0$ large enough. Thus we complete the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemma \ref{lem5.3}, we obtain that  $\{u_n\}$ is given by \eqref{e5.7} is bounded 
sequence of $I_\omega$ in $H^1_r(\mathbb{R}^3)$ for all $0<\omega<T_0^{-3}$.
 Moreover, by using \eqref{e5.6} and \eqref{e5.7}, we see that 
$$
I_\omega(u_n)\to c_{\omega,T_0}\geq \alpha>0,\quad\text{as } n\to \infty.
$$
Then, by the compactness of the embedding 
$H^1_r(\mathbb{R}^3)\hookrightarrow L^{s+1}(\mathbb{R}^3)(1<s<5)$, 
for any $0<\omega<T_0^{-3}$, problem \eqref{e1.1} has a solution 
$\tilde{u_1}$ satisfying $I_\omega(\tilde{u_1})>0$. 
Then, by Theorem \ref{thm5.1} and Lemma \ref{lem3.2}, we easily complete the proof.
\end{proof}

\subsection*{Acknowledgements}
This research was supported by 
the Natural Science Foun-dation of China 11271372, 
by the Hunan Provincial Natural Science Foundation of China 12JJ2004, 
and by the Mathematics and Interdisciplinary Sciences project of CSU.

The authors would like to thank the anonymous referee for his/her
 helpful comments and suggestions.


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\end{document}