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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 262, pp. 1--14.\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.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/262\hfil Positive ground state solution]
{Positive ground state solution for Kirchhoff equations with
subcritical growth and zero mass}

\author[Y. Duan, J. Liu, C.-L. Tang \hfil EJDE-2015/262\hfilneg]
{Yu Duan, Jiu Liu, Chun-Lei Tang}

\address{Yu Duan \newline
School of Mathematics  and  Statistics, Southwest University,
 Chongqing 400715, China. \newline
College of Science, Guizhou University of Engineering Science,
 Bijie, Guizhou 551700, China}
\email{duanyu3612@163.com}

\address{Jiu Liu \newline
School of Mathematics  and  Statistics, Southwest University,
 Chongqing 400715, China}
\email{jiuliu2011@163.com}

\address{Chun-Lei Tang (corresponding author)\newline
School of Mathematics  and  Statistics, Southwest University,
 Chongqing 400715, China}
\email{tangcl@swu.edu.cn, Pone +86 23 68253135, fax +86 23 68253135}

\thanks{Submitted September 7, 2015. Published October 8, 2015.}
\subjclass[2010]{35J20, 35J60, 35A01}
\keywords{Kirchhoff equation; subcritical growth; zero mass;
 Pohozaev identity}

\begin{abstract}
 In this article, we study the  Kirchhoff equation
 \begin{gather*}
 -\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx\Big)\Delta u=K(x)f(u),  \quad
 x\in \mathbb{R}^N,\\
 u\in D^{1,2}(\mathbb{R}^N),
 \end{gather*}
 where $a>0$, $b>0$ and $N\geq3$. Under suitable conditions on $K$ and $f$,
 we obtain four existence results and two nonexistence results, using
 variational methods.
\end{abstract}

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

\section{Introduction and statement of main results}

 We consider the Kirchhoff equation with zero mass
\begin{equation}\label{formula1}
\begin{gathered}
-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx\Big)\Delta u=K(x)f(u),  \quad
 x\in \mathbb{R}^N,\\
 u\in D^{1,2}(\mathbb{R}^N),
\end{gathered}
\end{equation}
where $a>0$, $b>0$ and $N\geq3$. The potential function
$K:\mathbb{R}^N\to \mathbb{R}$ satisfies:
\begin{itemize}
\item[(K1)] $K$ be a nonnegative function and
$K\in L^r(\mathbb{R}^N)\backslash\{0\}$,
where $r=\frac{2^*}{2^*-p}$, $2^*=\frac{2N}{N-2}$ and $2<p<2^*$;

\item[(K2)] $|(\nabla K(x),x)|\leq K(x)$ for a.e.
$x\in\mathbb{R}^N$, where $(\cdot,\cdot)$ denotes the scalar product
in $\mathbb{R}^N$.
\end{itemize}
The nonlinear term $f\in C(\mathbb{R}_+,\mathbb{R}_+)$ satisfies:
\begin{itemize}
\item[(F1)] $\limsup_{s\to 0^+}\frac{f(s)}{s^{p-1}}<+\infty$;

\item[(F2)] $\limsup_{s\to +\infty}\frac{f(s)}{s^{p-1}}<+\infty$;

\item[(F3)] $\lim_{s\to +\infty}\frac{f(s)}{s}=+\infty$.
\end{itemize}
When $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, the equation
\begin{equation}\label{formula3}
\begin{gathered}
-\Big(a+b\int_{\Omega}|\nabla u|^{2}dx\Big)\Delta u=f(x,u),  \quad x\in \Omega,\\
 u=0, \quad x\in \partial\Omega.
\end{gathered}
\end{equation}
is related to the stationary analogue of the Kirchhoff equation
\begin{equation}\label{formula4}
u_{tt}-\Big(a+b\int_{\Omega}|\nabla u|^{2}dx\Big)\Delta u=f(x,u)
\end{equation}
which was proposed by Kirchhoff \cite{K1} in 1883 as an extension of the
classical D'Alembert wave equation for free vibrations of elastic
strings. Kirchhoff's model takes into account the changes in length of
the string produced by transverse vibrations.
Some early classical studies of Kirchhoff equations were those of
Bernstein \cite{B} and Pohozaev \cite{P}. However, equation \eqref{formula4}
received great attention only after Lions \cite{L} proposed an abstract
framework for the problem. Some interesting results
can be found in \cite{AP}, \cite{CCS}, \cite{DS} and the references therein.

There are many recent articles studying the Kirchhoff equations with
subcritical growth in $\mathbb{R}^N$, see for example
\cite{FI,HU,I,LIG,LLS,LI,N,W,ZH} and so on.
But for the Kirchhoff equations with subcritical growth and zero mass,
to our best knowledge, there are very few results up to now except \cite{A}
and \cite{LLS1}. Azzollini \cite{A} obtained the existence of positive
radial solution under $K=1$ and $f$ satisfies the Berestycki-Lions
conditions \cite{BE}. In \cite{LLS1}, Li et al. obtain a positive solution
for $b>0$ small enough when the potential $K$ satisfies
\begin{itemize}
\item[(K3)] $K\in L^\infty(\mathbb{R}^N)$,
\end{itemize}
and other conditions similar to (K1), (K3), and the nonlinear term
$f$ satisfies (F3), and
\begin{itemize}
\item[(F5)] $\lim_{s\to +\infty}\frac{f(s)}{s^{2^*-1}}=0$,

\item[(F6)] $\lim_{s\to 0^+}\frac{f(s)}{s^{2^*-1}}=0$.
\end{itemize}
Obviously,  condition (F6) is stronger than (F1) and there exist functions
which satisfy (F1), but do not satisfy (F6), such as $f(s)=s^{p-1}$.
Thus, in the present paper, we will remove the assumption (F6) to study
equation \eqref{formula1}. Inspired by \cite{LLS1}, we will use the
monotonicity trick to investigate it. Set $F(s)=\int_{0}^{s}f(\tau)d\tau$.
Our results read as follows.

\begin{theorem} \label{thm1.1}
 Assume that $a>0$ and $b>0$. Suppose that {\rm (F1)--(F3)} hold.
Then there exists $b_0>0$ such that for any $b\in(0,b_0)$,
 equation \eqref{formula1} has a positive ground state solution, under
one of the following conditions:
\begin{itemize}
\item[(1)] $N\geq4$ and {\rm (K1)} holds,
\item[(2)] $N=3$, $2<p<4$ and {\rm (K1)} holds,
\item[(3)] $N=3$, $4\leq p<6$, {\rm (K1)} and {\rm (K2)} hold.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm1.2}
 Assume that $a>0$, $b>0$ and $N=3$. Suppose that {\rm (K1)} with $4\leq p<6$,
{\rm (F1)} and {\rm (F2)} hold. In addition, $f$ satisfies
\begin{itemize}
\item[(F4)]  $\lim_{s\to +\infty}\frac{F(s)}{s^4}=+\infty$ and
$sf(s)\geq4F(s)$ for all $s\geq0$.
\end{itemize}
Then equation \eqref{formula1} has a positive ground state solution.
\end{theorem}

\begin{theorem} \label{thm1.3}
 Assume that $a>0$ and $b>0$. Suppose that {\rm (F1), (F3), (F5)} hold.
Then there exists $b_0>0$ such that for any $b\in(0,b_0)$, equation
\eqref{formula1} has a positive ground state solution, under one of the
following conditions:
\begin{itemize}
\item[(1)] $N\geq4$, {\rm (K1)} and {\rm (K3)} hold,
\item[(2)] $N=3$, {\rm (K1)--(K3)} hold.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm1.4}
 Assume that $a>0$, $b>0$ and $N=3$. Suppose that (K1) with $4\leq p<6$, (K3),
{\rm (F1), (F4), (F5)} hold. Then \eqref{formula1} has a positive ground
 state solution.
\end{theorem}


\begin{remark} \label{rmk1.1} \rm
Just like the example which was given by Li et al. in \cite{LLS1},
when $p\in(\max\{2,\frac{22^*}{3}\},2^*)$, one can easily verify that the
function $K(x)=\frac{1}{1+|x|^\alpha}$,
$\alpha\in(\frac{32^*-3p}{2^*},1]$ satisfies $(K_1)-(K_3)$.
\end{remark}


For equation \eqref{formula1} with $b>0$ large enough, we have the following
nonexistence results.

\begin{theorem} \label{thm1.5}
 Assume that $a>0$, $b>0$, $N\geq3$ and $f$ satisfies {\rm (F1)} and {\rm (F2)}.
Suppose that when $N=3$, {\rm (K1)} with $2<p<4$ holds and when $N\geq4$,
{\rm (K1)} holds. Then there exists $B>0$ such that for any $b>B$,
equation \eqref{formula1} has only zero solution.
\end{theorem}

\begin{theorem} \label{thm1.6}
 Assume that $a>0$, $b>0$ and $N\geq4$. Suppose that
{\rm (K1), (K3), (F1), (F5)} hold. Then there exists $B>0$ such that
for any $b>B$, equation \eqref{formula1} has only zero solution.
\end{theorem}

Theorems \ref{thm1.1}--\ref{thm1.4} and \ref{thm1.5}--\ref{thm1.6} 
can be seen as an extension of the
results in \cite{LLS1}.
This  article is organized as follows.
In Section 2 we give some preliminary knowledge.
Section 3 is devoted to the proofs of Theorem \ref{thm1.1} and \ref{thm1.2}.
Finally, in Section 4 we complete the proofs of 
Theorems \ref{thm1.3}, \ref{thm1.4}, \ref{thm1.5}, and \ref{thm1.6}.



\section{Preliminaries}

 In what follows, we  use the following notation.

\noindent $\bullet$ $E:=D^{1,2}(\mathbb{R}^N)$ is the closure of the compactly
supported smooth functions with respect to the norm
\[
\|u\|=\Big(\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)^{1/2}.
\]
$\bullet$ $L^s(\mathbb{R}^N)$ is the usual Lebesgue space endowed with the norm
\[
|u|_s=\Big(\int_{\mathbb{R}^N}|u|^sdx\Big)^{1/s}, \quad
\forall  s\in[1,+\infty),\quad |u|_\infty
=\operatorname{ess\,sup}_{x\in\mathbb{R}^N}|u(x)|.
\]
$\bullet$ $S$ denotes the best constant of Sobolev embedding
$D^{1,2}(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$, that is,
\begin{equation}\label{formula14}
S|u|_{2^*}^2\leq\|u\|^2,\quad\text{for all } u\in D^{1,2}(\mathbb{R}^N).
\end{equation}
$\bullet$ $\langle\cdot,\cdot\rangle$ denotes the dual pairing.\\
$\bullet$ $E^{*}$ is the dual space of $E$.\\
$\bullet$ $C$, $C_i$ denote various positive constants.\\

Since we are looking for positive solution, we assume that
$f(s)=0$ for all $s\leq0$. By (F1) and (F2), there exists $C_1>0$ such that
\begin{gather}\label{formula9}
|f(s)|\leq C_1|s|^{p-1},\quad\text{for  all }s\in\mathbb{R},\\
\label{formula10}
|F(s)|\leq \frac{C_1}{p}|s|^{p},\quad\text{for  all } s\in\mathbb{R}.
\end{gather}
By (F1) and (F5), for any $\varepsilon>0$, there exists $C_\varepsilon>0$ such that
\begin{gather}\label{formula12}
|f(s)|\leq\varepsilon|s|^{2^*-1}+C_\varepsilon|s|^{p-1},\quad\text{for  all }
 s\in\mathbb{R}, \\
\label{formula13}
|F(s)|\leq\frac{\varepsilon}{2^*}|s|^{2^*}+C_\varepsilon|t|^{p},\quad
\text{for  all } s\in\mathbb{R}.
\end{gather}
By $f\in C(\mathbb{R}_+,\mathbb{R}_+)$ and (F3), for any $L>0$, there exists $C_L>0$ such that
\begin{equation}\label{formula11}
F(s)\geq L|s|^2-C_L,\ \mathrm{for \ all}\ s\in\mathbb{R}_+.
\end{equation}
By (F4), one has
\begin{equation}\label{formula15}
F(s)\geq L|s|^4-C_L,\ \mathrm{for \ all}\ s\in\mathbb{R}_+.
\end{equation}
The energy functional $I:E\to \mathbb{R}$ defined by
\begin{align*}
I(u)=\frac{a}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
+\frac{b}{4}\Big(\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)^2
-\int_{\mathbb{R}^N}K(x)F(u)dx.
\end{align*}
Obvious, $I$ is of class $C^1$ and has the derivative given by
$$
\langle I'(u),v\rangle=\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx\Big)
\int_{\mathbb{R}^N}(\nabla u,\nabla v)dx-\int_{\mathbb{R}^N}K(x)f(u)vdx,
$$
for all $u,v\in E$.  As well known, the critical point of the functional $I$
is solution of equation \eqref{formula1}.
For proving our theorems, we need the following proposition.

\begin{proposition} \label{prop2.1}
 Let $X$ be a Banach space equipped with a norm $\|\cdot\|_X$ and let
$J\subset \mathbb{R}^+$ be an interval. We
consider a family $\{\Phi_\lambda\}_{\lambda\in J}$ of $C^1$-functionals
on $X$ of the form
$$
\Phi_\lambda(u)=A(u)-\lambda B(u),\quad \forall \lambda\in J,
$$
where $B(u)\geq0$ for all $u\in X$ and such that either
$A(u)\to +\infty$ or $B(u)\to +\infty$, as $\|u\|_X\to +\infty$. We assume
that there are two points $v_1$, $v_2$ in $X$ such that
$$
c_\lambda=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\Phi_\lambda(\gamma(t))
>\max\{\Phi_\lambda(v_1),\Phi_\lambda(v_2)\}
$$
where
$$
\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=v_1,\gamma(1)=v_2\}.
$$
Then for almost every $\lambda\in J$, there is a bounded $(PS)_{c_\lambda}$
sequence for $\Phi_\lambda$, that is, there exists a sequence
$\{u_n\}\subset X$ such that
\begin{itemize}
\item[(i)] $\{u_n\}$ is bounded in $X$,
\item[(ii)] $\Phi_\lambda(u_n)\to  c_\lambda$,
\item[(iii)] $\Phi'_\lambda(u_n)\to 0$ in $X^*$, where
$X^*$ is the dual of $X$.
\end{itemize}
\end{proposition}

\begin{remark} \label{rmk2.2} \rm
The above result corresponds to \cite[Theorem 1.1]{J}
which is reminiscent of Struwe's monotonicity trick (see \cite{S})
and can be viewed as its generalization.
In \cite[Lemma 2.3]{J} it is also proved that under the assumptions
of Proposition \ref{prop2.1}, the map $\lambda\to  c_\lambda$ is continuous
from the left.
\end{remark}

Let $X:=E$, $J:=[1/2,1]$ and
$\Phi_\lambda(u):=I_\lambda(u)=A(u)-\lambda B(u)$, where
\begin{gather*}
A(u)=\frac{a}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx+\frac{b}{4}
\Big(\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)^2,\\
B(u)=\int_{\mathbb{R}^N}K(x)F(u)dx.
\end{gather*}
Then $I_1(u)=I(u)$. It is obvious that $B(u)\geq0$ for all $u\in E$
and $A(u)\to +\infty$ as $\|u\|\to +\infty$.

\section{Proofs of Theorem \ref{thm1.1} and Theorem \ref{thm1.2}}

First we give some lemmas.

\begin{lemma} \label{lem3.1}
 Assume that $a>0$, $b>0$ and $N\geq3$. Suppose that
{\rm (K1), (F1), (F2)} hold. Then there exist $\rho>0$ and $\alpha>0$
such that $I_{\lambda}(u)|_{\|u\|=\rho}\geq\alpha$ for all
$\lambda\in[1/2,1]$.
\end{lemma}

\begin{proof}
 By \eqref{formula10}, the H\"{o}lder and Sobolev inequalities, for all
$u\in E$ and all $\lambda\in[1/2,1]$, we have
\begin{align*}
I_{\lambda}(u)
&\geq \frac{a}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}K(x)F(u)dx\\
&\geq \frac{a}{2}\|u\|^2-\frac{C_1}{p}\int_{\mathbb{R}^N}K(x)|u|^pdx\\
&\geq \frac{a}{2}\|u\|^2-\frac{C_1}{p}|K|_r\Big(\int_{\mathbb{R}^N}|u|^{2^*}dx
 \Big)^{p/2^*}\\
&\geq C_2\|u\|^2-C_3\|u\|^{p}\\
&=\|u\|^2(C_2-C_3\|u\|^{p-2}).
\end{align*}
Since $p>2$, we can choose $\rho=\big(\frac{C_2}{2C_3}\big)^{\frac{1}{p-2}}$.
Then $I_\lambda(u)\geq\frac{C_2}{2}\big(\frac{C_2}{2C_3}\big)^{\frac{2}{p-2}}
:=\alpha$ for all $\|u\|=\rho$. The proof is complete.
\end{proof}

For $c_\lambda$ in Proposition \ref{prop2.1}, we have the following lemma.

\begin{lemma} \label{lem3.2}
 Assume that $a>0$, $b>0$ and $N\geq3$. Suppose that
{\rm (K1)} and {\rm (F1)--(F3)} hold. Then there exists $b_0>0$
such that for any $b\in(0,b_0)$, there are two points $v_1$, $v_2$ in $E$
such that $c_\lambda>\max\{I_\lambda(v_1),I_\lambda(v_2)\}$ for all
$\lambda\in[1/2,1]$.
\end{lemma}

\begin{proof}
 Choose a nonnegative function $\varphi\in C_0^\infty(\mathbb{R}^N)$ such
that $\|\varphi\|=1$, anf
\[
\int_{\operatorname{supp}\varphi}K(x)\varphi^2dx\neq0, \quad
\int_{\operatorname{supp}\varphi}K(x)dx\neq0.
\]
Let
\[
L=\frac{2a}{\int_{\operatorname{supp}\varphi}K(x)\varphi^2dx}
\]
in \eqref{formula11} and take
$$
t_0=\sqrt{2}\max\Big\{\rho,\Big(\frac{C_L}{a}
\int_{\operatorname{supp}\varphi}K(x)dx\Big)^{1/2}\Big\},
$$
where $\rho$ is given by Lemma \ref{lem3.1}. Then $\|t_0\varphi\|>\rho$ and for all
$\lambda\in[1/2,1]$, one has
\begin{align*}
I_\lambda(t_0\varphi)
&= \frac{at_0^2}{2}+\frac{bt_0^4}{4}
-\lambda\int_{\operatorname{supp}\varphi}K(x)F(t_0\varphi)dx\\
&\leq \frac{at_0^2}{2}+\frac{bt_0^4}{4}-\frac{1}{2}
 \int_{\operatorname{supp}\varphi}K(x)(Lt_0^2\varphi^2-C_L)dx\\
&= \frac{at_0^2}{2}+\frac{bt_0^4}{4}+\frac{C_L}{2}
 \int_{\operatorname{supp}\varphi}K(x)dx-\frac{Lt_0^2}{2}
 \int_{\operatorname{supp}\varphi}K(x)\varphi^2dx\\
&= \frac{bt_0^4}{4}+\frac{C_L}{2}\int_{\operatorname{supp}\varphi}K(x)dx
 -\frac{a}{2}t_0^2\\
&\leq \frac{bt_0^4}{4}-\frac{C_L}{2}\int_{\operatorname{supp}\varphi}K(x)dx,
\end{align*}
which implies that there exists $b_0>0$ such that for any $b\in(0,b_0)$,
one has $I_\lambda(t_0\varphi)<0$. Thus letting $v_1=0$ and $v_2=t_0\varphi$,
 by Lemma \ref{lem3.1} and definition of $c_\lambda$, we have
\begin{align*}
0<\alpha\leq c_1\leq c_\lambda\leq c_{\frac{1}{2}}<+\infty.
\end{align*}
and then $c_\lambda>I_\lambda(v_1)>I_\lambda(v_2)$.
\end{proof}

\begin{lemma} \label{lem3.3}
Assume that $a>0$, $b>0$ and $N=3$. Suppose that {\rm (K1)} with $4\leq p<6$,
{\rm (F1), (F2), (F4)} hold. Then there are two points $v_1$, $v_2$ in
 $E$ such that $c_\lambda>\max\{I_\lambda(v_1),I_\lambda(v_2)\}$
for all $\lambda\in[1/2,1]$.
\end{lemma}

\begin{proof}
Choose a nonnegative function $\varphi\in C_0^\infty(\mathbb{R}^3)$ such that
$\|\varphi\|=1$, and
\[
\int_{\operatorname{supp}\varphi}K(x)\varphi^4dx\neq0, \quad
\int_{\operatorname{supp}\varphi}K(x)dx\neq0.
\]
Let 
$L=\frac{b}{\int_{\operatorname{supp}\varphi}K(x)\varphi^4dx}$ in
\eqref{formula15}. Then for any $t>0$ and for all $\lambda\in[1/2,1]$, one has
\begin{align*}
I_\lambda(t\varphi)
&= \frac{at^2}{2}+\frac{bt^4}{4}-\lambda\int_{\operatorname{supp}\varphi}K(x)
 F(t\varphi)dx\\
&\leq \frac{at^2}{2}+\frac{bt^4}{4}-\frac{1}{2}\int_{\operatorname{supp}
 \varphi}K(x)(Lt^4\varphi^4-C_L)dx\\
&= \frac{at^2}{2}+\frac{bt^4}{4}+\frac{C_L}{2}\int_{\operatorname{supp}
 \varphi}K(x)dx-\frac{Lt^4}{2}\int_{\operatorname{supp}\varphi}K(x)\varphi^4dx\\
&= \frac{at^2}{2}+\frac{C_L}{2}\int_{\operatorname{supp}\varphi}K(x)dx
 -\frac{bt^4}{4},
\end{align*}
which indicates that there exists $t_0>0$ such that $\|t_0\varphi\|>\rho$
and $I_\lambda(t_0\varphi)<0$. Thus letting $v_1=0$ and $v_2=t_0\varphi$,
by Lemma \ref{lem3.1} and definition of $c_\lambda$, we have
\begin{align*}
0<\alpha\leq c_1\leq c_\lambda\leq c_{\frac{1}{2}}<+\infty.
\end{align*}
and then $c_\lambda>I_\lambda(v_1)>I_\lambda(v_2)$.
\end{proof}


\begin{lemma} \label{lem3.4}
Assume that $\{u_n\}$ is bounded in $L^s(\mathbb{R}^N)$, where $1<s<+\infty$.
Suppose that $u_n(x)\to  u(x)$ a.e. in $\mathbb{R}^N$. Then up to a subsequence,
$u_n\rightharpoonup u$ in $L^s(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
 Suppose that $|u_n|_s\leq M$ and $v\in L^{\frac{s}{s-1}}(\mathbb{R}^N)$ is fixed.
Then for any $\varepsilon>0$, there exists $r>0$ such that
\begin{align*}
\Big|\int_{|x|\geq r}u_nvdx\Big|\leq|u_n|_s
\Big(\int_{|x| \geq r}|v|^{\frac{s}{s-1}}dx\Big)^{\frac{s-1}{s}}\leq M\varepsilon.
\end{align*}
Similarly, combining the Fatou's lemma, we have
\begin{align*}
\Big|\int_{|x|\geq r}uvdx\Big|^s
&\leq |u|_s^s\Big(\int_{|x|\geq r}|v|^{\frac{s}{s-1}}dx\Big)^{s-1}\\
&= \int_{\mathbb{R}^N}\liminf_{n\to \infty}|u_n|^sdx
 \Big(\int_{|x|\geq r}|v|^{\frac{s}{s-1}}dx\Big)^{s-1}\\
&\leq \liminf_{n\to \infty}\int_{\mathbb{R}^N}|u_n|^sdx
 \Big(\int_{|x|\geq r}|v|^{\frac{s}{s-1}}dx\Big)^{s-1}\\
&\leq  M^s\varepsilon^s.
\end{align*}
Since $v\in L^{\frac{s}{s-1}}(B_r(0))$ with $B_r(0):=\{x\in\mathbb{R}^N:|x|<r\}$,
there exists $\delta>0$ such that for any $A\subset B_r(0)$, when
 $\mathrm{meas}A<\delta$, one has
$\big(\int_{A}|v|^{\frac{s}{s-1}}dx\big)^{\frac{s-1}{s}}<\varepsilon$.
Thus for all $n$, we have
\begin{align*}
\Big|\int_{A}u_nvdx\Big|\leq|u_n|_s\Big(\int_{A}|v|^{\frac{s}{s-1}}dx
\Big)^{\frac{s-1}{s}}<M\varepsilon.
\end{align*}
By  Vitali's theorem, one gets
\begin{align*}
\int_{B_r(0)}u_nvdx=\int_{B_r(0)}uvdx+o(1).
\end{align*}
Hence we have
\begin{align*}
\Big|\int_{\mathbb{R}^N}(u_n-u)vdx\Big|
\leq\Big|\int_{|x|\geq r}(u_n-u)vdx\Big|+\Big|\int_{B_r(0)}(u_n-u)vdx\Big|
\leq2M\varepsilon+o(1).
\end{align*}
By the arbitrariness of $\varepsilon$, we complete the proof.
\end{proof}

\begin{lemma} \label{lem3.5}
For any $\lambda\in[1/2,1]$, if $\{u_n\}\subset E$ is a bounded and nonnegative
Palais-Smale sequence of the function $I_\lambda$, there exists a nonnegative
 function $u\in E$ such that, up to a subsequence, $u_n\to  u$ in $E$.
\end{lemma}

\begin{proof}
Since $\{u_n\}$ is bounded and nonnegative in $E$, up to a subsequence,
there exists a nonnegative function $u\in E$ such that $u_n\rightharpoonup u$
in $E$, $u_n(x)\to  u(x)$ a.e. in $\mathbb{R}^N$ and there exists $d\geq0$
such that $d=\int_{\mathbb{R}^N}|\nabla u_n|^2dx+o(1)$. By \eqref{formula9}
 and the H\"{o}lder inequality, one has
\begin{align*}
\int_{\mathbb{R}^N}|f(u_n)(u_n-u)|^{\frac{2^*}{p}}dx
&\leq C_1\int_{\mathbb{R}^N}|u_n|^{\frac{2^*(p-1)}{p}}|u_n-u|^{\frac{2^*}{p}}dx\\
&\leq C_1\Big(\int_{\mathbb{R}^N}|u_n|^{2^*}dx\Big)^{\frac{p-1}{p}}
 \Big(\int_{\mathbb{R}^N}|u_n-u|^{2^*}dx\Big)^{1/p}\\
&\leq C.
\end{align*}
Combining $f(u_n(x))(u_n(x)-u(x))\to 0$ a.e. in $\mathbb{R}^N$ with 
Lemma \ref{lem3.4},
up to a subsequence, we get
$f(u_n)(u_n-u)\rightharpoonup0$ in $L^{\frac{2^*}{p}}(\mathbb{R}^N)$.
Since $K\in L^r(\mathbb{R}^N)$, we have
\begin{align*}
\int_{\mathbb{R}^N}K(x)f(u_n)(u_n-u)dx=o(1).
\end{align*}
Thus by $I'_\lambda(u_n)\to 0$ in $E^*$, one has
\begin{align*}
0&= \langle I_\lambda'(u_n),u_n-u\rangle+o(1)\\
&=  a\int_{\mathbb{R}^N}(\nabla u_n,\nabla(u_n-u))dx
 +b\int_{\mathbb{R}^N}|\nabla u_n|^2dx
 \int_{\mathbb{R}^N}(\nabla u_n,\nabla(u_n-u))dx\\
&\quad -\lambda\int_{\mathbb{R}^N}K(x)f(u_n)(u_n-u)dx+o(1)\\
&=  a(\|u_n\|^2-\|u\|^2)+bd(\|u_n\|^2-\|u\|^2)+o(1),
\end{align*}
which implies $\|u_n\|\to \|u\|$. Combining $u_n\rightharpoonup u$ in $E$,
we get $u_n\to  u$ in $E$. The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.6}
 According to Proposition \ref{prop2.1} and Lemma \ref{lem3.2}, for almost every
$\lambda\in[1/2,1]$, there exists a bounded sequence $\{u_n\}\subset E$
such that $I_\lambda(u_n)\to  c_\lambda$ and $I'_\lambda(u_n)\to 0$ in $E^*$.
 Define $u^{\pm}=\max\{\pm u,0\}$, then
\begin{align*}
o(1)&= \langle I'_\lambda(u_n),u_n^-\rangle\\
&= \Big(a+b\int_{\mathbb{R}^N}|\nabla u_n|^{2}dx\Big)
 \int_{\mathbb{R}^N}(\nabla u_n,\nabla u_n^-)dx
 -\lambda\int_{\mathbb{R}^N}K(x)f(u_n)u_n^-dx\\
&= -\Big(a+b\int_{\mathbb{R}^N}|\nabla u_n|^{2}dx\Big)
 \int_{\mathbb{R}^N}|\nabla u_n^-|^{2}dx,
\end{align*}
which implies $u_n^-\to 0$ in $E$. Thus one has
\begin{align*}
c_\lambda
&= I_\lambda(u_n)+o(1)\\
&= \frac{a}{2}\int_{\mathbb{R}^N}|\nabla u_n|^2dx
 +\frac{b}{4}\left(\int_{\mathbb{R}^N}|\nabla u_n|^2dx\right)^2
 -\lambda\int_{\mathbb{R}^N}K(x)F(u_n)dx+o(1)\\
&= \frac{a}{2}\Big(\int_{\mathbb{R}^N}|\nabla u_n^+|^2dx
 +\int_{\mathbb{R}^N}|\nabla u_n^-|^2dx\Big)\\
&\quad +\frac{b}{4}\Big(\int_{\mathbb{R}^N}|\nabla u_n^+|^2dx
+\int_{\mathbb{R}^N}|\nabla u_n^-|^2dx\Big)^2
 -\lambda\int_{\mathbb{R}^N}K(x)F(u_n^+)dx+o(1)\\
&= \frac{a}{2}\int_{\mathbb{R}^N}|\nabla u_n^+|^2dx
 +\frac{b}{4}\left(\int_{\mathbb{R}^N}|\nabla u_n^+|^2dx\right)^2
 -\lambda\int_{\mathbb{R}^N}K(x)F(u_n^+)dx+o(1)\\
&= I_\lambda(u_n^+)+o(1)
\end{align*}
and
\begin{align*}
0
&= \langle I'_\lambda(u_n),\varphi\rangle+o(1)\\
&= \Big(a+b\int_{\mathbb{R}^N}|\nabla u_n|^{2}dx\Big)
 \int_{\mathbb{R}^N}(\nabla u_n,\nabla \varphi)dx
 -\lambda\int_{\mathbb{R}^N}K(x)f(u_n)\varphi dx+o(1)\\
&= \Big(a+b\int_{\mathbb{R}^N}|\nabla u_n^+|^{2}dx
 +b\int_{\mathbb{R}^N}|\nabla u_n^-|^{2}dx\Big)
 \Big(\int_{\mathbb{R}^N}(\nabla u_n^+,\nabla \varphi)dx\\
&\quad -\int_{\mathbb{R}^N}(\nabla u_n^-,\nabla \varphi)dx\Big)
 -\lambda\int_{\mathbb{R}^N}K(x)f(u_n^+)\varphi dx+o(1)\\
&= \Big(a+b\int_{\mathbb{R}^N}|\nabla u_n^+|^{2}dx\Big)
 \int_{\mathbb{R}^N}(\nabla u_n^+,\nabla \varphi)dx
 -\lambda\int_{\mathbb{R}^N}K(x)f(u_n^+)\varphi dx+o(1)\\
&= \langle I'_\lambda(u_n^+),\varphi\rangle+o(1),
\end{align*}
uniformly for all $\varphi\in E$ and $\|\varphi\|=1$.
That is, $\{u_n^+\}$ is a bounded Palais-Smale sequence of
$I_\lambda$. By Lemma \ref{lem3.5}, there exists a nonnegative function $u\in E$
such that, up to a subsequence, $u_n^+\to  u$ in $E$.
Thus $I_\lambda(u)=c_\lambda$ and $I'_\lambda(u)=0$ in $E^*$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Set $\lambda_j\in[1/2,1]$ and $\lambda_j\to 1^-$. Then there exists
a sequence nonnegative functions $u_j\in E$ such that
$I_{\lambda_j}(u_j)=c_{\lambda_j}$ and $I'_{\lambda_j}(u_j)=0$.
 If $N\geq4$, $2<p<2^*\leq4$. By (K1), \eqref{formula9} and
$\langle I'_{\lambda_j}(u_j),u_j\rangle=0$, we have
\begin{equation}\label{formula8}
\begin{aligned}
a\|u_j\|^2+b\|u_j\|^4
&= \lambda_j\int_{\mathbb{R}^N}K(x)f(u_j)u_jdx\\
&\leq  C_1\int_{\mathbb{R}^N}K(x)|u_j|^pdx\\
&\leq  C_1|K|_r|u_j|^p_{2^*}\\
&\leq  C\|u_j\|^p,
\end{aligned}
\end{equation}
which implies $\|u_j\|\leq C$. If $N=3$ and $2<p<4$, by \eqref{formula8}
 with $N=3$, we have $\|u_j\|\leq C$. If $N=3$ and $4\leq p<6$, by
(K1), (K2), $I'_{\lambda_j}(u_j)=0$, one has the following Pohozaev
equality (see \cite[Lemma 2.2]{LLS1})
\[
\frac{a}{2}\|u_j\|^2+\frac{b}{2}\|u_j\|^4
= 3\lambda_j\int_{\mathbb{R}^3}K(x)F(u_j)dx
+\lambda_j\int_{\mathbb{R}^3}(\nabla K(x),x)F(u_j)dx.
\]
Combining (K2) with $I_{\lambda_j}(u_j)=c_{\lambda_j}$, we have
\begin{align*}
\frac{a}{2}\|u_j\|^2+\frac{b}{2}\|u_j\|^4
&= 3\lambda_j\int_{\mathbb{R}^3}K(x)F(u_j)dx+\lambda_j\int_{\mathbb{R}^3}(\nabla K(x),x)F(u_j)dx\\
&\geq 2\lambda_j\int_{\mathbb{R}^3}K(x)F(u_j)dx\\
&= a\|u_j\|^2+\frac{b}{2}\|u_j\|^4-2c_{\lambda_j},
\end{align*}
which implies that
\begin{align*}
2c_{\lambda_j}\geq\frac{a}{2}\|u_j\|^2.
\end{align*}
Hence $\|u_j\|\leq C$. Since
$I(u_j)=I_{\lambda_j}(u_j)+o(1)=c_{\lambda_j}+o(1)=c+o(1)$ and
$I'(u_j)=I'_{\lambda_j}(u_j)+o(1)=o(1)$ in $E^*$, according to Lemma \ref{lem3.5},
there exists a nonnegative function $u\in E$ such that $u_j\to  u$ in $E$.
Thereby $I(u_j)\to  I(u)=c$ and $I'(u_j)\to  I'(u)=0$ in $E^*$.
That is, $u$ is a nonnegative solution of equation \eqref{formula1}.
To obtain the ground state solution, we set $\pi=\inf_{u\in\Pi}I(u)$,
where $\Pi=\{u\in E\backslash\{0\}|I'(u)=0,u\geq0\}$ and then $\pi\leq c$.
 Obviously, $\pi>-\infty$. Since $\Pi\neq\emptyset$, there exist a nonnegative
sequence $u_n\in E$ such that $I'(u_n)=0$ and $I(u_n)\to \pi$.
With the same method, we can obtain $\{u_n\}$ is bounded in $E$ and then
there exists a nonnegative function $u\in E$ such that $u_n\to  u$ in $E$.
Hence we have $I(u_n)\to  I(u)=\pi$, $I'(u_n)\to  I'(u)=0$.
Because of the strongly maximum principle, we know $u>0$. So we complete
the proof of Theorem \ref{thm1.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 By Lemma \ref{lem3.1}, Lemma \ref{lem3.3} and the mountain pass theorem, there exists
a sequence $\{u_n\}\subset E$ such that $I(u_n)\to  c$ and
$I'(u_n)\to 0$ in $E^*$. By (F4), for $n$ large enough, we have
\begin{align*}
c+1+\|u_n\|
&= I(u_n)-\frac{1}{4}\langle I'(u_n),u_n\rangle\\
&= \frac{a}{4}\|u_n\|^2+\int_{\mathbb{R}^3}K(x)[\frac{1}{4}f(u_n)u_n-F(u_n)]dx\\
&\geq \frac{a}{4}\|u_n\|^2.
\end{align*}
Thus $\{u_n\}$ is bounded in $E$. By Remark \ref{rmk3.6}, $\{u_n^+\}$
is a bounded sequence satisfying $I(u_n^+)\to  c$ and $I'(u_n^+)\to 0$
in $E^*$. By Lemma \ref{lem3.5}, there exists a nonnegative function $u\in E$
such that, up to a subsequence, $u_n^+\to  u$ in $E$.
Thus $I(u)=c$ and $I'(u)=0$ in $E^*$. The rest of proof
is the same as Theorem \ref{thm1.1}. The proof is complete.
\end{proof}

\section{Proofs of Theorem \ref{thm1.3}, \ref{thm1.4}, \ref{thm1.5} and 
\ref{thm1.6}}


 We establish parallel steps as 
Lemmas \ref{lem3.1}, \ref{lem3.2}, \ref{lem3.3} and \ref{lem3.5}.

\begin{lemma} \label{lem4.1}
Assume that $a>0$, $b>0$ and $N\geq3$. Suppose that 
{\rm (K1), (K3), (F1), (F5)} hold. Then there exist $\rho>0$ and $\alpha>0$ 
such that $I_{\lambda}(u)|_{\|u\|=\rho}\geq\alpha$ for all 
$\lambda\in[1/2,1]$.
\end{lemma}

\begin{proof}
 By \eqref{formula13} with $\varepsilon=1$, the H\"{o}lder and Sobolev inequalities, 
for all $u\in E$ and all $\lambda\in[1/2,1]$, we have
\begin{align*}
I_{\lambda}(u)
&\geq \frac{a}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx
 -\int_{\mathbb{R}^N}K(x)F(u)dx\\
&\geq \frac{a}{2}\|u\|^2-\frac{1}{2^*}\int_{\mathbb{R}^N}K(x)|u|^{2^*}dx
 -C\int_{\mathbb{R}^N}K(x)|u|^pdx\\
&\geq \frac{a}{2}\|u\|^2-\frac{1}{2^*}|K|_\infty\int_{\mathbb{R}^N}|u|^{2^*}dx
 -C|K|_r|u|_{2^*}^p\\
&\geq C_2\|u\|^2-C_3\|u\|^{2^*}-C_4\|u\|^{p}\\
&= \|u\|^2(C_2-C_3\|u\|^{2^*-2}-C_4\|u\|^{p-2}).
\end{align*}
Since $p>2$, we can choose $\rho>0$ small enough such that 
$C_2-C_3\rho^{2^*-2}-C_4\rho^{p-2}>0$. Then there exists $\alpha>0$ such 
that $I_\lambda(u)\geq\alpha$ for all $\|u\|=\rho$. The proof is complete.
\end{proof}

The proofs of the following two lemmas are the same as Lemma \ref{lem3.2} and 
\ref{lem3.3}.

\begin{lemma} \label{lem4.2}
Assume that $a>0$, $b>0$ and $N\geq3$. Suppose that 
{\rm (K1), (K3), (F1), (F3), (F5)} hold. Then there exists $b_0>0$ such 
that for any $b\in(0,b_0)$, there are two points $v_1$, $v_2$ in $E$ such 
that $c_\lambda>\max\{I_\lambda(v_1),I_\lambda(v_2)\}$ for all $\lambda\in[1/2,1]$.
\end{lemma}

\begin{lemma} \label{lem4.3}
Assume that $a>0$, $b>0$ and $N=3$. Suppose that (K1) with $4\leq p<6$,
{\rm (K3), (F1), (F4), (F5)} hold. Then there are two points $v_1$, $v_2$ 
in $E$ such that $c_\lambda>\max\{I_\lambda(v_1),I_\lambda(v_2)\}$ for all 
$\lambda\in[1/2,1]$.
\end{lemma}

\begin{lemma} \label{lem4.4}
For any $\lambda\in[1/2,1]$, if $\{u_n\}\subset E$ is a bounded and nonnegative 
Palais-Smale sequence of the function $I_\lambda$, there exists a nonnegative 
function $u\in E$ such that, up to a subsequence, $u_n\to  u$ in $E$.
\end{lemma}

\begin{proof} 
Since $\{u_n\}$ is bounded and nonnegative in $E$, up to a subsequence, 
there exists a nonnegative function $u\in E$ such that $u_n\rightharpoonup u$ 
in $E$, $u_n(x)\to  u(x)$ a.e. in $\mathbb{R}^N$ and there exists $d\geq0$ 
such that $d=\int_{\mathbb{R}^N}|\nabla u_n|^2dx+o(1)$. 
Since $\{|u_n|^{p-1}|u_n-u|\}$ is bounded in $L^{\frac{2^*}{p}}(\mathbb{R}^N)$ 
and $|u_n(x)|^{p-1}|u_n(x)-u(x)|\to 0$ a.e. in $\mathbb{R}^N$, 
by Lemma \ref{lem3.4}, up to a subsequence, we have
$|u_n|^{p-1}|u_n-u|\rightharpoonup0$ in $L^{\frac{2^*}{p}}(\mathbb{R}^N)$.
 In view of $K\in L^r(\mathbb{R}^N)$, one has
\begin{align*}
\int_{\mathbb{R}^N}K(x)|u_n|^{p-1}|u_n-u|dx=o(1).
\end{align*}
Combining this with \eqref{formula12}, we have
\begin{align*}
&\Big|\int_{\mathbb{R}^N}K(x)f(u_n)(u_n-u)dx\Big|\\
&\leq \varepsilon\int_{\mathbb{R}^N}K(x)|u_n|^{2^*-1}|u_n-u|dx
 +C_\varepsilon\int_{\mathbb{R}^N}K(x)|u_n|^{p-1}|u_n-u|dx\\
&\leq \varepsilon|K|_\infty|u_n|_{2^*}^{2^*-1}|u_n-u|_{2^*}+o(1)
= C\varepsilon+o(1).
\end{align*}
Hence by $I'_\lambda(u_n)\to 0$ in $E^*$, we  deduce that
\begin{align*}
0&= \langle I_\lambda'(u_n),u_n-u\rangle+o(1)\\
&= a\int_{\mathbb{R}^N}(\nabla u_n,\nabla(u_n-u))dx
 +b\int_{\mathbb{R}^N}|\nabla u_n|^2dx\int_{\mathbb{R}^N}(\nabla u_n,\nabla(u_n-u))dx\\
&\quad -\lambda\int_{\mathbb{R}^N}K(x)f(u_n)(u_n-u)dx+o(1)\\
&= a(\|u_n\|^2-\|u\|^2)+bd(\|u_n\|^2-\|u\|^2)+o(1),
\end{align*}
which implies $\|u_n\|\to \|u\|$. Combining $u_n\rightharpoonup u$ in $E$, 
we get $u_n\to  u$ in $E$. The proof is complete.
\end{proof} 

According to Proposition \ref{prop2.1} and Lemma \ref{lem4.2}, for almost every 
$\lambda\in[1/2,1]$, there exists a bounded sequence $\{u_n\}\subset E$ 
such that $I_\lambda(u_n)\to  c_\lambda$ and $I'_\lambda(u_n)\to 0$ in $E^*$. 
By Remark \ref{rmk3.6}, we can assume that $u_n$ is nonnegative. 
Thus from Lemma \ref{lem4.4}, we know that there exists a nonnegative function 
$u\in E$ such that $u_n\to  u$ in $E$ and then for almost every 
$\lambda\in[1/2,1]$, $I_\lambda(u)=c_\lambda$ and $I'_\lambda(u)=0$.


\begin{proof}[Proof of Theorem \ref{thm1.3}] 
Set $\lambda_j\in[1/2,1]$ and $\lambda_j\to 1^-$. T
hen there exists a nonnegative sequence $\{u_j\}\subset E$ such that 
$I_{\lambda_j}(u_j)=c_{\lambda_j}$ and $I'_{\lambda_j}(u_j)=0$. 
If $N\geq4$, $2<p<2^*\leq4$. By (K1), (K3), \eqref{formula12} and 
$\langle I'_{\lambda_j}(u_j),u_j\rangle=0$, we have
\begin{align*}
a\|u_j\|^2+b\|u_j\|^4&= \lambda_j\int_{\mathbb{R}^N}K(x)f(u_j)u_jdx\\
&\leq \varepsilon\int_{\mathbb{R}^N}K(x)|u_j|^{2^*}dx
 +C_\varepsilon\int_{\mathbb{R}^N}K(x)|u_j|^pdx\\
&\leq \varepsilon|K|_\infty\int_{\mathbb{R}^N}|u_j|^{2^*}dx
 +C_\varepsilon|K|_r|u_j|^p_{2^*}\\
&\leq C\varepsilon\|u_j\|^{2^*}+ C_\varepsilon\|u_j\|^p,
\end{align*}
which implies $\|u_j\|\leq C$, for $\varepsilon=\frac{b}{2C}$. 
If $N=3$, by (K1), (K2), $I'_{\lambda_j}(u_j)=0$, one has the 
Pohozaev equality
\begin{align*}
\frac{a}{2}\|u_j\|^2+\frac{b}{2}\|u_j\|^4
&= 3\lambda_j \int_{\mathbb{R}^3}K(x)F(u_j)dx
 +\lambda_j\int_{\mathbb{R}^3}(\nabla K(x),x)F(u_j)dx.
\end{align*}
Combining $I_{\lambda_j}(u_j)=c_{\lambda_j}$, we have
\begin{align*}
\frac{a}{2}\|u_j\|^2+\frac{b}{2}\|u_j\|^4
&= 3\lambda_j\int_{\mathbb{R}^3}K(x)F(u_j)dx
 +\lambda_j\int_{\mathbb{R}^3}(\nabla K(x),x)F(u_j)dx\\
&\geq 2\lambda_j\int_{\mathbb{R}^3}K(x)F(u_j)dx\\
&= a\|u_j\|^2+\frac{b}{2}\|u_j\|^4-2c_{\lambda_j}
\end{align*}
which implies 
\begin{align*}
2c_{\lambda_j}\geq\frac{a}{2}\|u_j\|^2.
\end{align*}
Hence $\|u_j\|\leq C$. The rest of the proof is similar with Theorem \ref{thm1.1}.
\end{proof}


The proof of Theorem \ref{thm1.4} is same as that of  
Theorem \ref{thm1.2} and it is omitted.


\begin{proof}[Proof of Theorem \ref{thm1.5}]
 Suppose that $u\in E$ is a nonzero solution of  \eqref{formula1}. 
Then combining \eqref{formula14}, \eqref{formula9}, the H\"{o}lder and 
Young inequalities, we have
\begin{align*}
a\|u\|^2+b\|u\|^4
&= \int_{\mathbb{R}^N}K(x)f(u)udx\\
&\leq C_1\int_{\mathbb{R}^N}K(x)|u|^pdx\\
&\leq C_1|K|_r|u|_{2^*}^p\\
&\leq C_1S^{-\frac{p}{2}}|K|_r\|u\|^p\\
&= \big(\frac{2a}{4-p}\big)^{\frac{4-p}{2}}\|u\|^{4-p}C_1S^{-\frac{p}{2}}|K|_r
\big(\frac{4-p}{2a}\big)^{\frac{4-p}{2}}\|u\|^{2p-4}\\
&\leq a\|u\|^2+\frac{p-2}{2}\big[C_1S^{-\frac{p}{2}}|K|_r
\big(\frac{4-p}{2a}\big)^{\frac{4-p}{2}}\big]^{\frac{2}{p-2}}\|u\|^4\\
&< a\|u\|^2+b\|u\|^4,
\end{align*}
for any $b>B:=\frac{p-2}{2}[C_1S^{-\frac{p}{2}}|K|_r(\frac{4-p}{2a})
^{\frac{4-p}{2}}]^{\frac{2}{p-2}}$, which is a contradiction. 
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.6}]
 Suppose that $u\in E$ is a nonzero solution of  \eqref{formula1}. 
Then combining \eqref{formula14}, \eqref{formula12} and  H\"{o}lder's inequality, 
we have
\begin{equation}\label{formula16}
\begin{aligned}
a\|u\|^2+b\|u\|^4
&= \int_{\mathbb{R}^N}K(x)f(u)udx\\
&\leq \varepsilon\int_{\mathbb{R}^N}K(x)|u|^{2^*}dx
 +C_\varepsilon\int_{\mathbb{R}^N}K(x)|u|^pdx\\
&\leq \varepsilon|K|_\infty S^{-\frac{2^*}{2}}\|u\|^{2^*}
 +C_\varepsilon|K|_rS^{-\frac{p}{2}}\|u\|^p.
\end{aligned} 
\end{equation}
When $N=4$, $2^*=4$. Choose 
$\varepsilon=\frac{b}{2|K|_\infty S^{-\frac{2^*}{2}}}$.
 Using \eqref{formula16} and the Young inequality, one has
\begin{align*}
a\|u\|^2+\frac{b}{2}\|u\|^4
&\leq C|K|_rS^{-\frac{p}{2}}\|u\|^p\\
&= \big(\frac{2a}{4-p}\big)^{\frac{4-p}{2}}\|u\|^{4-p}CS^{-\frac{p}{2}}|K|_r
\big(\frac{4-p}{2a}\big)^{\frac{4-p}{2}}\|u\|^{2p-4}\\
&\leq a\|u\|^2+\frac{p-2}{2}\big[CS^{-\frac{p}{2}}|K|_r
\big(\frac{4-p}{2a}\big)^{\frac{4-p}{2}}\big]^{\frac{2}{p-2}}\|u\|^4\\
&< a\|u\|^2+\frac{b}{2}\|u\|^4,
\end{align*}
for any $b>B:=(p-2)\big[CS^{-\frac{p}{2}}|K|_r\big(\frac{4-p}{2a}
\big)^{\frac{4-p}{2}}\big]^{\frac{2}{p-2}}$, which is a contradiction. 
When $N>4$, $2<2^*<4$. Choose $\varepsilon=1$. Using \eqref{formula16}
 and the Young inequality, one has
\begin{align*}
a\|u\|^2+b\|u\|^4&\leq |K|_\infty S^{-\frac{2^*}{2}}\|u\|^{2^*}
+C|K|_rS^{-\frac{p}{2}}\|u\|^p\\
&= \big(\frac{a}{4-2^*}\big)^{\frac{4-2^*}{2}}\|u\|^{4-2^*}|K|_\infty
  S^{-\frac{2^*}{2}}\big(\frac{4-2^*}{a}\big)^{\frac{4-2^*}{2}}\|u\|^{22^*-4}\\
&\quad +\big(\frac{a}{4-p}\big)^{\frac{4-p}{2}}\|u\|^{4-p}CS^{-\frac{p}{2}}|K|_r
 \big(\frac{4-p}{a}\big)^{\frac{4-p}{2}}\|u\|^{2p-4}\\
&\leq a\|u\|^2+\frac{2^*-2}{2}
\big[|K|_\infty S^{-\frac{2^*}{2}}\big(\frac{4-2^*}{a}\big)
 ^{\frac{4-2^*}{2}}\big]^{\frac{2}{2^*-2}}\|u\|^4\\
&\quad +\frac{p-2}{2}\big[CS^{-\frac{p}{2}}|K|_r\big(\frac{4-p}{a}
 \big)^{\frac{4-p}{2}}\big]^{\frac{2}{p-2}}\|u\|^4\\
&< a\|u\|^2+b\|u\|^4,
\end{align*}
for any $b>B:=\frac{2^*-2}{2}\big[|K|_\infty S^{-\frac{2^*}{2}}\big(\frac{4-2^*}{a}
\big)^{\frac{4-2^*}{2}}\big]^{\frac{2}{2^*-2}}
+\frac{p-2}{2}\big[CS^{-\frac{p}{2}}|K|_r\big(\frac{4-p}{a}
 \big)^{\frac{4-p}{2}}\big]^{\frac{2}{p-2}}$, which is a contradiction. 
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the
National Natural Science Foundation of China (No. 11471267), and by
the Fundamental Research Funds for the Central Universities (No. XDJK2013C006).


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