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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 89, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/89\hfil $p$-Kirchhoff type problem]
{$p$-Kirchhoff type problem with a general critical nonlinearity}

\author[H. Zhang, B. Lin \hfil EJDE-2018/89\hfilneg]
{Huixing Zhang, Baiquan Lin}

\address{Huixing Zhang \newline
School of Mathematics and
School of Safety Engineering,
China University of Mining and Technology,
Xuzhou, Jiangsu 221116, China}
\email{huixingzhangcumt@163.com}

\address{Baiquan Lin \newline
School of Safety Engineering,
China University of Mining and Technology,
Xuzhou, Jiangsu 221116, China}
\email{lbq21405@126.com}

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted September 5, 2017. Published April 11, 2018.}
\subjclass[2010]{35B25, 35B33, 35J61}
\keywords{$p$-Kirchhoff type problem; critical growth; variational methods}

\begin{abstract}
 In this article, we consider the $p$-Kirchhoff type problem
 $$
 \Big(1+\lambda\int_{\mathbb{R}^N}|\nabla u|^p
 +\lambda b\int_{\mathbb{R}^N}|u|^p\Big)(-\Delta_p u+b|u|^{p-2}u)
 =f(u), x\in\mathbb{R}^N,
 $$
 where $\lambda>0$, the nonlinearity $f$ can reach critical growth.
 Without the Ambrosetti-Robinowitz condition or the monotonicity condition
 on $f$, we prove the existence of positive solutions for the $p$-Kirchhoff
 type problem. In addition, we also study the asymptotic behavior of the
 solutions with respect to the parameter $\lambda\to 0$.
\end{abstract}

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

\section{Introduction and statement of results}

In this article, we study the $p$-Kirchhoff type problem
\begin{equation}\label{eq1.1}
 \Big(1+\lambda\int_{\mathbb{R}^N}|\nabla u|^p
 +\lambda b\int_{\mathbb{R}^N}|u|^p\Big)(-\Delta_p u+b|u|^{p-2}u)
 =f(u)\quad \text{in }\mathbb{R}^N,
\end{equation}
where $b,\lambda>0$, $\Delta_pu:=\text{div}(|\nabla u|^{p-2}\nabla u)$ with
$1<p<N$ and the nonlinearity\ $f$\ may be critical. Problem \eqref{eq1.1}
 with $p=2$ reduces to the Kirchhoff type problem
\begin{equation} \label{eq1.2}
(1+\lambda\int_{\mathbb{R}^N}|\nabla u|^2+\lambda b\int_{\mathbb{R}^N}|u|^2)
(-\Delta u+bu)=f(u)\quad \text{in } \mathbb{R}^N.
\end{equation}
In the previous decades, the Kirchhoff type problem \eqref{eq1.2} has been
object of intensive research as its strong relevance in applications.
From a physical point of view, problem \eqref{eq1.2} on bounded domain
$\Omega \subset \mathbb{R}^N$ is related to the stationary analogue of
the equation
\begin{equation} \label{eq1.3}
\rho\frac{\partial^2 u}{\partial t^2}
 -\Big(\frac{P_0}{h}+\frac{E}{2L}\int_0^L |\frac{\partial u}{\partial x}|^2\,\mathrm{d} x
 \Big)\frac{\partial^2 u}{\partial x^2}=0,
\end{equation}
which is proposed by Kirchhoff in \cite{Kirchhoff} as an existence of the
classical D'Alembert's wave equations for free vibration of elastic strings.
After Lions \cite{Lions} introduced a functional analysis approach to
equation \eqref{eq1.3}, he gave the equation
\begin{equation} \label{eq1.4}
u_{tt}-\Big(a+b\int_{\Omega}|\nabla u|^2\,\mathrm{d} x\Big)\Delta u
 =f(x,u),\quad x\in\Omega,\; u=0,\; x\in\partial\Omega.
\end{equation}
When Kirchhoff's model takes into account the change of the string length
caused by oscillations, $u$ represents the displacement, $f(x,u)$ denotes
the external force, $b$ is the initial tension while $a$ is related to
the intrinsic properties of the string, such as Young's modulus.
Moreover, problem \eqref{eq1.2} on bounded domain appears in many mathematical
biological contexts. In \cite{Alves}, Kirchhoff type problem models some
biological systems, where $u$ describes a process which depends on the
average of itself, such as population density.

In recent years, Kirchhoff type problems on $\mathbb{R}^N$ have been
studied widely by the variational methods and results can be seen
in \cite{Huang,Li,Liu1,Liu2}. Especially, in \cite{Li}, Li et al.\
 considered problem \eqref{eq1.2} under the following conditions
\begin{itemize}
\item[(A1)] $f\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ and $|f(t)|\leq C_1(|t|+|t|^{p-1})$ for $t\ge0$ and some $p\in(2,2^{\ast})$;

\item[(A2)] $\lim_{t\to0}f(t)/t=0$;

\item[(A3)] $\lim_{t\to\infty}\sup f(t)/t=\infty$.

\end{itemize}

\begin{theorem}[see\cite{Li}] \label{thmA}
 Assume that $N\geq3$ and {\rm (A1)--(A3)} hold. Then there exists $\lambda_{0}>0$
such that for any $\lambda\in[0,\lambda_{0}]$, problem \eqref{eq1.2}
 has at least one positive solution.
\end{theorem}

Subsequently, Liu, Liao and Tang \cite{Liu2} studied problem \eqref{eq1.2}
 under some weaker conditions than the ones in \cite{Li}.
In \cite{Li,Liu2}, the authors only considered problem \eqref{eq1.2}
 with subcritical growth. For the $p$-Kirchhoff type problem \eqref{eq1.1},
there are also many results, see for example \cite{Autuori,Chen1,Chen2,Cheng}.
 Autuori, Colasuonno and Pucci \cite{Autuori} obtained two nontrivial solutions
of possibly degenerate nonlinear eigenvalue problems involving the $p$-poly-harmonic
Kirchhoff operator in bounded domains. Using the Nehari manifold method,
Chen and Zhu \cite{Chen2} obtained positive solutions to the problem
$$
\Big[a+\lambda\Big(\int_{\mathbb{R}^N}(|\nabla u|^p+b|u|^p)dx\Big)^{\tau}\Big]
(-\Delta_{p}u+b|u|^{p-2}u)=|u|^{m-2}u+\mu|u|^{q-2}u,\ x\in\mathbb{R}^N.
$$
For fractional $p$-Kirchhoff problems, we refer to \cite{Binlin} and the
references therein.

The papers cited above were all focused on $p$-Kirchhoff type problem with
subcritical growth. Many of them need usual compactness conditions.
Compared to $p$-Kirchhoff problems with subcritical growth, there are few
results in term of $p$-Kirchhoff type problem involving critical growth.
In \cite{Jian}, the author only considered the $p$-Kirchhoff type problem with
specified critical growth term, not involving general critical growth.
To the best of our knowledge, without usual compactness condition,
there are few results conducted on problem \eqref{eq1.1} with general
nonlinearity $f$ reaching critical growth.

\subsection*{Main results}
In this article, we study $p$-Kirchhoff type problem \eqref{eq1.1} with critical
growth. Throughout the paper, $f\in C(\mathbb{R}_{+},\mathbb{R}_{+})$ with
$\mathbb{R}_{+}=[0,+\infty)$ and satisfies
\begin{itemize}
\item [(A4)] $\lim_{s\to0}f(s)/s^{p-1}=0$;
\item [(A5)] $\lim_{s\to\infty}\sup f(s)/s^{p^*-1}\le1$,\ where\ $p^*=Np/(N-p)$;
\item [(A6)] There are $\alpha>0$ and $q\in(p,p^*)$ such that
$f(s)\ge\alpha s^{q-1}$ for all $s\ge0$.
\end{itemize}

Condition (A5) implies that $f$ has a critical growth at infinity and the
limit of $f(s)/s^{p^*-1}$ at $+\infty$ is not necessary to exist.


Let $S$ and $C_s$ denote the best constants of Sobolev embeddings
$D^{1,p}(\mathbb{R}^N)\hookrightarrow L^{p^*}(\mathbb{R}^N)$ and
$W^{1,p}(\mathbb{R}^N)\hookrightarrow L^s(\mathbb{R}^N)$, namely,
\begin{gather*}
S\Big(\int_{\mathbb{R}^N}|u|^{p^*}\Big)^{p/p^*}
\le\int_{\mathbb{R}^N}|\nabla u|^p\quad \text{for all }
  u\in D^{1,p}(\mathbb{R}^N),\\
C_s\left(\int_{\mathbb{R}^N}|u|^s\right)^{p/s}
\le\int_{\mathbb{R}^N}(|\nabla u|^p+b|u|^p)\quad \text{for all }
 u\in W^{1,p}(\mathbb{R}^N).
\end{gather*}

Our main results read as follows.

\begin{theorem} \label{thm1.1}
Assume {\rm (A4)--(A6)} hold. Then there exists $\lambda^{\ast}>0$ such that,
for any $\lambda\in (0,\lambda^{\ast})$, problem \eqref{eq1.1} possesses
a nontrivial positive radial solution $u_\lambda$, provided that
$$
\alpha > S^{\frac{N(p-q)}{p^2}}C_q^{p/q}m^{\frac{N(p-q)}{p^2}
+\frac{q}{p}}\left(\frac{N}{p}-\frac{N}{q}\right)^{\frac{q-p}{p}}.
$$
\end{theorem}

\begin{theorem} \label{thm1.2}
Assume {(A4)--(A6)} hold. As $\lambda\to0$, $\{u_\lambda\}$ converges to $u$
in $W_r^{1,p}(\mathbb{R}^N)$ (necessarily along a subsequence), where $u$
is a ground state solution of
$$
-\Delta_p u+b|u|^{p-2}u=f(u)\ \text{in}\ \mathbb{R}^N.
$$
\end{theorem}

For $p=2$ in Theorem \ref{thm1.1}, Li et al.\ \cite{Li} and Liu et al.\ \cite{Liu2}
considered problem \eqref{eq1.2}, but they only studied Kirchhoff type problem
\eqref{eq1.2} with the general nonlinearity involving subcritical growth.
Theorem \ref{thm1.1} is concerned with a nonlinearity $f$ reaching critical growth,
which makes the problem much more complicated.

\subsection*{Main difficulties and ideas}

To prove our results by variational methods, the difficulties are two-fold.
The first difficulty is due to the appearance of
$\int_{\mathbb{R}^N}(|\nabla u|^p+b|u|^p)$, which implies that \eqref{eq1.1}
is no longer a pointwise identity. Namely, such a phenomenon causes some
 mathematical difficulties. The second difficulty lies in obtaining the
boundedness of the Palais-Smale sequence (in short (PS) sequence) to the
energy functional without usual Ambrosetti-Rabinowtiz condition.
To overcome these difficulties, we adopt a local deformation argument
from Byeon and Jeanjean \cite{Byeon1} to obtain a bounded (PS) sequence.
Then we use similar ideas in \cite{Zhang2} to make a crucial modification
on the min-max value as the presence of nonlocal term.

The rest of this is organized as follows. Section 2 is devoted to the
limit problem. In Section 3, we define a min-max level and construct
a bounded (PS) sequence. Finally, we give the proof of Theorem \ref{thm1.1}.

\noindent\textbf{Notation}

$\bullet$ $\|u\|_s :=\big(\int_{\mathbb{R}^N}|u|^s\big)^{1/s}$ for
$s\in[1,\infty)$ and $u\in L^s(\mathbb{R}^N)$.

 $\bullet$ Let $W^{1,p}(\mathbb{R}^N)$ be the Sobolev space equipped with
the norm
\[
\|u\|:=\Big(\int_{\mathbb{R}^N}(|\nabla u|^p+b|u|^p)\Big)^{1/p}
\]
and $W_r^{1,p}(\mathbb{R}^N)=\{u\in W^{1,p}(\mathbb{R}^N): u(x)=u(|x|)\}$.

\section{Limit problem }

When $\lambda=0$, problem \eqref{eq1.1} reduces to the problem
\begin{equation}\label{eq1.5}
-\Delta_p u+b|u|^{p-2}u=f(u)\quad \text{in } \mathbb{R}^N,
\end{equation}
which is called the limit problem of problem \eqref{eq1.1}.
For $\lambda>0$ small, we may view the problem \eqref{eq1.1} as a
corresponding perturbation problem to \eqref{eq1.5}. In general,
if problem \eqref{eq1.5} is well-behaved, then we may expect that the
 perturbed problem \eqref{eq1.1} possesses a solution in some neighborhood
of solutions to problem \eqref{eq1.5}. Indeed, the idea plays a critical
role in establishing the existence of positive solutions to problem \eqref{eq1.1}.

We define
$$
J(u):=\frac{1}{p}\int_{\mathbb{R}^N}(|\nabla u|^p+b|u|^p)
-\int_{\mathbb{R}^N}F(u),\ u\in W_r^{1,p}(\mathbb{R}^N),
$$
where $F(t)=\int_0^t f(s)\,\mathrm{d} s$ and the mountain pass value
$
m:=\inf_{\gamma\in\Gamma}\max_{0\le t\le1}J(\gamma(t)),
$
where
$$
\Gamma=\{\gamma\in C([0,1],W_r^{1,p}(\mathbb{R}^N)):\gamma(0)=0,J(\gamma(1))<0\}.
$$
J. Zhang et al.\ \cite{Zhang1} show that $m$ is the least energy of problem
\eqref{eq1.5} and can be achieved by a radially symmetric function. Let $W_r$
denotes the set of positive ground state solutions $U$ of \eqref{eq1.5}
satisfying $U(0)=\max_{x\in \mathbb{R}^N}U(x)$. Then, $W_r\subset W_r^{1,p}(\mathbb{R}^N)$
and $W_r\not=\emptyset$.

\begin{lemma}[\cite{Zhang1}] \label{lem2.1}
Under the assumptions in Theorem \ref{thm1.1}, $W_r$ is compact in $W_r^{1,p}(\mathbb{R}^N)$.
\end{lemma}



\section{Proof of main results}

Since we only seek positive solutions of problem \eqref{eq1.1}, we may assume
$f(s)=0$ for all $s<0$. In addition, we work in $W_r^{1,p}(\mathbb{R}^N)$
because problem \eqref{eq1.1} is autonomous. Define
$$
\Phi_\lambda(u)=\frac{1}{p}\|u\|^p+\frac{\lambda}{2p}\|u\|^{2p}
-\int_{\mathbb{R}^N}F(u),\ u\in W^{1,p}(\mathbb{R}^N).
$$
By {\rm (A4)--(A6)}, $\Phi_\lambda\in C^1(W^{1,p}(\mathbb{R}^N),\mathbb{R})$
and for all $u,v\in W^{1,p}(\mathbb{R}^N)$,
$$
\langle\Phi_\lambda'(u),v\rangle=(1+\lambda\|u\|^p)
\int_{\mathbb{R}^N}(|\nabla u|^{p-2}\nabla u\cdot \nabla v+b|u|^{p-2}uv)
-\int_{\mathbb{R}^N}f(u)v.
$$
It is standard to verify that the critical points of $\Phi_\lambda$ are
weak solutions of \eqref{eq1.1}.


\subsection*{Minimax level}
Set $W_t(x)=W(x/t)$, $W\in W_r$, by the Pohoz\v{a}ev identity, we have
\begin{align*}
J(W_t)
&=\frac{1}{p}t^{N-p}\int_{\mathbb{R}^N}|\nabla W|^p
 -t^N\int_{\mathbb{R}^N}\Big(F(W)-\frac{b|W|^p}{p}\Big)\\
&=\Big(\frac{1}{p}t^{N-p}-\frac{N-p}{Np}t^N\Big)\int_{\mathbb{R}^N}|\nabla W|^p.
\end{align*}
It is clear that $J(W_t)\to-\infty$ as $t\to\infty$ and $J(W_{t^*})<-3$ for some
$t^*>1$. Let $A_\lambda=\max_{t\in[0,t^*]}\Phi_\lambda(W_t)$. By
$$
\int_{\mathbb{R}^N}|\nabla W|^p=Nm,
$$
we know
$$
\lim_{\lambda\to0}A_\lambda=\lim_{\lambda\to0}\max_{t\in[0,t^*]}J(W_t)=m.
$$

To get a uniformly bounded set of the mountain pathes, we have the following result.

\begin{lemma} \label{lem3.1}
There exist $\lambda^*>0$ and $C_2>0$, such that for any $\lambda\in(0,\lambda^*)$,
$\Phi_\lambda(W_{t^*})<-3$, $\|W_t\|\le C_2$, $t\in(0,t^*]$ and $\|W\|\le C_2$
for any $W\in W_r$.
\end{lemma}

\begin{proof}
For any $W\in W_r$, by Lemma \ref{lem2.1}, there exists $C_3>0$ such that $\|W\|\le C_3$
and
\begin{align*}
\|W_t\|^p
&=t^{N-p}\|\nabla W\|_p^p+bt^N\| W\|_p^p\\
&\le(t^{N-p}+bt^N)\| W\|^p\\
&\le((t^*)^{N-p}+b(t^*)^N)C_3^p.
\end{align*}
Let $C_2=\max\{C_3,((t^*)^{N-p}+b(t^*)^N)^{1/p}C_3\}$, then
 $\|W_t\|\le C_2$, $ t\in(0,t^*]$. In addition,
$$
\Phi_\lambda(W_{t^*})=J(W_{t^*})+\frac{\lambda}{2p}\|W_{t^*}\|^{2p}
\le J(W_{t^*})+\frac{\lambda}{2p}C_2^{2p}.
$$
By $J(W_{t^*})<-3$, there exists $\lambda^*>0$ such that
$\Phi_\lambda(W_{t^*})<-3$ for $\lambda\in(0,\lambda^*)$.
\end{proof}

Next, we define a minmax value $B_\lambda$ given by
$B_\lambda=\inf_{\gamma\in\Gamma_\lambda}\max_{s\in[0,t^*]}\Phi_\lambda(\gamma(s))$
where
$$
\Gamma_\lambda=\{\gamma\in C([0,t^*],W_r^{1,p}(\mathbb{R}^N)):\gamma(0)=0,
\gamma(t^*)=W_{t^*},\|\gamma(t)\|\le C_2+2\}.
$$
It is clear that $\Gamma_\lambda\neq\emptyset$ and $B_\lambda\le A_\lambda$
for $\lambda\in(0,\lambda^*)$.

\begin{lemma} \label{lem3.2}
$B_\lambda\to m$ as $\lambda\to0$.
\end{lemma}

\begin{proof}
Obviously, $B_\lambda\le A_\lambda\to m$ as $\lambda\to 0$. Notice that
$\Phi_\lambda(u)\ge J(u)$ for $u\in W_r^{1,p}(\mathbb{R}^N)$ and for any
$\gamma\in\Gamma_\lambda, \tilde{\gamma}(\cdot)=\gamma(t^*)\in\Gamma$.
Thus, $B_\lambda\ge m$. So, $\lim_{\lambda\to 0}B_\lambda=m$.
\end{proof}

For $c,d>0$, set
\begin{gather*}
\Phi_\lambda^c=\{u\in W_r^{1,p}(\mathbb{R}^N):\Phi_\lambda(u)\le c\}, \\
W^d=\{u\in W_r^{1,p}(\mathbb{R}^N):\inf_{v\in W_r}\|u-v\|\le d\}.
\end{gather*}
Clearly, $W^d\neq\emptyset$ for all $d>0$. In the following, we look for
a solution $u\in W^d$ of problem \eqref{eq1.1} for $\lambda>0$ small enough.

\begin{lemma} \label{lem3.3}
 There exist $C'>0$ and $\lambda_*>0$ such that for any $\lambda\in(0,\lambda_*)$
and $W\in\Phi_\lambda^{A_\lambda}\cap(W^d\backslash W^{d/2})$, we have
$\|\Phi_\lambda'(W)\|\ge C'$, provided that
\begin{equation} \label{eq3.1}
0<d<\min\{1,(Nm)^{1/p},\frac{1}{4}(\frac{p^{\ast}}{2p}
s^{p^{\ast}/p})^{\frac{1}{p^{\ast}-p}}\}.
\end{equation}
\end{lemma}

\begin{proof}
It suffices to prove that for $d$ small with \eqref{eq3.1} and any
 $\{W_{\lambda_i}\}\subset W^d$ with
\begin{gather*}
\lim_{i\to\infty}\Phi_{\lambda_i}(W_{\lambda_i})\le m, \\
\lim_{i\to\infty}\|\Phi_{\lambda_i}'(W_{\lambda_i})\|\to0,
\end{gather*}
there exists $W_0\in W_r$ such that $W_{\lambda_i}\to W_0$ in
$W_r^{1,p}(\mathbb{R}^N)$, where $\lim_{i\to 0}\lambda_i=0$.
For convenience, we replace $\lambda_i$ by $\lambda$.
Because $W_\lambda\in W^d$, $W_\lambda=u_\lambda+v_\lambda$, where
$u_\lambda\in W_r$ and $v_\lambda\in W^{1,p}(\mathbb{R}^N)$,
such that $u_\lambda\to u_0$ strongly in $W_r^{1,p}(\mathbb{R}^N)$,
$v_\lambda\to v_0$ weakly in $W^{1,p}(\mathbb{R}^N)$ and
$v_\lambda\to v_0$ a.e. in $\mathbb{R}^N$.
Set $W_0=u_0+v_0$, then $W_0\in W^d$ and $W_\lambda\to W_0$ weakly in
$W^{1,p}(\mathbb{R}^N)$. From
$\lim_{\lambda\to\infty}\|\Phi_\lambda'(W_\lambda)\|=0$,
we get that $J'(W_\lambda)\to0$ as $\lambda\to0$. So $J'(W_0)=0$.
 We claim that $W_0\not\equiv0$. On the contrary, if $W_0\equiv0$,
then $\|u_0\|=\|v_0\|\le d$. By the Pohoz\v{a}ev identity and $u_0\in W_r$,
we obtain that $\|\nabla u_0\|_p=(Nm)^{1/p}$.
However, by \eqref{eq3.1}, $\|\nabla u_0\|_p\le \|u_0\|\le d<(Nm)^{1/p}$.
This is a contradiction. So $W_0\not\equiv0$ and $J(W_0)\ge m$.
Moreover, by \cite[Theorem 2.1]{Boccardo}(also\cite[Lemma 2.8]{Pucci})
and its remark, since $W_\lambda$ satisfies $J'(W_\lambda)=o(1)$, we know
$\nabla W_\lambda\to\nabla W_0$ a. e. in $\mathbb{R}^N$. Then, we have as $\lambda\to0$,
$$
\Phi_\lambda(W_\lambda)=J(W_\lambda)+o(1)=J(W_0)+J(W_\lambda-W_0)+o(1).
$$
Thus $J(W_\lambda-W_0)\le o(1)$. Together with the Sobolev's embedding
$D^{1,p}(\mathbb{R}^N)\hookrightarrow L^{p^{\ast}}(\mathbb{R}^N)$,
by (A4)--(A6), we have
$$
\|W_\lambda-W_0\|^p/2p\le\|W_\lambda-W_0\|_{p^{\ast}}^{p^{\ast}}/p^{\ast}
\le S^{\frac{p}{p^{\ast}}}\|\nabla(W_\lambda-W_0)\|_{p}^{p^{\ast}}/p^{\ast}.
$$
If $\|W_\lambda-W_0\|\not\to0$ as $\lambda\to0$, then
$\|W_\lambda-W_0\|\ge(\frac{p^{\ast}}{2p}
s^{p^{\ast}/p})^{\frac{1}{p^{\ast}-p}}$. On the other hand,
$$
\|W_\lambda-W_0\|\le \|u_\lambda-u_0\|+\|v_\lambda\|
+\|v_0\|\le o(1)+2d
\le \frac{1}{2}(\frac{p^{\ast}}{2p}s^{p^{\ast}/p})^{\frac{1}{p^{\ast}-p}}.
$$
This is a contradiction. Thus, $W_\lambda\to W_0$ strongly in
$W_r^{1,p}(\mathbb{R}^N)$. The proof is completed.
\end{proof}

\begin{lemma} \label{lem3.4}
 Assume there exists $C_4>0$, for small $\lambda>0$ such that
$\Phi_\lambda(\gamma(s))\ge B_\lambda-C_4$. Then
$\gamma(s)\in W^{d/2}$, where $\gamma(s)=W(\cdot/s)$, $s\in(0,t^*]$.
\end{lemma}

\begin{proof}
It follows from the Pohoz\v{a}ev's identity that for $s\in(0,t^*]$,
\[
\Phi_\lambda(\gamma(s))
=\Big(\frac{1}{p}s^{N-p}-\frac{N-p}{Np}s^N\Big)
\int_{\mathbb{R}^N}|\nabla W|^p+o(\lambda)=J(\gamma(s))+o(\lambda).
\]
Noting that $m=\max_{s\in(0,t^*]}J(\gamma(s))=J(\gamma(1))$, for $C_5>0$ small,
$\gamma(s)=W(\cdot/s)\in W^{d/2}$ for $|s-1|\le C_5$. Since
$B_\lambda\to m$ as $\lambda\to 0$, there exists $C_4>0$, for
$\lambda>0$ small enough, such that $\Phi_\lambda(\gamma(s))\ge B_\lambda-C_4$.
 Furthermore, $|s-1|\le C_5$ and $\gamma(s)\in W^{d/2}$.
\end{proof}

Next, we use the local deformation argument to get a bounded (PS) sequence.

\begin{lemma} \label{lem3.5}
 For $\lambda>0$ small, there exists a sequence 
$\{u_n\}\subset \Phi_\lambda^{A_\lambda}\cap W^d$
with $\lim_{n\to\infty}\Phi_\lambda'(u_n)\to0$.
\end{lemma}

\begin{proof}
Assume by contradiction, there is $\beta(\lambda)>0$ such that
$|\Phi_\lambda'(u)|\ge \beta(\lambda)$, $u\in\Phi_\lambda^{A_\lambda}\cap W^d$
for some small $\lambda>0$. Similar arguments in \cite{Willem} show that there
exists a pseudo-gradient vector field $\Psi_\lambda$ in
 $W_r^{1,p}(\mathbb{R}^N)$ on a neighborhood $D_\lambda$ of
$\Phi_\lambda^{A_\lambda}\cap W^d$ such that
$\|\Phi_\lambda(u)\|\le 2\min\{1,|\Phi_\lambda'(u)|\}$
and
$\langle\Phi_\lambda'(u),\Psi_\lambda(u)\rangle
\ge \min\{1,|\Phi_\lambda'(u)|\}|\Phi_\lambda'(u)|$.
Denote $\delta_\lambda$ be a Lipschitz continuous function on
$W_r^{1,p}(\mathbb{R}^N)$ such that $\delta_\lambda\in[0,1]$ and
\[
\delta_\lambda(u)
=\begin{cases}
1, & u\in\Phi_\lambda^{A_\lambda}\cap W^d\\
0, & u\in W_r^{1,p}(\mathbb{R}^N)\backslash D_\lambda.
\end{cases}
\]
Define $\xi_\lambda$ be a Lipschitz continuous function on $\mathbb{R}$
such that $\xi_\lambda\in[0,1]$ and
\[
\xi_\lambda(t)=\begin{cases}
1,& |t-B_\lambda|\le C_4/2,\\
0,& |t-B_\lambda|\ge C_4,
\end{cases}
\]
where $C_4$ is given in Lemma \ref{lem3.4}. Set
\[
{E_\lambda(u)}
=\begin{cases}
-\delta_\lambda(u)\xi_\lambda(\Phi_\lambda(u))\Psi_\lambda(u),& u\in D_\lambda,\\
0,& u\in W_r^{1,p}(\mathbb{R}^N)\backslash D_\lambda.
\end{cases}
\]
Then, the initial-value problem
\begin{gather*}
\frac{d}{dt}Y_\lambda(u,t)=E_\lambda(Y_\lambda(u,t)),\\
Y_\lambda(u,0)=u,
\end{gather*}
admits a unique global solution
$Y_\lambda:W_r^{1,p}(\mathbb{R}^N)\times\mathbb{R}_+\to W_r^{1,p}(\mathbb{R}^N)$
which satisfies
\begin{itemize}
\item[(i)] $Y_\lambda(u,t)=u$, if $t=0$ or $u\in D_\lambda$ or
 $|\Phi_\lambda(u)-B_\lambda|\ge C_4$;

\item[(ii)] $\|\frac{d}{dt}Y_\lambda(u,t)\|\le2$, for
 $(u,t)\in W_r^{1,p}(\mathbb{R}^N)\times\mathbb{R}_+$;

\item[(iii)] $\frac{d}{dt}\Phi_\lambda(Y_\lambda(u,t))\le0$.
\end{itemize}
As in \cite{Chen3}, we get that for any $s\in(0,t^*]$, there is
$t_s>0$ such that
$Y_\lambda(\gamma(s),t_s)\in \Phi_\lambda^{B_\lambda-C_4/2}$,\
where $\gamma(s)=W(\cdot/s)$, $s\in (0,t^*]$.
Let $\gamma_0(s)=Y_\lambda(\gamma(s),{t_*}(s))$, where
$t_*(s)=\inf\{t\ge0,\, Y_\lambda(\gamma(s),t)\in \Phi_\lambda^{B_\lambda-C_4/2}\}$.
Then we can prove that $\gamma_0(s)$ is continuous in $[0,t^*]$ and
$\|\gamma_0(s)\|\le C_2+2$. Therefore, $\gamma_0\in\Gamma_\lambda$ with
$\max_{t\in[0,t^*]}\Phi_\lambda(\gamma_0(t))\le B_\lambda-C_4/2$,
which contradicts the definition of $B_\lambda$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
For fixed $d>0$ small which satisfies $d<S^{N/p^2}/3$, by Lemma \ref{lem3.5},
there exist $\lambda^\ast>0$ with $\lambda\in(0,\lambda^\ast)$ and
$\{u_n\}\subset\Phi_\lambda^{A_\lambda}\cap W^d$ such
that $\Phi_\lambda(u_n)\le A_\lambda, \Phi_\lambda'(u_n)\to0$ as $n\to\infty$.
We may assume $\lim_{n\to\infty}\|u_n\|^p:=\kappa
\le\left(d+\sup_{u\in W_r}\|u\|\right)^p$ and
$u_n\to u_\lambda$ weakly in $W_r^{1,p}(\mathbb{R}^N)$, then by \cite[Corollary 1.26]{Willem},
up to a subsequence, $u_n\to u_\lambda$ strongly in $L^t(\mathbb{R}^N), t\in(p,p^\ast)$
and a. e. in $\mathbb{R}^N$. Since $u_n\in W^d$, there exist $U_n\in W_r$ and
$w_n\in W_r^{1,p}(\mathbb{R}^N)$ such that $u_n=U_n+w_n$ and $\|w_n\|\le d$.
By Lemma \ref{lem2.1}, for some $U\in W_r$, $U_n\to U$ strongly in $W_r^{1,p}(\mathbb{R}^N)$.
Let $v_n=u_n-u_\lambda$, then $\|v_n\|\le 3d$ for $n$ large.
\smallskip

\noindent\textbf{Step 1.} For any $\delta>1$, up to a subsequence, it holds
$$
\int_{\mathbb{R}^N}f(u_n)u_n\le\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda+\delta\int_{\mathbb{R}^N}|v_n|^{p^\ast}+o_n(1).
$$
Obviously, there exists $s_0>1$ such that $f(s)\le\delta s^{p^\ast}$ for all
$s\ge s_0$. Choose $\chi(s)\in C(\mathbb{R})$ such that $\chi(s)=0$ if $s\le1$,
$\chi(s)=f(s)/s^{p^\ast}$ if $s\ge s_0$ and $\chi(s)\in[0,\delta]$ for any $s\in\mathbb{R}$.
Let $g(s)=f(s)-\chi(s)s^{p^\ast}, s\ge0$, then $\lim_{s\to0^+}g(s)/s^{p-1}\to0$
 and $\lim_{s\to+\infty}g(s)/s^{p^\ast}\to0$. It follows from the compactness lemma
of Strauss\cite{Strauss} that
$$
\int_{\mathbb{R}^N}g(u_n)u_n=\int_{\mathbb{R}^N}g(u_\lambda)u_\lambda+o_n(1).
$$
Meanwhile, similar to Brezis-Lieb Lemma \cite[Lemma 1.32]{Willem}, we have
$$
\int_{\mathbb{R}^N}\chi(u_n)|u_n|^{p^\ast}=\int_{\mathbb{R}^N}\chi(u_n)|v_n|^{p^\ast}
+\int_{\mathbb{R}^N}\chi(u_\lambda)|u_\lambda|^{p^\ast}+o_n(1).
$$
Therefore,
\begin{align*}
\int_{\mathbb{R}^N}f(u_n)u_n
&=\int_{\mathbb{R}^N}g(u_n)u_n+\int_{\mathbb{R}^N}\chi(u_n)|u_n|^{p^\ast}\\
&=\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda+\int_{\mathbb{R}^N}\chi(u_n)|v_n|^{p^\ast}+o_n(1)\\
&\le\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda+\delta\int_{\mathbb{R}^N}|v_n|^{p^\ast}+o_n(1).
\end{align*}
\smallskip

\noindent\textbf{Step 2.} We show that $u_n\to u_\lambda$ strongly in $D^{1,p}(\mathbb{R}^N)$
 as $n\to\infty$. In fact, $u_\lambda$ satisfies
$$
(1+\lambda\kappa)(-\Delta_p u+b |u|^{p-2}u)=f(u),\,\,\,u\in W^{1,p}(\mathbb{R}^N).
$$
Similar as that in Lemma \ref{lem3.3}, $\nabla u_n\to\nabla u_\lambda$ a. e. in $\mathbb{R}^N$ as $n\to\infty$.
It follows from the Brezis-Lieb Lemma that
$$
\|u_n\|^p=\|v_n\|^p+\|u_\lambda\|^p+o(1).
$$
By Step 1 and $\langle\Phi_\lambda'(u_n),u_n\rangle\to0$, we have
$$
(1+\lambda\kappa)(\|v_n\|^p+\|u_\lambda\|^p)
\le\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda+\delta\int_{\mathbb{R}^N}|v_n|^{p^\ast}+o_n(1).
$$
Since
$$
(1+\lambda\kappa)\|u_\lambda\|^p=\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda,
$$
we have
$$
(1+\lambda\kappa)\|v_n\|^p\le\delta\int_{\mathbb{R}^N}|v_n|^{p^\ast}+o_n(1).
$$
If $\|\nabla v_n\|_p\not\to0$ as $n\to\infty$, then it follows from Sobolev's
 embedding that
$$
\|\nabla v_n\|_p^p\le\delta\int_{\mathbb{R}^N}|v_n|^{p^\ast}+o_n(1)
\le \delta S^{-p^\ast/p}\|\nabla v_n\|_p^{p^\ast}+o_n(1),
$$
which implies
$$
\liminf_{n\to\infty}\|\nabla v_n\|_p\ge\left(\delta^{-1}S^{p^\ast/p}\right)^{1/(p^\ast-p)}.
$$
Then
$$
\liminf_{n\to\infty}\|\nabla v_n\|_p\ge S^{N/p^2},
$$
which is impossible since $d<S^{N/p^2}/3$. Thus, $\|\nabla v_n\|_p\to0$ as $n\to\infty$.
\smallskip

\noindent\textbf{Step 3.} $u_n\to u_\lambda$ strongly in $W^{1,p}(\mathbb{R}^N)$.
In fact, by Step 2, we have
$$
(1+\lambda\kappa)(-\Delta_p u_\lambda+b |u_\lambda|^{p-2}u_\lambda)=f(u_\lambda),\quad
u_\lambda\in W^{1,p}(\mathbb{R}^N).
$$
By Step 1, $f(u_n)u_n\to f(u_\lambda)u_\lambda$ strongly in $L^1(\mathbb{R}^N)$.
Thus, by $\langle\Phi_\lambda'(u_n),u_n\rangle\to0$, we get
\begin{align*}
(1+\lambda\kappa)\|u_n\|^p
&=\int_{\mathbb{R}^N}f(u_n)u_n+o_n(1)\\
&=\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda+o_n(1)\\
&=(1+\lambda\kappa)\|u_\lambda\|^p+o_n(1).
\end{align*}
So, $\|u_n\|\to\|u_\lambda\|$ as $n\to\infty$. Therefore, $u_n\to u_\lambda$ strongly
in $W^{1,p}(\mathbb{R}^N)$, which implies that $\Phi_\lambda'(u_\lambda)=0$ and
$u_\lambda\in\Phi_\lambda^{A_\lambda}\cap W^d$. For $d$ small enough, $u_\lambda\not=0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 For $\lambda>0$ small enough, problem \eqref{eq1.1} admits a positive solution
 $u_\lambda$ with $u_\lambda\in\Phi_\lambda^{A_\lambda}\cap W^d$. That is,
$u_\lambda\in W^d$ and
$\Phi_\lambda'(u_\lambda)=0,\,\,\Phi_\lambda(u_\lambda)\le A_\lambda$.
Obviously, $\{u_\lambda\}$ is bounded in $W_r^{1,p}(\mathbb{R}^N)$. Up to a subsequence,
we assume that for some $u\in W_r^{1,p}(\mathbb{R}^N)$, $u_\lambda\to u$ weakly in
$W_r^{1,p}(\mathbb{R}^N)$, strongly in $L^t(\mathbb{R}^N)$ for $t\in(p,p^\ast)$ and a. e. in
$\mathbb{R}^N$ as $\lambda\to0$. Similar to Theorem \ref{thm1.1}, for $d<S^{N/p^2}/3$ given and small,
$u\not\equiv0$ and $u_\lambda\to u$ strongly in $W_r^{1,p}(\mathbb{R}^N)$ as $\lambda\to0$.
It implies that $u\in W^d$ and
$J'(u)=0,\,\,J(u)\le \lim_{\lambda\to0}A_\lambda=m$.
Since $m$ is the least energy of problem \eqref{eq1.5}, $ J(u)\ge m $.
It follows that $J(u)=m$, i. e. $u$ is a ground state solution of
problem \eqref{eq1.5}. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This research was funded by the Priority Academic Program Development of
Jiangsu Higher Education Institutions(PAPD), the National Key R\&D
Program of China (2016YFC0801402),
by the Jiangsu Planned Projects for Postdoctoral Research Funds (1402156C)
and by the Fundamental Research Funds for the Central Universities (2015XKMS072).


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\end{document}