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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 292, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/292\hfil $p$-Laplacian equations of Kirchhoff type]
{Positive solutions for $p$-Laplacian equations of Kirchhoff
type problem with a parameter}

\author[Q. Zhang, J. Huang \hfil EJDE-2017/292\hfilneg]
{Qi Zhang, Jianping Huang}

\address{Qi Zhang \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083, China}
\email{zq8910@csu.edu.cn}

\address{Jianping Huang (corresponding author) \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083, China}
\email{huangjianping@csu.edu.cn}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted September 11, 2017. Published November 27, 2017.}
\subjclass[2010]{35B30, 35B09, 35J50, 35J62}
\keywords{Positive solution; $p$-Laplacian equation; iterative technique}

\begin{abstract}
 In this article, we consider the existence and non-existence of positive
 solutions for the Kirchhoff type equation
 \begin{gather*}
 -\Big(a+\lambda M \Big(\int_{\Omega}|\nabla u|^{p}dx\Big)\Big)\Delta_pu= f(u),
 \quad \text{in } \Omega,\\
 u=0, \quad \text{on } \partial\Omega,
 \end{gather*}
 where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with a smooth boundary
 $\partial\Omega$, $a$ is a positive constant, $N\geq3$, $\lambda\geq0$,
 $2\leq p<N$, $M$ and $f$ are positive continuous functions.
 Under some weak assumptions on $f$, we show that the above problem has
 at least one positive solution when $\lambda$ is small and has no nonzero
 solution when $\lambda$ is large. Our argument is based on iterative
 technique and 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 main results}

The well-known nonlinear Kiffchhoff type equation has attracted massive
attention as it stems from interesting physical problems, see \cite{r1,r7,r3}.
The pioneer research on Kirchhoff type problem belongs to Pohozaev \cite{r8}
and Bernstein \cite{r10}. But only after the work of Lions \cite{r9},
in which an abstract functional framework to the equation was set, the equation
received extensive attention.

 In this article, we are interested in the existence of positive solutions
for the nonlinear Kirchhoff equation
 \begin{equation}\label{a1}
\begin{gathered}
 -\Big(a+\lambda M\Big(\int_{\Omega}|\nabla u|^{p}dx\Big)\Big)\Delta_pu
 = f(u),\quad \text{in } \Omega,\\
 u=0, \quad \text{on } \partial\Omega,
 \end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$, $N\geq3$, $a>0$
is a positive constant, $\lambda\geq0$ is a parameter,
$\Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ with $2\leq p<N$,
and $M: \mathbb{R}_{+}\to \mathbb{R}_{+}$ with $ \mathbb{R}_{+}=[0, +\infty)$. Moreover,
the nonlinearity $f(t)$ satisfies the following basic assumptions:
\begin{itemize}
\item[(A1)] $f$ is Lipschitz continuous and
$\lim_{t\to 0^{+}}\frac{f(t)}{t^{p-1}}=0$;

\item[(A2)] $\lim_{t\to+\infty}\frac{f(t)}{t^{p^{\ast}-1}}=0$, where
$p^{\ast}=\frac{pN}{N-p}$;

\item[(A3)] $\lim_{t\to+\infty}\frac{f(t)}{t^{p-1}}=+\infty$.
\end{itemize}

Since we are only interested in positive solutions,
without loss of generality, we suppose that $f(t)\equiv0$ for $t<0$.

Problem \eqref{a1} has been widely researched in recent years, especially
on the existence of positive solutions, multiple solutions and sign-changing
solutions, see \cite{r24, r20, r25, r23, r13, r19}. For example, Ourraoui \cite{r19}
considered problem \eqref{a1} involving critical Sobolev exponent. They got
their results via the variational principle of Ekeland.
 Correa et al \cite{r20} also studied problem \eqref{a1}.
They established sufficient conditions on $M$ and the nonlinearity $f$ under
which \eqref{a1} possesses positive solutions. Later, based on the
fountain theorem, Huang et al in \cite{r13} proved the existence and multiplicity
of solutions of problem \eqref{a1} when the nonlinearity is concave-convex.

The generalization of problem \eqref{a1} to unbounded domain also attracted
 much attention. For some interesting results, we refer to
\cite{r16, r17, r18, r21, r22}. Chen, Song and Xiu \cite{r16} studied the
 following general case:
\begin{equation}\label{a2}
\begin{gathered}
\begin{aligned}
& M\Big(\int_{\mathbb{R}^{N}}(|\nabla u|^{p}+V(x)|u|^{p})dx\Big)
(-\Delta u+V(x)|u|^{p-2}u) \\
&=f(x, u)+g(x),\quad\text{in } \mathbb{R}^{N},
\end{aligned}\\
 u(x)\to0,\quad \text{as } |x|\to+\infty.
 \end{gathered}
\end{equation}
Under different assumptions on the nonlinear term $f(x, u)$,
 multiple solutions of problem \eqref{a2} was constructed by applying
the Mountain Pass Theorem, Ekeland's variational principle and Krasnoselskii's
genus theory in \cite{r15}. Cheng and Dai \cite{r21} also considered
a class of generalized form of problem \eqref{a1}.
They used a cut-off function to get the bounded Palais-Smale sequences
and proved the existence of a positive solution. In addition,
when $M(t)=t$, Chen and Zhu \cite{r17} utilized the Nehari manifold
method to study problem \eqref{a1}. They obtained that there exists at
least a positive ground state solution.

 In the special case of $p=2$, there are much more works than that of general $p$.
For example, Li et al \cite{r4} studied the existence of a positive solution
to the nonlinear Kirchhoff type problem
\begin{equation}\label{a3}
\Big(a+\lambda\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+\lambda\int_{\mathbb{R}^{N}}u^{2}dx\Big)
(-\Delta u+bu)=f(u),\quad \text{in } \mathbb{R}^{N}.
\end{equation}
Where $N\geq3$ and $a$, $b$ are positive constants. Under the condition
$\lim_{t\to+\infty}f(t)/t=+\infty$ and $\lambda$ is sufficient small,
a positive solution of problem \eqref{a3} was obtained by using a cut-off
function and monotonicity trick. But they didn't show whether there still
exists at least one positive solution when $\lambda$ is not so small.
 More recently, Liu, Liao and Tang \cite{r14} investigated problem \eqref{a3}
further. They considered two cases where the nonlinearity respectively
satisfied asymptotically linear and superlinear conditions at the infinity.
The most important is, they proved that if $\lambda$ is large enough
the problem \eqref{a3} has no nonzero solution. For more interesting results,
 we refer to \cite{r27, r30, r32, r28, r29,  r31} and the references therein.

In this article, motivated by the papers \cite{ r21, r14,r4}, we discuss
the existence and non-existence of positive solutions of problem \eqref{a1}.
We adopt the method in \cite{r12}, which studied the solution for semilinear
 elliptic equation. More precisely, first of all, we will use monotonicity
tricks introduced in \cite{r2, r6} and iterative technique to establish
the existence of positive solutions for equation \eqref{a1} whenever
$\lambda$ is sufficient small. Secondly, we will show that if $\lambda$
is large enough, equation \eqref{a1} has no nonzero solution.

To state our main results clearly, we firstly introduce some Sobolev spaces
and norms. Let $W^{1, p}_{0}(\Omega)$ be the usual Sobolev space.
We denote the usual norm of $L^{s}(\Omega)$ by $\|u\|_{s}$ for all
$p\leq s\leq p^{\ast}$. Define $H:=\{u\in W^{1, p}_{0}(\Omega): u(x)=u(|x|)\}$,
equipped with the norm
 $$
 \|u\|=\Big(\int_{\Omega}|\nabla u|^{p}dx\Big)^{1/p},\quad \forall u\in H.
 $$
It is clear that the embedding $H\hookrightarrow L^{s}(\Omega)$ for 
$p<s< p^{\ast}$ is compact and continuous for $p\leq s\leq p^{\ast}$, 
namely, there exists constants $\gamma_{s}>0$ such that 
$\|u\|_{s}\leq \gamma_{s}\|u\|$ for $p\leq s\leq p^{\ast}$. 
Here and in the sequel, $C_{i}$ denote positive constants, 
$i=1, 2, 3, \dots$.
The following theorem is the first main result in the paper.

\begin{theorem} \label{thm1.1}
Assume that $\Omega$ is convex and $\lambda\geq0$ is a parameter. 
If the conditions {\rm (A1)--(A3)} hold. Then for any positive continuous
function $M$, there exists $\lambda_{0}$ such that for any 
$\lambda\in[0, \lambda_{0})$, problem \eqref{a1} has at least one positive solution.
\end{theorem}

\begin{remark} \label{rmk1.2}\rm
 We note that for the special case $p=2$ and $\lambda=0$, the above result 
has been established in \cite{r26} and \cite{r11} respectively. 
Besides, Cheng and Dai \cite{r21} also obtained the result under the 
conditions (A1), (A3) and the assumption
\begin{itemize}
\item[(A4)] there exist constants $C>0$ and $q\in(p, p^{\ast})$ such that
$$
|f(t)|\leq C(|t|^{p-1}+|t|^{q-1}),\quad \forall t\in\mathbb{R}_{+}.
$$
\end{itemize}
\end{remark}

Evidently, the condition (A4) is stronger than our condition (A2).
 Thus, our result can be regarded as an extension of these papers mentioned above.

 Nevertheless, if the parameter $\lambda>0$ is big enough, and 
$\Omega$ is unbounded, in addition to the following assumptions:
 \begin{itemize}
\item[(A5)] there exists a $\tau>0$ such that $M(t)=t^{\tau}$ with 
$p(\tau+1)/\tau< N$;

\item[(A6)] $f\in C(\mathbb{R}, \mathbb{R})$ and $\lim_{t\to 0}\frac{f(t)}{t^{p-1}}=0$;

\item[(A7)] $\limsup_{|t|\to+\infty}\frac{|f(t)|}{t^{p^{*}-1}}<+\infty$;
\end{itemize}
we can obtain the following results:

 \begin{theorem} \label{thm1.3}
Assume that $\Omega=\mathbb{R}^{N}$ with $N\geq 3$ and $\lambda>0$ is a parameter.
If {\rm (A5)--(A7)} hold, then there exists $\Theta>0$ such that for 
any $\lambda>\Theta$, problem \eqref{a1} has no nontrivial solution.
\end{theorem}

 The main results are proved in the sections below. In Section 2, 
some preliminary concepts and results are presented. 
In Section 3, we prove Theorems \ref{thm1.1} and \ref{thm1.3}.

\section{Preliminary results}

Throughout this section, we suppose $T, S>0$ and $\varphi\in H$ with 
$\|\varphi\|\leq S$. For given $\varphi\in H$ and $\theta\in[\frac{1}{p}, 1]$, 
we study the energy functional $\Phi_{\varphi, \theta}:H\to\mathbb{R}$ define by
\begin{equation} \label{b1}
\Phi_{\varphi, \theta}(u)
=\frac{a}{p}\int_{\Omega}|\nabla u|^{p}dx
 +\frac{\lambda}{p}M(\|\varphi\|^{p})\int_{\Omega}|\nabla u|^{p}dx
 -\theta\int_{\Omega}F(u)dx
 \end{equation}
for all $u\in H$, where $F(u)=\int^{u}_{0}f(t)dt$. Obviously, the functional 
$\Phi_{\varphi, \theta}$ is well defined and 
$\Phi_{\varphi, \theta}\in C^{1}(H, \mathbb{R})$. Further, for any $u, v\in H$, we have
\begin{equation}\label{b2}
\langle \Phi'_{\varphi, \theta}(u),v\rangle 
=\left(a+\lambda M(\|\varphi\|^{p})\right)
\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\,dx -\theta\int_{\Omega}f(u)v\,dx.
\end{equation}
In the process of our argument, we will  use the following proposition.

\begin{proposition}[\cite{r2, r6}] \label{prop2.1}
 Let $(X, \|\cdot\|_{X})$ be a Banach space and $I\subset\mathbb{R}_{+}$ an interval. 
Consider the family of $C^{1}$ functionals on $X$
$$
J_{\mu}(u)=A(u)-\mu B(u),\quad \mu\in I,
$$
with $B$ nonnegative and either $A(u)\to+\infty$ or $B(u)\to+\infty$ as 
$\|u\|_{X}\to+\infty$ and such that $J_{\mu}(0)=0$.

For any $\mu\in I$, we set
$$
\Gamma_{\mu}=\{\gamma\in C([0, 1], X): \gamma(0)=0, J_{\mu}(\gamma(1))<0\}.
$$
If for every $\mu\in I$ the set $\Gamma_{\mu}$ is nonempty and 
$c_{\mu}=\inf_{\gamma\in\Gamma_{\mu}}\max_{t\in [0, 1]}J_{\mu}(\gamma(t))>0$,
then for almost every $\mu\in I$ there is a sequence $\{u_{n}\}\subset X$ such that
\begin{itemize}
\item[(1)] $\{u_{n}\}$ is bounded;

\item[(2)] $J_{\mu}(u_{n})\to c_{\mu}$;

\item[(3)] $J'_{\mu}(u_{n})\to 0$ in the dual $X^{-1}$ of $X$.
\end{itemize}
\end{proposition}

To apply Proposition \ref{prop2.1}, in our case, we let
$$
A_{\varphi}(u)=\frac{a}{p}\|u\|^{p}+\frac{\lambda}{p}M(\|\varphi\|^{p})\|u\|^{p},\quad
B(u)=\int_{\Omega}F(u)dx.
$$

The following Poho\v{z}aev equality is crucial to the proof of the boundedness 
of the Palais-Smale sequence.

\begin{lemma} \label{lem2.2} 
If $u\in H$ is a critical point of $\Phi_{\varphi, \theta}$, namely, $u$ 
is a week solution of
 \begin{equation}\label{b3}
\begin{gathered}
 -\left(a+\lambda M(\|\varphi\|^{p})\right)\Delta_pu
 = \theta f(u),\quad \text{in } \Omega,\\
 u=0, \quad \text{on } \partial\Omega,
 \end{gathered}
 \end{equation}
 then the following Poho\v{z}aev type identity holds
\begin{equation}\label{b4}
\begin{aligned}
&[a+\lambda M(\|\varphi\|^{p})]\Big[\big(\frac{N}{p}-1\big)\|u\|^{p}
+\big(1-\frac{1}{p}\big)\int_{\partial\Omega}|\nabla u|^{p}(x\cdot \nu)d\sigma\Big]\\
&=\theta N\int_{\Omega}F(u)dx.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Because $u\in H$ is a week solution of \eqref{b3}, by the standard regularity 
results, we get that $u\in W^{2, p}_{0}(\Omega)\cap W^{1, p}_{0}(\overline{\Omega})$.
 Setting
$$
g(u)=\frac{\theta f(u)}{a+\lambda M(\|\varphi\|^{p})}.
$$
Then, it is obvious that $u\in H$ is also a solution of $-\Delta _pu=g(u)$.
Applying the Poho\v{z}aev identity in \cite{r5}, we have
 \begin{equation}\label{b5}
 \big(1-\frac{1}{p}\big)\int_{\partial\Omega}|\nabla u|^{p}(x\cdot\nu)d\sigma
=\big(1-\frac{N}{p}\big)\int_{\Omega}g(u)udx+N\int_{\Omega}G(u)dx,
 \end{equation}
where $G(t)=\int^{t}_{0}g(s)ds$,
we obtain the conclusion.
\end{proof}

 \begin{lemma} \label{lem2.3}
 If {\rm (A3)} holds, then there exist $\lambda_{0}=\lambda_{0}(T, S)>0$ 
and $u_{0}\in H$, such that
 $\Phi_{\varphi, \theta}(u_{0})<0$ for every $\lambda\in[0, \lambda_{0})$.
\end{lemma}

\begin{proof}
For given $T>0$, there exists a constant $\lambda_{0}=\lambda_{0}(T, S)$, 
such that
 \begin{equation}\label{b6}
\lambda\max_{\tau\in[0, S^{p}]}M(\tau)\leq T
 \end{equation}
 whenever $\lambda\in[0, \lambda_{0})$. Choose $\phi\in H$
with $\phi\geq0$ and $\|\phi\|=1$. In view of (A3), we have that for 
any $C_1>0$ with $C_1>(a+T)/\int_{\Omega}|\phi|^{p}dx$,
 there exists $C_2>0$ such that
\begin{equation}\label{b7}
F(t)\geq C_1|t|^{p}-C_2, \quad t\in \mathbb{R}_{+}.
 \end{equation}
Thus, for any $\lambda\in[0, \lambda_{0})$, we have
\begin{equation}\label{b8}
\begin{aligned}
 \Phi_{\varphi, \theta}(t\phi)
&=\frac{1}{p}(a+\lambda M(\| \varphi\|^{p}))
 \int_{\Omega}|\nabla (t\phi)|^{p}dx-\theta\int_{\Omega}F(t\phi)dx \\
& \leq  \frac{t^{p}}{p}(a+T)-\theta\int_{\Omega}F(t\phi)dx \\
&\leq  \frac{t^{p}}{p}\Big(a+T-C_1
\int_{\Omega}|\phi|^{p}dx\Big)+\frac{C_2|\Omega|}{p}.
\end{aligned}
\end{equation}
Hence, we can choose $t>0$ large enough such that 
$\Phi_{\varphi, \theta}(t\phi)<0$, the proof is completed.
\end{proof}

 \begin{lemma} \label{lem2.4}
 Under assumptions {\rm (A1)} and {\rm (A2)}, there exists positive constants
 $\alpha, \beta$ such that
 \begin{equation}\label{b9}
 \Phi_{\varphi, \theta}(u)\geq\alpha, \quad \forall u\in H,\quad \|u\|\leq\beta.
\end{equation}
\end{lemma}

 \begin{proof}
 Using (A1) and (A2), for $\varepsilon\in(0, 1/(2\gamma^{p}_p))$,
there exists a constants $C_3(\varepsilon)>0$ such that
 \begin{equation}\label{b10}
 F(t)\leq\frac{\varepsilon}{p}t^{p}+C_3(\varepsilon)t^{p^{\ast}}, \quad t\in\mathbb{R}_{+}.
 \end{equation}
 Furthermore, for $u\in H$, by the Sobolev embedding, 
 \begin{equation}\label{b11}
\begin{aligned}
 \Phi_{\varphi, \theta}(u)
&=\frac{1}{p}(a+\lambda M(\|\varphi\|^{p}))\|u\|^{p}-\theta\int_{\Omega}F(u)dx \\
&\geq  \frac{a}{p}\|u\|^{p}-\int_{\Omega}(\frac{\varepsilon}{p}|u|^{p}
 +C_3(\varepsilon)|u|^{p^{\ast}})dx  \\
&\geq  \frac{a}{2p}\|u\|^{p}-C_3(\varepsilon)\gamma^{p^{\ast}}_{p^{\ast}}
\|u\|^{p^{\ast}}.
\end{aligned}
\end{equation}
Hence, choosing 
\[
\beta:=\|u\| =\Big(\frac{1}{2p^{\ast}C_3(\varepsilon)\gamma^{p^{\ast}}_{p^{\ast}}}
\Big)^{\frac{1}{p^{\ast} -p}},
\]
 one has $\Phi_{\varphi, \theta}(u)\geq\alpha$, where 
$\alpha=\beta^{p}(\frac{a}{2p}-\frac{1}{2p^{\ast}})$, 
which is independent of $\varphi$ and $\theta$.
 \end{proof}
 
 \begin{lemma} \label{lem2.5}
 If {\rm (A1)--(A3)} hold, then there exist $\lambda_{0}=\lambda_{0}(T, S)>0$ 
and a sequence $\{\theta_{k}\}\subset I$ satisfying $\theta_{k}\to 1$ 
(as $k\to+\infty$), such that $\Phi_{\varphi, \theta_{k}}$ has a nontrivial 
critical point $u_{\varphi, \theta_{k}}$ for $\lambda\in[0, \lambda_{0})$.
\end{lemma}

\begin{proof}
Set $I=[\frac{1}{p}, 1]$, from Proposition \ref{prop2.1}, there is 
$\{\theta_{k}\}\subset I$ with $\theta_{k}\to 1$ as $k\to+\infty$, 
and corresponding sequence $\{u_{n, \varphi, \theta_{k}}\}\subset H$ such that
\begin{gather*}
\{u_{n, \varphi, \theta_{k}}\} \text{ is bounded and } 
\Phi_{\varphi, \theta_{k}}(u_{n, \varphi, \theta_{k}})\to c_{\varphi, \theta_{k}}; \\
\Phi'_{\varphi, \theta_{k}}(u_{n, \varphi, \theta_{k}})\to 0\quad \text{in} \ H^{-1};
\end{gather*}
where $c_{\varphi, \theta_{k}}=\inf_{\gamma\in 
\Gamma_{\varphi, \theta_{k}}}\sup_{u\in\gamma([0, 1])}\Phi_{\varphi, \theta_{k}}(u)$ 
and
\[
\Gamma_{\varphi, \theta_{k}}=\{\gamma\in C([0, 1], H)|\gamma(0)=0, 
\Phi_{\varphi, \theta_{k}}(\gamma(1))<0\}.
\]
Up to a subsequence, we can assume that there exists $u_{\varphi, \theta_{k}}$ 
in $H$ such that
\begin{equation}\label{b12}
 \begin{gathered}
 u_{n, \varphi, \theta_{k}}\rightharpoonup u_{\varphi, \theta_{k}}, \quad
  \text{in } H;\\
 u_{n, \varphi, \theta_{k}}\to u_{\varphi, \theta_{k}},\quad
 \text{on } L^{s}(\Omega), \; \forall s\in(p, p^{\ast});\\
 u_{n, \varphi, \theta_{k}}\to u_{\varphi, \theta_{k}},\quad \text{a.e. on }  \Omega.
 \end{gathered}
 \end{equation}
From (A1) and (A2), for any $0<\varepsilon<\frac{1}{p}$, there 
exists $C_{\varepsilon}>0$ such that
 \begin{equation}\label{b13}
 |f(t)|\leq\varepsilon|t|^{p-1}+\varepsilon|t|^{p^{\ast}-1}
+C_{\varepsilon}|t|^{k_{0}-1},\quad k_{0}\in (p, p^{\ast}).
 \end{equation}
Then, from H\"{o}lder's inequality, we have
\begin{align*}
 & \big|\int_{\Omega}f(u_{n, \varphi, \theta_{k}})(u_{n, \varphi, 
\theta_{k}}-u_{\varphi, \theta_{k}})dx\big| \\
 &\leq \int_{\Omega}|f(u_{n, \varphi, \theta_{k}})||u_{n, \varphi, \theta_{k}}
-u_{\varphi, \theta_{k}}|dx \\
 &\leq \varepsilon\|u_{n, \varphi, \theta_{k}}\|^{p-1}_p
 \|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|_p
 +\varepsilon\|u_{n, \varphi, \theta_{k}}\|^{p^{\ast}-1}_{p^{\ast}}
\|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|_{p^{\ast}} \\
 &\quad +C_{\varepsilon}\|u_{n, \varphi, \theta_{k}}\|^{k_{0}-1}_{k_{0}}
 \|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|_{k_{0}} \\
 &\leq  \varepsilon\gamma^{p}_p\|u_{n, \varphi, \theta_{k}}\|^{p-1}
\|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|
 + \varepsilon\gamma^{p^{\ast}}_{p^{\ast}}\|u_{n, \varphi, \theta_{k}}\|^{p^{\ast}-1}\|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\| \\
 &\quad +C_{\varepsilon}\gamma^{k_{0}-1}_{k_{0}}\|u_{n, \varphi, \theta_{k}}\|^{k_{0}-1}
 \|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|_{k_{0}},
 \end{align*}
which implies that
\begin{equation}\label{b14}
\int_{\Omega}f(u_{n, \varphi, \theta_{k}})(u_{n, \varphi, \theta_{k}}
-u_{\varphi, \theta_{k}})dx\to 0.
\end{equation}
Similar to the argument above, we can also conclude that
\begin{gather*}
\langle \Phi'_{\varphi, \theta_{k}}(u_{\varphi, \theta_{k}}), 
u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\rangle \to 0, \\
 \int_{\Omega}f(u_{\varphi, \theta_{k}})(u_{n, \varphi, \theta_{k}}-u_{\varphi, 
\theta_{k}})dx\to 0 ,\quad \text{as } n\to+\infty.
\end{gather*}
From this and \eqref{b14}, one has
\begin{align*}
&\langle \Phi'_{\varphi, \theta_{k}}(u_{n, \varphi, \theta_{k}})
-\Phi'_{\varphi, \theta_{k}}(u_{\varphi, \theta_{k}}), u_{n, \varphi, \theta_{k}}
-u_{\varphi, \theta_{k}} \rangle \\
&= a\int_{\Omega}(|\nabla u_{n, \varphi, \theta_{k}}|^{p-2}\nabla u_{n, \varphi, 
 \theta_{k}}-|\nabla u_{\varphi, \theta_{k}}|^{p-2}\nabla u_{\varphi, 
 \theta_{k}})\cdot \nabla(u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}})dx \\
&\quad +\lambda M(\|\varphi\|^{p})\int_{\Omega}\Big(|\nabla u_{n, \varphi, 
 \theta_{k}}|^{p-2}\nabla u_{n, \varphi, \theta_{k}}-|\nabla u_{\varphi, 
 \theta_{k}}|^{p-2}\nabla u_{\varphi, \theta_{k}}\Big)\\
&\quad \cdot   \nabla(u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}})dx \\
&\quad +\theta_{k}\int_{\Omega}[f(u_{n, \varphi, \theta_{k}})
 -f(u_{\varphi, \theta_{k}})](u_{n, \varphi, \theta_{k}}
 -u_{\varphi, \theta_{k}})dx 
\to 0.
\end{align*}
Combining this with the standard inequality in $\mathbb{R}^{N}$ given by
\begin{equation}\label{b15}
(|\zeta|^{p-2}\zeta-|\eta|^{p-2}\eta, \zeta-\eta)
\geq \begin{cases}
 C_p|\zeta-\eta|^{p},  &p\in[2,+\infty),\\
 C_p|\zeta-\eta|^{2}(|\zeta|+|\eta|)^{p-2}, &1<p<2.
 \end{cases}
\end{equation}
We have that $\|u_{n, \varphi, \theta_{k}}-u_{\varphi, \theta_{k}}\|\to 0$, 
that is, $u_{n, \varphi, \theta_{k}}\to u_{\varphi, \theta_{k}}$ in $H$.

It follows from the above discussion that there exist 
$\lambda_{0}=\lambda_{0}(T, S)>0$ and a sequences $\{\theta_{k}\}$ with 
$\theta_{k}\to 1$ such that
$$
\Phi_{\varphi, \theta_{k}}(u_{\varphi, \theta_{k}})
=c_{\varphi, \theta_{k}}\quad \text{and}\quad
 \langle\Phi'_{\varphi, \theta_{k}}(u_{\varphi, \theta_{k}}), u_{\varphi, 
\theta_{k}}\rangle=0,
$$
if $\lambda\in [0,\lambda_{0})$. The proof is complete.
\end{proof}

 \begin{lemma} \label{lem2.6}
 Let $u_{\varphi, \theta_{k}}$ be a critical point of $\Phi_{\varphi, \theta_{k}}$ 
at level $c_{\varphi, \theta_{k}}$. Then for $S>0$ sufficiently large, 
there exists $\lambda_{0}=\lambda_{0}(T, S)$ such that for any 
$\lambda\in[0, \lambda_{0})$, subject to a subsequence, 
$\|u_{\varphi, \theta_{k}}\|\leq S$ for all $k\in \mathbb{N}$.
\end{lemma}

\begin{proof}
On the one hand, since $u_{\varphi, \theta_{k}}$ be a critical point of 
$\Phi_{\varphi, \theta_{k}}$, then from \eqref{b4}, $u_{\varphi, \theta_{k}}$ 
satisfies the following Poho\v{z}aev identity
\begin{equation}\label{b16}
\begin{aligned}
&[a+\lambda M(\|\varphi\|^{p})]\Big[\big(\frac{N}{p}-1\big)
\|u_{\varphi, \theta_{k}}\|^{p}+\big(1-\frac{1}{p}\big)
\int_{\partial\Omega}|\nabla u_{\varphi, \theta_{k}}|^{p}(x\cdot \nu)d\sigma\Big] \\
&=\theta_{k}N\int_{\Omega}F(u_{\varphi, \theta_{k}})dx.
\end{aligned}
\end{equation}
We assume $\mu_1$ is an eigenvalue of the operator $-\Delta _p$, and let 
$\phi_1>0$, $x\in \Omega$ be an eigenfunction corresponding to $\mu_1$, 
in view of (A3), we have that for any $\kappa>0$ with $\kappa>2\mu_1 (a+T)$, 
there exists $C_{4}(\kappa)>0$ such that
\begin{equation}\label{b17}
\begin{aligned}
\mu_1(a+\lambda M(\|\varphi\|^{p}))\int_{\Omega}u^{p-1}_{\varphi, 
\theta_{k}}\phi_1dx
&=\theta_{k}\int_{\Omega}f(u_{\varphi, \theta_{k}})\phi_1dx \\
&\geq \kappa\int_{\Omega}u^{p-1}_{\varphi, \theta_{k}}\phi_1dx-C_{4}(\kappa)
\end{aligned}
\end{equation}
 and $\int_{\Omega}u^{p-1}_{\varphi, \theta_{k}}\phi_1dx\leq C_5(T)$
for a constant $C_5(T)>0$. Combining this with the results in \cite{r11}, 
there is a constant $C_6(T)>0$ such that 
$|\nabla u_{\varphi, \theta_{k}}|^{p}\leq C_6(T)$, $x\in \partial\Omega$. 
Thus, by \eqref{b16}, there exists a constant $C_7(T)>0$ such that
\begin{equation}\label{b18}
\begin{aligned}
&\big(\frac{N}{p}-1\big)\|u_{\varphi, \theta_{k}}\|^{p}
-\frac{\theta_{k} N}{a+\lambda M(\|\varphi\|^{p})}\int_{\Omega}
F(u_{\varphi, \theta_{k}})dx\\
&=-\big(1-\frac{1}{p}\big)\int_{\partial\Omega}|\nabla 
u_{\varphi, \theta_{k}}|^{p}(x\cdot \nu)d\sigma \\
&\geq -C_7(T).
\end{aligned}
\end{equation}

On the other hand, from Lemmas \ref{lem2.3} and \ref{lem2.5}, there is a constant 
$C_8(T)>0$ such that
\begin{equation}\label{b19}
\begin{aligned}
c_{\varphi, \theta_{k}}
&=\Phi_{\varphi, \theta_{k}}(u_{\varphi, \theta_{k}})\\
&\leq \max_{t\geq0}\Phi_{\varphi, \theta_{k}}(t\phi) \\
&\leq \max_{t\geq0} \Big\{\frac{t^{p}}{p}(a+T)
 -\frac{1}{p}\int_{\Omega}F(t\phi)dx\Big\} 
\leq C_8(T).
\end{aligned}
 \end{equation}
So, one has
\begin{equation}\label{b20}
\frac{1}{p}\|u_{\varphi, \theta_{k}}\|^{p}
-\frac{\theta_{k}}{a+\lambda M(\|\varphi\|^{p})}\int_{\Omega}
F(u_{\varphi, \theta_{k}})dx=\frac{c_{\varphi, \theta_{k}}}
{a+\lambda M(\|\varphi\|^{p})}\leq C_8(T).
\end{equation}
It follows from \eqref{b18} and \eqref{b20} that 
$\|u_{\varphi, \theta_{k}}\|^{p}\leq N C_8(T)+C_7(T)$.
Consequently, for given $T>0$, if we take $S=( NC_8(T)+C_7(T))^{1/p}$, then 
$\|u_{\varphi, \theta_{k}}\|\leq S$.
\end{proof}

 From Lemma \ref{lem2.6}, for any $k$, if $\varphi=\varphi_{0}\equiv0$, then we
 know that $\Phi_{\varphi_{0}, \theta_{k}}$ has a nontrivial critical 
point and we denote it by $u_{1, k}$ with $\|u_{1, k}\|\leq S$. 
Let $\varphi=u_{1, k}$, then $\Phi_{u_{1, k}, \theta_{k}}$ has a 
nontrivial critical point $u_{2, k}$ with $\|u_{2, k}\|\leq S$. 
Therefore, by induction, we can obtain a sequence $u_{m, k}$ with 
$\|u_{m, k}\|\leq S$, $m=1, 2, \dots$.

\section{Proof of the main results}
 
\begin{proof}[Proof of Theorem \ref{thm1.1}]
In view of $u_{n, k}$ with $\|u_{n, k}\|\leq S$, for all $n, k\in \mathbb{N}$.
 For fixed $k$, up to a subsequence, we assume that $u_{n, k}\rightharpoonup u_{k}$ 
in $H$, $u_{n, k}\to u_{k}$ on $L^{s}(\Omega)$ for all $s\in(p, p^{\ast})$ and 
$u_{n, k}(x)\to u_{k}(x)$ a.e. in $\Omega$, we also have $\|u_{k}\|\leq S$. 
Then, one has
\begin{equation}\label{c1}
\begin{aligned}
&\langle \Phi'_{u_{n-1, k},\theta_{k}}(u_{k}), u_{n, k}
 -u_{k}\rangle \\
&=\left(a+\lambda M(\|u_{n-1, k}\|^{p})\right)
 \int_{\Omega}|\nabla u_{k}|^{p-2}\nabla u_{k}\cdot\nabla( u_{n, k}-u_{k})dx \\ 
&\quad -\theta_{k}\int_{\Omega}f(u_{k})( u_{n, k}-u_{k})dx 
\to 0, \quad \text{as } n\to+\infty,
\end{aligned}
\end{equation}
and
\begin{equation}\label{c2}
\begin{aligned}
&\langle \Phi'_{u_{n-1, k},\theta_{k}}(u_{n, k})-\Phi'_{u_{n-1, k},
\theta_{k}}(u_{k}), u_{n, k}-u_{k}\rangle \\
&=\left(a+\lambda M(\|u_{n-1, k}\|^{p})\right)\int_{\Omega}
 \Big(|\nabla u_{n, k}|^{p-2}\nabla u_{n, k}-|\nabla u_{k}|^{p-2}\nabla u_{k}\Big)\\
&\quad \cdot\nabla( u_{n, k}-u_{k})dx 
-\theta_{k}\int_{\Omega}[f(u_{n, k})-f(u_{k})](u_{n, k}-u_{k})dx \\
&\to 0, \quad \text{as } n\to+\infty.
\end{aligned}
 \end{equation}
It follows from \eqref{c2} and \eqref{b15} that $u_{n, k}\to u_{k}$ as 
$n\to+\infty$. And for every $v\in H$, one has
\begin{equation}\label{c3}
\begin{aligned}
 0&=\lim_{n\to+\infty}\langle \Phi'_{u_{n-1, k},\theta_{k}}(u_{n, k}), v\rangle \\
 &=\lim_{n\to+\infty}\Big[\left(a+\lambda M(\|u_{n-1, k}\|^{p})\right)
 \int_{\Omega}|\nabla u_{n, k}|^{p-2}\nabla u_{n, k}\cdot\nabla vdx \\
&\quad -\theta_{k}\int_{\Omega}f(u_{n, k})vdx\Big] \\
&= \left(a+\lambda M(\|u_{k}\|^{p})\right)\int_{\Omega}|\nabla u_{k}|^{p-2}
 \nabla u_{k}\cdot\nabla vdx-\theta_{k}\int_{\Omega}f(u_{k})vdx \\
&=\langle \Phi'_{u_{k},\theta_{k}}(u_{k}), v\rangle
\end{aligned}
\end{equation}
and
\begin{equation}\label{c4}
\begin{aligned}
 \Phi_{u_{k},\theta_{k}}(u_{k})
&=\frac{1}{p}(a+\lambda M(\|u_{k}\|^{p}))\|u_{k}\|^{p}
 -\theta_{k}\int_{\Omega}F(u_{k})dx \\
&=\lim_{n\to+\infty}\Big[\frac{1}{p}(a+\lambda M(\|u_{n-1, k}\|^{p}))
 \|u_{n, k}\|^{p}-\theta_{k}\int_{\Omega}F(u_{n, k})dx\Big] \\
&=\lim_{n\to+\infty}\Phi_{u_{n-1, k},\theta_{k}}(u_{n, k}).
\end{aligned}
 \end{equation}
From Lemma \ref{lem2.4}, we have
$\Phi_{u_{n-1, k},\theta_{k}}(u_{n, k})=c_{u_{n-1, k},\theta_{k}}\geq\alpha$. 
Hence $\Phi'_{u_{k},\theta_{k}}(u_{k})=0$, and 
$\Phi_{u_{k},\theta_{k}}(u_{k})\geq\alpha$ follows directly from \eqref{c3} 
and \eqref{c4}.

 On the other side, because of $\|u_{k}\|\leq S$, $k\in \mathbb{N}$, without lose 
of generality, we may suppose that $u_{k}\rightharpoonup u$ in $H$, $u_{k}\to u$ 
on $L^{s}(\Omega)$ for all $s\in(p, p^{\ast})$ and $u_{k}(x)\to u(x)$ a.e. in 
$\Omega$. Together with the boundedness of $\lambda M(\|u_{k}\|^{p})$, we get
\begin{equation}\label{c5}
\begin{aligned}
&\langle \Phi'_{u_{k},\theta_{k}}(u), u_{k}-u\rangle \\
&=\left(a+\lambda M(\|u_{k}\|^{p})\right)
 \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla (u_{k}-u)
 -\theta_{k}\int_{\Omega}f(u)(u_{k}-u)dx \\
&\to 0, \quad \text{as } n\to+\infty,
\end{aligned}
 \end{equation}
and
\begin{equation}\label{c6}
\begin{aligned}
& \langle \Phi'_{u_{k},\theta_{k}}(u_{k})- \Phi'_{u_{k},\theta_{k}}(u), u_{k}
-u\rangle \\
&=\left(a+\lambda M(\|u_{k}\|^{p})\right)\int_{\Omega}
 (|\nabla u_{k}|^{p-2}\nabla u_{k}-|\nabla u|^{p-2}\nabla u)
 \cdot\nabla( u_{k}-u)dx \\
&\quad - \theta_{k}\int_{\Omega}[f(u_{k})-f(u)](u_{k}-u)dx.
\end{aligned}
 \end{equation}
From \eqref{b15}, we also get that $u_{k}\to u$ in $H$ as $k\to+\infty$. 
Therefore, $\forall\ \omega\in H$, one has
\begin{equation}\label{c7}
\begin{aligned}
 0&=\lim_{n\to+\infty}\langle \Phi'_{u_{k},\theta_{k}}(u_{k}), \omega\rangle \\
 &=\lim_{n\to+\infty}\Big[\left(a+\lambda M(\|u_{k}\|^{p})\right)
 \int_{\Omega}|\nabla u_{k}|^{p-2}\nabla u_{k}\cdot\nabla \omega dx
 -\theta_{k}\int_{\Omega}f(u_{k})\omega dx\Big] \\
 &=\left(a+\lambda M(\|u\|^{p})\right)
\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla \omega dx
-\int_{\Omega}f(u)\omega dx,
\end{aligned}
 \end{equation}
which shows that $u$ is a solution of \eqref{a1}. Furthermore,
\begin{equation}\label{c8}
\begin{aligned}
 & \frac{1}{p}(a+\lambda M(\|u\|^{p}))\|u\|^{p}-\int_{\Omega}F(u)dx \\
 &=\lim_{k\to+\infty}\Big[\frac{1}{p}(a+\lambda M(\|u_{k}\|^{p}))\|u_{k}\|^{p}
 -\theta_{k}\int_{\Omega}F(u_{k})dx\Big] \\
 &=\lim_{k\to+\infty}\Phi_{u_{k},\theta_{k}}(u_{k}).
\end{aligned}
 \end{equation}
Combining this with $\Phi_{u_{k},\theta_{k}}(u_{k})\geq\alpha>0$, we know that 
$u$ is nontrivial. By the strong maximum principle, we further obtain that $u$ 
is positive in $\Omega$. Hence, the proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.3}]
 From (A6) and (A7), for any $\varepsilon>0$, there exists $C_{\varepsilon}>0$ 
such that
 $$
 |f(t)|\leq \varepsilon |t|^{p-1}+C_{\varepsilon}|t|^{p^{\ast}-1}.
 $$
Arguing by contradiction, assume that the problem \eqref{a1} has a nontrivial 
solution $u\in H$, one has
 \begin{equation}\label{c9}
(a+\lambda M(\|u\|^{p}))\|u\|^{p}
=a\|u\|^{p}+\lambda \|u\|^{p(\tau+1)} 
\leq \frac{\varepsilon}{p}\|u\|^{p}+C_{\varepsilon}\|u\|^{p^{\ast}}.
 \end{equation}
By $p(\tau+1)/\tau< N$, we deduce that $p^{\ast}<p(\tau+1)$. 
Choosing $\varepsilon=a/2$, it follows from \eqref{c9} and the Young inequality that
\begin{equation}\label{c10}
\begin{aligned}
& \frac{a}{p}\|u\|^{p}+\lambda \|u\|^{p(\tau+1)} \\
&\leq C_{\varepsilon}\|u\|^{p^{\ast}} \\
&=\Big(\frac{a\tau}{p(\tau+1)-p^{\ast}}\Big)^{\frac{p(\tau+1)-p^{\ast}}{p\tau}}
\|u\|^{\frac{p(\tau+1)-p^{\ast}}{\tau}}
 \Big(\frac{p(\tau+1)-p^{\ast}}{a\tau}\Big)^{\frac{p(\tau+1)-p^{\ast}}{p\tau}}\\
&\quad\times  C_{\varepsilon}\|u\|^{\frac{(p^{\ast}-p)(\tau+1)}{\tau}} \\
&\leq\frac{a}{p}\|u\|^{p}+\frac{p^{\ast}-p}{p\tau}
\Big(\frac{p(\tau+1)-p^{\ast}}{a\tau}\Big)^{\frac{p(\tau+1)-p^{\ast}}{p^{\ast}-p}}
C^{\frac{p\tau}{p^{\ast}-p}}_{\varepsilon} \|u\|^{p(\tau+1)}.
\end{aligned}
 \end{equation}
 Define 
\[
\Theta:=\frac{p^{\ast}-p}{p\tau}
\Big(\frac{p(\tau+1)-p^{\ast}}{a\tau}\Big)^{\frac{p(\tau+1)-p^{\ast}}{p^{\ast}-p}}
C^{\frac{p\tau}{p^{\ast}-p}}_{\varepsilon}.
\]
 Consequently, for any $\lambda>\Theta$, the problem \eqref{a1} has no 
nontrivial solution.
\end{proof}

\subsection*{Acknowledgments} 
 This research was partially support by innovative project of graduate 
students of Central South University (No. 2017zzts306), 
innovative project of Central South University (2017CX017) and the 
National Natural Science Foundation of China (No. 11671405).

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\end{document}