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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 188, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/188\hfil Infinitely many solutions]
{Infinitely many solutions for Kirchhoff type problems with nonlinear
Neumann boundary conditions}

\author[W.-B. Wang, W. Tang \hfil EJDE-2016/188\hfilneg]
{Wei-Bing Wang, Wei Tang}

\address{Wei-Bing Wang (corresponding author)\newline
Department of Mathematics, 
Hunan University of Science and Technology,
 Xiangtan, Hunan 411201, China}
\email{wwbing2013@126.com}

\address{Wei Tang \newline
Department of Mathematics, 
Hunan University of Science and Technology,
Xiangtan, Hunan 411201, China}
\email{tangwei280584@163.com}

\thanks{Submitted December 18, 2015. Published July 13, 2016.}
\subjclass[2010]{35J60, 35J20}
\keywords{Kirchhoff type equation; weak solution; critical point}

\begin{abstract}
 In this article, we study a Kirchhoff type problem with nonlinear
 Neumann boundary conditions on a bounded domain. By using variational
 methods, we prove the existence of infinitely many solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this work, we study the multiplicity of solutions for the
 elliptic problem
\begin{equation}\label{E11}
\begin{gathered}
-\Big[M\Big(\int_\Omega|\nabla u|^pdx\Big)\Big]^{p-1}\Delta_p
u= f(x,u),\quad \text{in }\Omega,\\
|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda k(u)+\mu
g(u),\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $M(t)=a+bt$, $p>N$, $a>0,b\geq 0$, $\Omega$ is a nonempty
bounded open subset of $\mathbb{R}^N$ with a boundary of class
$C^1$, $\frac{\partial u}{\partial\nu}$ is the outer unit normal
derivative, $\Delta_pu:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian operator, $\lambda,\mu$ are positive real parameters,
the functions $f,k,g$ satisfy  hypotheses  stated as follows:
\begin{itemize}

\item[(H1)] $k,g\in C(\mathbb{R},\mathbb{R})$  and  there exist two
positive constants $\rho,\rho^*$ such that
$$
|K(u)|+|G(u)|\leq \rho^*(|u|^\rho+1)
$$
for all $u\in \mathbb{R}$, where
$K(u)=\int_0^uk(s)ds$, $G(u)=\int_0^ug(s)ds$.


\item[(H2)] $f\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ and
there exist two positive constants $a_1,a_2$ such that
$$
a_1|u|^p\leq -F(x,u)\leq a_2|u|^p
$$
for all $(x,u)\in \Omega\times\mathbb{R}$, where
$F(x,u)=\int_0^uf(x,s)ds$.

\item[(H3)] $f\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ and
there exist two positive constants $\varrho, \varrho^*$ such that
$$
 |F(x,u)|\leq \varrho^*(|u|^\varrho+1)
$$
for all $(x,u)\in \Omega\times\mathbb{R}$.


\item[(H4)] there exist two positive constants $b_1,b_2$ such that
$$
b_1|u|^p\leq -G(u)\leq b_2|u|^p
$$
for all $u\in \mathbb{R}$.
\end{itemize}

Problem \eqref{E11} is the nonlocal problem, which is related to
the  model introduced by Kirchhoff \cite{K},
\begin{equation}\label{E12}
\rho\frac{\partial^2u}{\partial
t^2}-\Big(\frac{\rho_0}h+\frac{E}{2L}\int_0^L|\frac{\partial
u}{\partial x}|^2dx\Big)\frac{\partial^2u}{\partial x^2}=0,
\end{equation}
which extends the classical D'Alembert's wave equation by
considering the effects of the changes in the length of the strings
during the vibrations. Interest of the mathematicians on the
nonlocal problems has increased because they represent a variety of
relevant physical and engineering situations\cite{GMB,L}.  Many
interesting results for Kirchhoff type problems were obtained and we
refer to\cite{CO,A,APS,CY,FC,HZ,HMT,KP,JJS,SJ,LH} and references
therein for an overview on these subjects. Relatively speaking,
Kirchhoff type problems with nonlinear boundary conditions have
rarely been considered. In addition, when involving the existence of
infinitely many solutions, most results assume that nonlinear term is
 odd in order to apply some variant of
 the classical Lusternik-Schnirelmann theory and  only a few papers
 deal with nonlinearities having
no symmetry properties\cite{MB,BB,B1}.


The main purpose of this article is to establish the existence
of infinitely many solutions for  \eqref{E11} without the
assumption of symmetry property, by adopting the framework of
Bonanno and Molica Bisci \cite{BB}.


\section{Preliminaries}

Let $X$ be a reflexive real Banach  space and
$\mathcal{I}_\lambda:X\to\mathbb{ R}$  a functional
satisfying the structure hypothesis:
\begin{itemize}
\item[(H5)] $\mathcal{I}_\lambda(u)=\Psi(u)-\lambda\Phi(u)$ for all
$u\in X$, where $\Psi,\Phi:X\to \mathbb{R}$ are two
functions of class $C^1$ on $X$ with $\Psi$ coercive, i.e.
$\lim_{\|u\|\to+\infty}\Psi(u)=+\infty$, and $\lambda$ is a
real parameter.
\end{itemize}
Provided that $\inf_X\Psi<r$, put
\begin{gather*}
\phi_{\mathcal{I_\lambda}}(r):=\inf_{u\in
\Psi^{-1}(]-\infty,r[)}\frac{\big(\sup_{u\in(
\Psi^{-1}]-\infty,r[)}\Phi(u)\big)-\Phi(u)}{r-\Psi(u)},
\\
\gamma:=\liminf_{r\to +\infty}\phi_{\mathcal{I_\lambda}}(r),\quad
\delta:=\liminf_{r\to (\inf_X\Psi)^+}\phi_{\mathcal{I_\lambda}}(r).
\end{gather*}
When $\gamma=0$( or $\delta=0$) we agree to read $\frac{1}{\gamma}$
(or $\frac{1}{\delta})$ as $+\infty$.
Our main tool is a smooth  version of critical point  theorem which
are recalled below, see \cite{BB}.

\begin{theorem}\label{thm21}
Assume that  {\rm (H5)} holds. Then:
\begin{itemize}
\item[(a)]  For each $r>\inf_X\Psi$ and every $\lambda\in
]0,1/\phi_{\mathcal{I_\lambda}}(r)[$, the restriction of the functional
$\mathcal{I}_\lambda$ to $\Psi^{-1}(]-\infty,r[)$ has a global
minimum, which a s critical point (local  minimum) of
$\mathcal{I}_\lambda$ in $X$.

\item[(b)] If  $\gamma<+\infty$, then, for each $\lambda\in ]0,1/\gamma[$,
the following alternative holds:either
\begin{itemize}
\item[(1)] $\mathcal{I}_\lambda$ possess a global minimum, or
 \item[(2)]   there is a sequence $\{u_n\}$ of critical points of
$\mathcal{I}_\lambda$ such that \\
$\lim_{n\to \infty}\Psi(u_n)=+\infty$.
\end{itemize}

\item[(c)] If  $\delta<+\infty$, then, for each $\lambda\in ]0,1/\delta[$,
the following alternative holds: either
\begin{itemize}
\item[(1)]  there is a global minimum of $\Phi$ which
 is a local minimum of $\mathcal{I}_\lambda$, or 
\item[(2)]  there is a sequence $\{u_n\}$ of pairwise distinct
critical points of $\mathcal{I}_\lambda$ such that
$\lim_{n\to \infty}\Psi(u_n)=\inf_X\Psi$, which weakly
converges to a global minimum of $\Phi$.
\end{itemize}
\end{itemize}
\end{theorem}

Let $W^{1,p}(\Omega)$ be the usual Sobolev space endowed with the norm
$$
\|u\|:=\Big(\int_\Omega(|\nabla u|^p+|u|^p)dx\Big)^{1/p}
$$
or
$$
\|u\|_*:=\Big(\int_\Omega|\nabla u|^pdx+\int_{\partial\Omega}|
u|^pd\sigma\Big)^{1/p},
$$
where $d\sigma$ is the measure on the boundary. Clearly, $\|\cdot\|$
is equivalent to $\|\cdot\|_*$. Let
\begin{equation}\label{E21}
\mathrm{k}:=\sup_{u\in W^{1,p}(\Omega)\setminus\{0\}}
\frac{\max_{x\in\overline{\Omega}} |u(x)|}{\|u\|},\quad
\mathrm{k}_*:=\sup_{u\in W^{1,p}(\Omega)\setminus\{0\}}
\frac{\max_{x\in\overline{\Omega}} |u(x)|}{\|u\|_*}
\end{equation}
Since $p>N$, the embedding $W^{1,p}(\Omega)\hookrightarrow C(\overline{\Omega})$
is compact, and thus $0<\mathrm{k},\mathrm{k}_*<\infty$,
\begin{equation}\label{E22}
|u(x)|\leq \mathrm{k}\|u\|,\quad
|u(x)|\leq \mathrm{k}_*\|u\|_*\,\quad\text{for } u\in W^{1,p}(\Omega) .
\end{equation}
A weak solution of problem \eqref{E11}, we mean that a function
$u\in W^{1,p}(\Omega)$ satisfies
\begin{align*}
&\Big[M\Big(\int_\Omega |\nabla
u|^pdx\Big)\Big]^{p-1} \int_\Omega |\nabla u|^{p-2}\nabla u\nabla vdx\\
&-\int_\Omega f(x,u)vdx
-\int_{\partial\Omega}(\lambda k(u)
+\mu g(u))v d\sigma=0
\end{align*}
for every $v\in W^{1,p}(\Omega)$.

Define the functionals on $W^{1,p}(\Omega)$ by
\begin{gather*}
\Gamma(u)=\frac{1}{p}\int_0^{\int_\Omega |\nabla
u|^pdx}M^{p-1}(s)ds
=\begin{cases}
\frac{1}{bp^2}\big[\big(a+b\int_\Omega |\nabla
u|^pdx\big)^{p}-a^p\big], & b>0,\\
\frac{a^{p-1}}{p}\int_\Omega |\nabla u|^pdx , & b=0,
\end{cases}
\\
\psi(u)=\Gamma(u)-\int_\Omega F(x,u)dx,\quad
\varphi(u)=\int_{\partial\Omega}\big(K(u)+\frac{\mu}{\lambda}G(u)\big)d\sigma,
\\
\psi_*(u)=\Gamma(u)-\mu\int_{\partial\Omega}G(u)d\sigma,\quad
\varphi_*(u)=\int_{\partial\Omega}K(u)d\sigma+\frac{1}{\lambda}\int_\Omega
F(x,u)dx,
\\
J_\lambda(u)=\psi(u)-\lambda\varphi(u),\quad
I_\lambda(u)=\psi_*(u)-\lambda\varphi_*(u).
\end{gather*}

Conditions (H1) and (H2)  (or (H1) and (H3)) and $p>N$ guarantee that
$\psi,\varphi$ (or $\psi_*,\varphi_*$) are well defined  and of
class $C^1$. Moreover,
\begin{align*}
\langle J_\lambda'(u),v\rangle
&=\langle I_\lambda'(u),v\rangle \\
&=\Big[M\Big(\int_\Omega |\nabla u|^pdx\Big)\Big]^{p-1}
\int_\Omega |\nabla u|^{p-2}\nabla u\nabla vdx \\
&\quad -\int_\Omega f(x,u)vdx -\int_{\partial\Omega}(\lambda k(u)
 +\mu g(u))v d\sigma.
\end{align*}
Hence, the critical points of $J_\lambda$ or $I_\lambda$ are the
weak solutions of \eqref{E11}.

\section{Main results}
Put
$$
a_3=\big\{\frac{a^{p-1}}{p},a_1\big\},\quad
b_3=\big\{\frac{a^{p-1}}{p},b_1\big\},\quad
|\Omega|=\int_{\Omega}dx, \quad |\partial\Omega|=\int_{\partial\Omega}d\sigma.
$$
Moreover, let
\begin{gather*}
A_\infty:=\liminf_{\xi\to+\infty}\frac{\max_{|t|\leq  \xi}K(t)}{\xi^p},\quad
A_0:=\liminf_{\xi\to 0^+}\frac{\max_{|t|\leq  \xi}K(t)}{\xi^p}, \\
B_\infty:=\limsup_{\xi\to+\infty}\frac{ K(\xi)}{\xi^p},\quad
B_0:=\limsup_{\xi\to0^+}\frac{ K(\xi)}{\xi^p},\\
G_\infty:=\limsup_{\xi\to+\infty}\frac{\max_{|t|\leq  \xi}G(t)}{\xi^p},\quad
G_0:=\limsup_{\xi\to0^+}\frac{\max_{|t|\leq  \xi}G(t)}{\xi^p}. \\
F_\infty:=\limsup_{\xi\to+\infty}\frac{\int_\Omega \max_{|t|\leq
 \xi}F(x,t)dx}{\xi^p},\quad
F_0:=\limsup_{\xi\to 0^+}\frac{\int_\Omega \max_{|t|\leq
 \xi}F(x,t)dx}{\xi^p}
\end{gather*}
We present our main results as follows.

\begin{theorem} \label{thm31}
 Assume that {\rm (H1), (H2)}  hold and there exist two real sequences
 $\{\alpha_n\},\{\beta_n\}$ with $\lim_{n\to\infty}\beta_n=+\infty$
 and a positive constant $\rho>0$ such that
\begin{gather*}
|\alpha_n|<\frac{1}{\mathrm{k}}\Big(\frac{a_3}{a_2|\Omega|}\Big)^{1/p}
\beta_n,\quad  G(t)\geq 0,\quad \forall t\geq \rho,\\
\mathcal{A}_\infty:=\lim_{n\to\infty}\frac{\max_{|t|\leq
 \beta_n}K(t)-K(\alpha_n)}{\beta_n^p-\mathrm{k}^pa_2a_3^{-1}
 |\Omega|\alpha_n^p}<\frac{a_3B_\infty}{\mathrm{k}^p|\Omega|a_2},\\
\mathcal{G}_\infty:=\lim_{n\to\infty}\frac{\max_{|t|
\leq  \beta_n}G(t)  -G(\alpha_n)}{\beta_n^p
 -\mathrm{k}^pa_2a_3^{-1}|\Omega|\alpha_n^p}<+\infty.
 \end{gather*}
Then for each $\lambda \in \Lambda:=]\lambda_1,\lambda_2[$, where
$$
\lambda_1=\frac{|\Omega|a_2}{|\partial\Omega|B_\infty},\quad
\lambda_2=\frac{a_3}{\mathrm{k}^p|\partial\Omega|\mathcal{A}_\infty},
$$
there exists $\mu_\lambda>0$, where
$$
\mu_\lambda=\frac{1}{\mathcal{G}_\infty}
\Big(\frac{a_3}{\mathrm{k}^p| \partial\Omega|}-\lambda \mathcal{A}_\infty\Big),
$$
such that for all $\mu \in[0,\mu_\lambda[$,
\eqref{E11} has an unbounded sequence of weak solutions.
\end{theorem}

\begin{proof}
First, we observe that owing to the condition
$\mathcal{A}_\infty< a_3B_\infty/(\mathrm{k}^p|\Omega|a_2)$, the interval
$\Lambda$ is  non-empty. Moreover, for each fixed
$\bar{\lambda} \in\Lambda$ and taking into  account
 that $\mathcal{A }_\infty\bar{\lambda}<a_3/\mathrm{k}^p|\partial\Omega|$,
one has $0<\mu_{\bar{\lambda}}<\infty$. Our aim is to
apply Theorem \ref{thm21}. For this end, we show $\gamma<+\infty$,
where $\gamma$ is defined in Theorem \ref{thm21}.

By Assumption (H2), we have
\begin{equation}\label{E}
a_3\| u\|^p\leq \psi (u)\leq
\max\big\{\frac{(2a)^{p-1}}{p},a_2\big\}\|u\|^p
+\frac{(2b)^{p-1}}{p}\|u\|^{p^2}.
\end{equation}
Put $r_n=\beta_n^pa_3/\mathrm{k}^p$ for all $n \in \mathbb{N}$, by
\eqref{E22} and \eqref{E}, one has
\begin{gather*}
\psi^{-1}(]-\infty,r_n])\subseteq \{u\in W^{1,p}(\Omega):
\|u\|_\infty\leq \beta_n\},\\
\psi(\alpha_n)=-\int_\Omega F(x,\alpha_n)dx\leq
a_2|\Omega|\alpha_n^p<r_n.
\end{gather*}
Hence,
\begin{align*}
\phi_{J_\lambda}(r_n)
&=\inf_{u\in \psi^{-1}(]-\infty,r_n[)}
\frac{\big(\sup_{u\in(
\psi^{-1}]-\infty,r_n[)}\varphi(u)\big)-\varphi(u)}{r_n-\psi(u)}\\
&\leq\inf_{\psi(u)<r_n}\frac{|\partial\Omega|\max_{|t|\leq
 \beta_n}[K(t)+\frac{\mu}{\bar{\lambda}}G(t)] -\varphi(u)}{r_n-\psi(u)}\\
&\leq\frac{|\partial\Omega|\max_{|t|\leq
 \beta_n}[K(t)+\frac{\mu}{\bar{\lambda}}G(t)]
 -\varphi(\alpha_n)}{r_n-\psi(\alpha_n)}\\
&\leq \frac{|\partial\Omega|\max_{|t|\leq
 \beta_n}[K(t) -K(\alpha_n)]}{r_n-a_2|\Omega|\alpha_n^p}+\frac{\mu}{\bar{\lambda}}\frac{|\partial\Omega|\max_{|t|\leq
 \beta_n}[G(t) -G(\alpha_n)]}{r_n-a_2|\Omega|\alpha_n^p}\\
&\leq \frac{\mathrm{k}^p|\partial
\Omega|}{a_3}\Big(\mathcal{A}_\infty+\frac{\mu}{\bar{\lambda}}
\mathcal{G}_\infty\Big)<+\infty.
 \end{align*}
Since $\mu \in[0,\mu_{\bar{\lambda}}[$,
$$
\gamma\leq\liminf_{n\to+\infty}\phi_{J_\lambda}(r_n)
< \frac{ \mathrm{k}^p |\partial
\Omega|}{a_3}\Big(\mathcal{A}_\infty
+\frac{\mu_{\bar{\lambda}}}{\bar{\lambda}}\mathcal{G}_\infty\Big)
=\frac{1}{\bar{\lambda}}<+\infty;
$$
that is, $0<\lambda_1<\bar{\lambda}<1/\gamma$. The condition (b) of
Theorem \ref{thm21} can be applied and either $J_{\bar{\lambda}}$ has a
global minimum or there exists a sequence $\{u_n\}$ of weak
solutions of the problem \eqref{E11} such that
$\|u_n\|\to\infty$ as $n\to\infty$.

Now, we verify that $J_{\bar{\lambda}}$ is unbounded from below.
First, assume that $B_\infty=+\infty$. Accordingly, fixed $C$ with
$C>a_2|\Omega|/\bar{\lambda}|\partial\Omega|$ and $\{c_n\}$ be a
sequence of positive numbers with $c_n\to+\infty$ as
$n\to\infty$ such that
$$
K(c_n)>C c_n^p,\quad  n \text{ sufficiently large}.
$$
Taking the sequence $\{v_n\}\subseteq W^{1,p}(\Omega)$ such that
 $v_n(x)=c_n, x\in \bar{\Omega}$, for the sufficiently large $n$, one has
\begin{align*}
J_{\bar{\lambda}}(v_n)
&=-\int_\Omega F(x,v_n)dx-\bar{\lambda}\int_{\partial\Omega}K(v_n)d\sigma
 -\mu\int_{\partial\Omega}G(v_n)d\sigma\\
&\leq a_2|\Omega|c_n^p-\bar{\lambda}\int_{\partial\Omega}K(v_n)d\sigma
\leq (a_2|\Omega|-C\bar{\lambda}|\partial\Omega|)c_n^p;
\end{align*}
that is, $J_{\bar{\lambda}}\to-\infty$ as $n\to\infty$.

Next, assume that $B_\infty<+\infty$. Since
$\bar{\lambda}>\lambda_1=a_2|\Omega|/|\partial\Omega|B_\infty$, we fix
$0<\varepsilon<B_\infty-\frac{a_2|\Omega|}{\bar{\lambda}|\partial\Omega|}$.
Let $\{c_n\}$ be a sequence of positive numbers with
$c_n\to+\infty$ as $n\to\infty$ such that
$$
(B_\infty-\varepsilon)c_n^p<K(c_n)<(B_\infty+\varepsilon)c_n^p,\quad
 n \text{ sufficiently large}
$$
Arguing as before and by choosing $v_n\equiv c_n$, for the
sufficiently large $n$, one has
\begin{align*}
J_{\bar{\lambda}}(v_n)
&=-\int_\Omega F(x,v_n)dx-\bar{\lambda}\int_{\partial\Omega}K(v_n)d\sigma
-\mu\int_{\partial\Omega}G(v_n)d\sigma\\
&\leq a_2|\Omega|c_n^p-(B_\infty-\varepsilon)\bar{\lambda}
 |\partial\Omega|c_n^p\to-\infty \quad \text{as } n\to\infty.
\end{align*}
Hence,  $J_{\bar{\lambda}}$ is unbounded from blew and the proof is
complete.
\end{proof}

\begin{corollary}\label{coro1}
 Assume that {\rm (H1), (H2)} hold. Further suppose that
$G_\infty<+\infty$, $A_\infty<  a_3B_\infty/(\mathrm{k}^p|\Omega|a_2)$
and there exists $\rho>0$ such that
 $G(t)\geq 0$ for all $t\geq\rho$.
Then for each $\lambda \in ]\lambda_3,\lambda_4[$, where
$$
\lambda_3=\frac{|\Omega|a_2}{|\partial\Omega|B_\infty},\quad
\lambda_4=\frac{a_3}{\mathrm{k}^pA_\infty|\partial\Omega|},
$$
there exists $\tilde{\mu}_\lambda>0$, where
$$
\tilde{\mu}_\lambda=\frac{1}{G_\infty}
\Big(\frac{a_3}{\mathrm{k}^p| \partial\Omega|}-\lambda A_\infty\Big),
$$
such that for all
$\mu \in[0,\tilde{\mu}_\lambda[$, \eqref{E11} has an unbounded sequence
of weak solutions.
\end{corollary}

\begin{proof}
Let $\{\beta_n\}$ be a sequence of positive numbers which approaches
infinity such that
$$
A_\infty=\lim_{\xi\to+\infty}\frac{\max_{|t|\leq
 \beta_n}K(t)}{\beta_n^p}.
$$
Taking $\alpha_n=0$ for every $n\in\mathbb{N}$ and noting that
$$
\mathcal{G}_\infty=\lim_{n\to+\infty}\frac{\max_{|t|\leq
 \beta_n}G(t)}{\beta_n^p}\leq G_\infty,
$$
from Theorem \ref{thm31}, we obtain the conclusion.
\end{proof}

Applying part (c) of Theorem \ref{thm21}, we get the following
theorem.

\begin{theorem} \label{thm32}
 Assume that {\rm (H1), (H2)} hold and there exist two real sequences
 $\{\alpha_n\},\{\beta_n\}$ with $\lim_{n\to\infty}\beta_n=0$
and positive constant $\rho>0$ such that
\begin{gather*}
|\alpha_n|<\frac{1}{\mathrm{k}}\Big(\frac{a_3}{a_2|\Omega|}\Big)^{1/p}\beta_n,\quad
G(t)\geq 0,\quad \forall 0\leq t\leq \rho,\\
\mathcal{A}_0:=\lim_{n\to\infty}\frac{\max_{|t|
\leq  \beta_n}K(t)-K(\alpha_n)}{\beta_n^p-\mathrm{k}^p
 a_2a_3^{-1}|\Omega|\alpha_n^p}<\frac{a_3B_0}{\mathrm{k}^p|\Omega|a_2},\\
\mathcal{G}_0:=\lim_{n\to\infty}\frac{\max_{|t|\leq\beta_n}
G(t)-G(\alpha_n)}{\beta_n^p-\mathrm{k}^pa_2a_3^{-1}|\Omega|\alpha_n^p}<+\infty.
\end{gather*}
Then for each $\lambda \in ]\lambda_4,\lambda_5[$, where
$$
\lambda_4=\frac{|\Omega|a_2}{|\partial\Omega|B_0},\quad
\lambda_5=\frac{a_3}{\mathrm{k}^p|\partial\Omega|\mathcal{A}_0},
$$
there exists $\bar{\mu}_\lambda>0$, where
$$
\bar{\mu}_\lambda=\frac{1}{\mathcal{G}_0}
\Big(\frac{a_3}{\mathrm{k}^p| \partial\Omega|}-\lambda \mathcal{A}_0\Big),
$$
such that for all $\mu \in[0,\bar{\mu}_\lambda[$,
\eqref{E11} has a sequence of weak solutions, which converges strongly
to zero.
\end{theorem}

\begin{corollary}\label{coro3}
 Assume that {\rm (H1), (H2)} hold. Further suppose that
$G_0<+\infty$, $A_0<  a_3B_0/(\mathrm{k}^p|\Omega|a_2)$
and there exists $\rho>0$ such that
 $G(t)\geq 0$ for all $0<t\leq\rho$.
Then for each $\lambda \in ]\lambda_7,\lambda_8[$, where
$$
\lambda_7=\frac{|\Omega|a_2}{|\partial\Omega|B_0},\quad
\lambda_8=\frac{a_3}{\mathrm{k}^pA_0|\partial\Omega|},
$$
there exists $\hat{\mu}_\lambda>0$, where
$$
\hat{\mu}_\lambda=\frac{1}{G_0}
\Big(\frac{a_3}{\mathrm{k}^p| \partial\Omega|}-\lambda A_0\Big),
$$
 such that for all $\mu \in[0,\hat{\mu}_\lambda[$,
\eqref{E11} has a sequence of weak solutions, which
converges strongly to zero.
\end{corollary}

Now, we consider the case when (H1), (H3), (H4) hold.

\begin{theorem} \label{thm33}
 Assume that {\rm (H1), (H3), (H4)} hold and
$F(x,u)\geq 0$ for $x\in \overline{\Omega},u\geq r>0$.
If there exists the positive  constant $\bar{\lambda} $ such that
$$
\bar{\lambda}B_\infty>b_2,\quad
F_\infty<\frac{b_3}{\mathrm{k}_*^p}-\bar{\lambda}|\partial\Omega|A_\infty,
$$
then \eqref{E11} has an unbounded sequence of weak solutions for
$\lambda=\bar{\lambda}$, $\mu=1$.
\end{theorem}

\begin{proof}
Let $\lambda=\bar{\lambda},\mu=1$ and $\{\beta_n\}$ be a sequence of
positive numbers with $\beta_n\to+\infty$ as
$n\to\infty$ such that
$$
A_\infty= \lim_{n\to+\infty}\frac{\max_{|t|\leq
 \beta_n}K(t)}{\beta_n^p}.
$$
By Assumption (H4), we have
\begin{equation}\label{E34}
b_3\| u\|_*^p\leq \psi_* (u)
\leq \max\big\{\frac{(2a)^{p-1}}{p},b_2\Big\}\|u\|_*^p
+\frac{(2b)^{p-1}}{p}\|u\|_*^{p^2}.
\end{equation}
Put $r_n=\beta_n^pb_3/\mathrm{k}_*^p$ for all $n \in \mathbb{N}$, by
\eqref{E22} and \eqref{E34}, one has
$$
\psi_*^{-1}(]-\infty,r_n])\subseteq \{u\in W^{1,p}(\Omega): \|u\|_\infty\leq
\beta_n\}.
$$
Hence,
\begin{align*}
\phi_{I_\lambda}(r_n)
&=\inf_{u\in
\psi_*^{-1}(]-\infty,r_n[)}\frac{\big(\sup_{u\in(
\psi_*^{-1}]-\infty,r_n[)}\varphi_*(u)\big)-\varphi_*(u)}{r_n-\psi_*(u)}\\
&\leq\frac{\sup_{u\in(
\psi_*^{-1}]-\infty,r_n[)}\varphi_*(u)}{r_n}\\
&\leq \frac{\max_{\{u\in X: \|u\|_\infty\leq \beta_n\}}\varphi_*(u)}{r_n}\\
&\leq \frac{\mathrm{k}_*^p}{b_3}\Big(\frac{|\partial
\Omega|\max_{|t|\leq
 \beta_n}K(t) }{\beta_n^p}+\frac{1}{\bar{\lambda}}\frac{
 \int_\Omega \max_{|t|\leq \beta_n}F(x,t)dx}{\beta_n^p}\Big)\\
&\leq \frac{\mathrm{k}_*^p}{b_3}\Big(|\partial
\Omega|A_\infty+\frac{1}{\bar{\lambda}}F_\infty\Big)
<\frac{1}{\bar{\lambda}}.
 \end{align*}
The rest proof is  similar to that of Theorem \ref{thm31} and we omit
it.  \end{proof}

\begin{theorem} \label{thm34}
 Assume that {\rm (H1), (H3), (H4)} hold and $F(x,u)\geq 0$ for
$x\in \bar{\Omega}, 0\leq u\leq r(r>0)$.
 If there exists the positive  constant $\bar{\lambda} $ such that
$$
\bar{\lambda}B_0>b_2,\quad
F_0<\frac{b_3}{\mathrm{k}_*^p}-\bar{\lambda}|\partial\Omega|A_0,
$$
then \eqref{E11} has a sequence of weak solutions for
$\lambda=\bar{\lambda},\mu=1$, which  converges strongly to zero.
\end{theorem}


\section{Examples}

In this section, we  present the two examples which provide the
problems that admit infinitely many solutions.

\begin{example} \label{exmp4.1} \rm
Consider the  differential equation
\begin{equation}\label{E41}
\begin{gathered}
-\Big[1+b\int_0^1|u'|^2dx\Big]u''(x)
+u(u\cos u+2\sin u+4)=0,\quad  x\in (0,1),\\
-u'(0)=\lambda k(u(0))+\mu g(u(0)),\quad u'(1)=\lambda k(u(1))+\mu g(u(1)),
\end{gathered}
\end{equation}
where $b\geq 0$,
\[
 k(u)=\begin{cases}
u^2 \sin(\ln u),&  u>0,\\
0,&  u\leq 0,
\end{cases}
\qquad g(u)=u-\sin u.
\]
Then
\begin{gather*}
f(u)=-u(u\cos u+2\sin u+4),\quad -F(u)=u^2(2+\sin u), \\
K(u)=\begin{cases}
\frac{u^{3}}{8}[3\sin(\ln u)-\cos(\ln u)], &  u>0,\\
0, &  u\leq 0,
\end{cases}
\\
G(u)=\frac{1}{2}u^2+\cos u-1,\quad
 |\Omega|=| \partial\Omega|=1, \quad
 a_1=1,\quad a_2=3, \quad a_3=\frac{1}{2}.
\end{gather*}
According to \cite[Remark 1]{B},  one has the estimate
$\mathrm{k}\leq \sqrt{2}$.
 Taking $\alpha_n=e^{(2n+1)\pi}$, $\beta_n=e^{2(n+1)\pi}$, we easily
obtain that
$$
\mathcal{A}_\infty=0,\quad   B_\infty=+\infty, \quad
  \mathcal{G}_\infty=\frac{e^{2\pi}-1}{2(e^{2\pi}-6\mathrm{k}^2)}.
$$
By Theorem \ref{thm31},  \eqref{E41} has an unbounded sequence of weak
solutions for
$$
\lambda>0,\quad
0\leq \mu<\frac{e^{2\pi}-6\mathrm{k}^2}{\mathrm{k}^2(e^{2\pi}-1)}.
$$
\end{example}

\begin{example} \label{exmp4.2} \rm
Consider the  differential equation
\begin{equation}\label{E42}
\begin{gathered}
-\Big[M\Big(\int_\Omega|\nabla
u|^pdx\Big)\Big]^{p-1}\Delta_p
u=c(x)|u|^{\rho-1}u,\quad \text{in }\Omega,\\
|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda
k(u)-|u|^{p-2}u , \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $M(t)=a+bt$, $p>N$, $a>0,b\geq 0$, $1<\rho <p$,
$c\in C(\overline{\Omega})$ and  $c(x)\geq 0$, $\Omega$ is a nonempty
bounded open subset of $\mathbb{R}^N$ with a boundary of class
$C^1$,
\begin{gather*}
k_1=2,\quad k_{n+1}=k^{12}_n, \quad l_n=k^{10}_n,n\in \mathbb{N},\\
k(t)=\begin{cases}
(l^{p-0.5}_{n}-k^{p+0.5}_{n})(1-|l_n-t|), & l_n-1\leq t\leq l_n+1,n\geq1,\\
(k^{p+0.5}_{n+1}-l^{p-0.5}_{n})(1-|k_{n+1}-t|),
 & k_{n+1}-1\leq t\leq k_{n+1}+1,n\geq 1,\\
0,&\text{otherwise}.
\end{cases}
\end{gather*}
Noting that
\[
K(k_{n}+1)=k_n^{p+0.5}-k_1^{p+0.5},\quad
K(l_{n}+1)=l_n^{p-0.5}-k_1^{p+0.5},\quad  n\geq 2,
\]
we have
\[
\lim_{n\to\infty}
\frac{K(k_{n}+1)}{(k_{n}+1)^p}=+\infty,\quad
\lim_{n\to\infty} \frac{K(l_{n}+1)}{(l_{n}+1)^p}=0.
\]
Hence, $A_\infty=0$, $B_\infty=+\infty$. It is easy to check that
$F_\infty=0$. By Theorem \ref{thm33},  \eqref{E42} has an unbounded
sequence of weak solutions for all $\lambda>0$.
\end{example}


\subsection*{Acknowledgments}
The authors express their gratitude to the reviewers for careful
reading and helpful suggestions which led to an improvement of the
original manuscript. The work is supported by Hunan Provincial
Natural Science Foundation of China (2015JJ2068) and NNSF of
China (11501190).


\begin{thebibliography}{99}

\bibitem{CO} C. O. Alves, F. J. S. A. Corr\^ea, T. F. Ma;
\emph{Positive solutions for a quasilinear elliptic equation of
Kirchhoff type}, Comput. Math. Appl., 49 (2005), no.1, 85-93.


\bibitem{A} G. Anello;
\emph{A uniqueness result for a nonlocal equation of Kirchhoff type
and some related open problem}, J. Math. Anal. Appl., 373
(2011), no. 1, 248-251.

\bibitem{APS} G. Autuori, P. Pucci, M. C. Salvatori;
\emph{Asymptotic Stability for anisotropic Kirchhoff system}, J.
Math. Anal. Appl., 352 (2009), no. 1, 149-165.

\bibitem{GMB} G. Molica Bisci;
\emph{Variational problems on the Sphere, Recent Trends
in Nonlinear Partial Differential Equations II. Stationary
problems,} Contemp. Math., 595 (2013), 273-291.

\bibitem{MB}G. Molica Bisci, P. Pizzimenti;
\emph{ Sequences of weak solutions for non-local elliptic problems with 
Dirichlet boundary condition,}
Proc. Edinb. Math. Soc., 57 (2014), no.3, 779-809.

\bibitem{BB} G. Bonanno, G. Molica Bisci;
\emph{Infinitely many solutions for a boundary value problem with
discontinuous nonlinearities}, Bound. Value Probl., 2009 (2009), 1-20.


\bibitem{B} G. Bonanno, P. Candito; 
\emph{Three solutions to a Neumann problem for
elliptic equations involving the p-Laplacian,} Arch. Math.
80(2003), no. 4, 424-429.

\bibitem{CY} C. Chen, Y. Kuo,  T. Wu;
\emph{The Nehari manifold for a Kirchhoff type problem involving
sign-changing weight functions}, J. Differ. Equations., 250
(2011), no. 4, 1876-1908.

\bibitem{FC} F. J\'ulio,  S. A. Corr\^ea, G. M. Figueiredo;
\emph{On an elliptic equation of  $p$-Kirchhoff type via variational methods,}
Bull. Aust. Math. Soc., 74 (2006), no. 2, 263-277.

\bibitem{FE} M. Ferrara, G. Molica Bisci;
\emph{Existence results for elliptic problems with Hardy potential,} 
Bull. Sci. Math., 138 (2014), no. 7, 846-859.

\bibitem{HZ} X. He, W. Zou;
\emph{Existence and concentration behavior of positive solutions for
a Kirchhoff equation in $ {\mathbb {R}}^3$}, J. Differ. Equations.
252(2012), no. 2, 1813-1834.

\bibitem{L} J. L. Lions;
\emph{On some questions in boundary value problems of mathematical physics},
in Contemporary developments in continuum mechanics and partial differential 
equations, North-Holland Mathematics Studies, Volume 30, pp. 284-346 
(North-Holland, Amster-dam, 1978).

\bibitem{HMT} A. Hamydy, M. Massar, N. Tsouli;
\emph{Existence of solutions for p-Kirchhoff type problems with
critical exponent}, Electron. J. Diff. Eq., Vol. 2011 (2011), no.105,
1-8.

\bibitem{KP} K. Perera, Z. Zhang;
\emph{Nontrivial solutions of Kirchhoff-type problems via the Yang
index}, J. Differ. Equations. 221 (2006), no. 1, 246-255.

\bibitem{K} G. Kirchhoff;
\emph{Mechanik}, Teubner, leipzig, Germany, 1883.

\bibitem{B1}B. Ricceri;
\emph{A general variational principle and some
of its applications}, J. Comput. Appl. Math.,
113 (2000), no. 1-2, 401-410.

\bibitem{JJS} J. Sun, C. Tang;
\emph{Existence and multipicity of solutions for Kirchhoff type
equations}, Nonlinear Anal. 74 (2011), no. 4, 1212-1222.

\bibitem{SJ}  J. Sun, S. Liu;
\emph{Nontrivial solutions of Kirchhoff type problems},
 Appl. Math. Lett., 25 (2012), no. 3, 500-504.

\bibitem{LH}  W. Liu, X. He;
\emph{Multiplicity of high energy solutions for superlinear
Kirchhoff equations}, J. Appl. Math. Comput. 39 (2012), no. 1-2,
473-487.

\end{thebibliography}

\end{document}