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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 268, pp. 1--9.\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-2015/268\hfil Existence of infinitely many solutions]
{Existence of infinitely many solutions for perturbed Kirchhoff type elliptic
problems with Hardy potential}

\author[M. Xu, C. Bai \hfil EJDE-2015/268\hfilneg]
{Mei Xu, Chuanzhi Bai}

\address{Mei Xu \newline
Department of Mathematics,
Huaiyin Normal University,
Huai, Jiangsu 223300, China}
 \email{13952342299@163.com}

\address{Chuanzhi Bai (Corresponding  author)  \newline
Department of Mathematics,
Huaiyin  Normal University\\
Huai, Jiangsu 223300,  China}
\email{czbai@hytc.edu.cn}

\thanks{Submitted May 23, 2015. Published October 16, 2015.}
\subjclass[2010]{35J40, 58E05}
\keywords{Infinitely many solutions; critical points theory; Hardy potential;
\hfill\break\indent   $p$-biharmonic type operators}

\begin{abstract}
 In this article, by using critical point theory, we show the
 existence of infinitely many weak solutions for a fourth-order
 Kirchhoff type elliptic problems with Hardy potential.
\end{abstract}

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

\section{Introduction}

This article concerns the existence of infinitely
many weak solutions for the $p$-biharmonic equation with Hardy
potential of Kirchhoff type
\begin{equation}\label{eq1.1}
\begin{gathered}
M \Big(\int_{\Omega} |\Delta u|^p dx\Big)  \Delta_p^2 u  -
\frac{a}{|x|^{2p}} |u|^{p-2} u  =  \lambda f(x, u) + \mu g(x, u)  \quad \text{in }  \Omega    \\
   u = \Delta u = 0   \quad \text{on }   \partial \Omega,
\end{gathered}
 \end{equation}
 where $\Omega$  is a bounded  domain in $\mathbb{R}^N$
($N \ge 3$)  containing the origin and with smooth boundary  
$\partial \Omega$, $1 < p < \frac{N}{2}$, 
$\Delta_p^2 u = \Delta(|\Delta u|^{p-2} \Delta u)$ is an operator of 
fourth order, the so-called
$p$-biharmonic operator, $\lambda, \mu$ are two positive
parameters, $M : [0, +\infty[ \to \mathbb{R}$ is a continuous
function, and $f, g : \overline{\Omega} \times \mathbb{R} \to
\mathbb{R}$ are two continuous functions.

Kirchhoff \cite{K} first introduced a model given by the equation
\begin{equation}\label{eq1.2}
 \rho \frac{\partial^2 u}{\partial t^2} - \Big(\frac{\rho_0}{h}
 + \frac{E}{2L} \int_0^L |\frac{\partial u}{\partial x}|dx\Big)
 \frac{\partial^2 u}{\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. After that, many authors studied
the following nonlocal elliptic boundary value problem
\begin{equation}\label{eq1.3}
\begin{gathered}
 - M \Big(\int_{\Omega} |\nabla u|^2 dx\Big)  \Delta u(x)  
=  f(x, u)  \quad  \text{in }  \Omega    \\
  u = 0  \quad \text{on }  \partial \Omega.
\end{gathered}
 \end{equation}
Problems like this are  called the Kirchhoff type problems.
In recent years, many interesting results for problem of Kirchhoff
type were obtained \cite{A, Fi, Ha, He, L,  M, W}. Recently, using
the variational methods, Graef, Heidarkham and Kong \cite{Gr}
studied the existence of at least three weak solutions to the
Kirchhoff-type problem
\begin{equation}\label{eq1.4}
\begin{gathered}
- K \Big(\int_{\Omega} |\nabla u|^2 dx\Big)  \Delta u(x)  
=  \lambda f(x, u) + \mu g(x, u)  \quad  \text{in }   \Omega    \\
   u = \Delta u = 0  \quad  \text{on }   \partial \Omega.
\end{gathered}
 \end{equation}
 In \cite{F1}, using variational methods and critical point theory,
Ferrara, Khademloo and Heidarkhani established the multiplicity
results of nontrivial and nonnegative solutions for
 the following perturbed   fourth-order Kirchhoff type elliptic
problem
\begin{equation}\label{eq1.5}
\begin{gathered}
  \Delta_p^2 u  - [M(\int_{\Omega} |\nabla u|^p dx)]^{p-1} \Delta_p u + \rho
  |u|^{p-2} u = \lambda f(x, u) \quad   \text{in }    \Omega    \\
   u = \Delta u = 0  \quad   \text{on }  \partial \Omega.
\end{gathered}
 \end{equation}

On the other hand,  singular elliptic problems have been
intensively studied in recent years, see for example, 
\cite{G2, G1, R} and the references.  
 Ferrara and Molica Basic \cite{F2}
studied the existence of solutions for the  elliptic
problem with Hardy potential
\begin{equation}\label{eq1.6}
\begin{gathered}
- \Delta_p u  =
\mu \frac{|u|^{p-2} u}{|x|^p}  +  \lambda f(x, u)  \quad   \text{in }  \Omega    \\
   u = \Delta u = 0  \quad   \text{on }  \partial \Omega.
\end{gathered}
 \end{equation}
Huang and Liu \cite{Hu}  studied the sign-changing solutions
for $p$-biharmonic equations with Hardy potential
\begin{equation}\label{eq1.7}
\begin{gathered}
  \Delta_p^2 u  -
\frac{a}{|x|^{2p}} |u|^{p-2} u  =  f(x, u) \quad   \text{in }  \Omega    \\
   u = \Delta u = 0  \quad   \text{on }  \partial \Omega,
\end{gathered}
 \end{equation}
 by using the method of invariant sets of descending flow.

Motivated by the papers \cite{ F1, F2, B1, B2, B3, Gr, Hu}, in
this paper, we look for the existence of infinitely many solutions
of problem \eqref{eq1.1}. Precisely, under appropriate hypotheses
on the nonlinear term $f, g$, the existence of two intervals
$\Lambda$ and $J$ such that, for each $\lambda \in \Lambda$ and
$\mu \in J$, BVP \eqref{eq1.1} admits a sequence of pairwise
distinct solutions is proved.   Our analysis is mainly based on a
recent critical  point theorem in \cite{Bo}.

This article is organized as follows.  
In section 2, we present some
necessary preliminary facts that will be needed in the paper. In
section 3,  we establish our main two existence results.


\begin{remark} \label{rmk1.1} \rm
 If $M(\cdot) \equiv 1$,  then
Kirchhoff type problem \eqref{eq1.1} reduces to the 
$p$-biharmonic equation with Hardy potential
\begin{gather*}
  \Delta_p^2 u  - \frac{a}{|x|^{2p}} |u|^{p-2} u  
=  \lambda f(x, u) + \mu g(x, u),  \quad   \text{in } \Omega    \\
   u = \Delta u = 0,  \quad   \text{on }  \partial \Omega.
\end{gather*}
\end{remark}

\section{Preliminaries}

Let $X$ be the space $W^{2, p}(\Omega) \cap W_0^{1, p}(\Omega)$
endowed with the norm
 $$ 
\|u\| = \Big(\int_{\Omega} |\Delta u|^p dx\Big)^{1/p}. 
$$
We recall Rellich inequality \cite{D},  which says that
\begin{equation}\label{eq2.1}
 \int_{\Omega} \frac{|u(x)|^p}{|x|^{2p}} dx \le \frac{1}{H}  \int_{\Omega}
|\Delta u|^p dx,
\end{equation}
 where the best constant is
\begin{equation}\label{eq2.2}
 H = \Big(\frac{(p-1)N(N-2p)}{p^2}\Big)^p.
\end{equation}

Define the functionals $\Phi, \Psi : X \to \mathbb{R}$ by
\begin{equation}\label{eq2.3}
\begin{gathered}
\Phi(u) = \frac{1}{p} \widehat{M}(\|u\|^p) - \frac{a}{p}
\int_{\Omega} \frac{|u(x)|^p}{|x|^{2p}} dx, \\
 \Psi(u) = \int_{\Omega} \left[F(x, u(x)) + \frac{\mu}{\lambda}
G(x, u(x))\right] dx,
\end{gathered}
 \end{equation}
 where
\begin{gather*}
\widehat{M}(t) = \int_0^t M(s) ds, \quad  t \ge 0, \\
F(x, t) = \int_0^t f(x, \xi) d \xi,
\quad G(x, t) = \int_0^t g(x, \xi) d \xi, \quad
 (x, t) \in \Omega \times \mathbb{R}. 
\end{gather*}
In this article, we  assume that the following condition holds, 
\begin{itemize}
\item[(H1)]  $M : [0, +\infty[ \to \mathbb{R}$ is a continuous function.
And there are two positive constants $m_0$, $m_1$ such that
\begin{equation}\label{eq2.4}
m_0 \le M(t) \le m_1, \quad \forall t \ge 0.
\end{equation}
\end{itemize}
It is easy to show that the functionals $\Phi$ and $\Psi$ are well
defined and continuously Gateaux differentiable and whose
derivative are
\begin{equation}\label{eq2.5}
\begin{aligned}
\Phi'(u)(v)
&=  M \Big(\int_{\Omega} |\Delta u(x)|^p dx\Big) 
\int_{\Omega} |\Delta u(x)|^{p-2} \Delta u(x) \Delta v(x) dx \\
 &\quad - a \int_{\Omega} \frac{|u(x)|^{p-2}}{|x|^{2p}} u(x) v(x)dx,
\end{aligned}
 \end{equation}
and
\begin{equation}\label{eq2.6}
 \Psi'(u)(v) = \int_{\Omega}
[f(x, u(x)) + \frac{\mu}{\lambda} g(x, u(x))] v(x) dx,
\end{equation}
for every $u, v \in X$.

Set $p^* = \frac{pN}{N-p}$.  By the Sobolev embedding theorem
there exist a positive constant $c$ such that
$$  
\|u\|_{L^{p^*}(\Omega)} \le c \|u\|, \quad \forall u \in X, 
$$
 where
\begin{equation}\label{eq2.7}
 c:= \pi^{-\frac{1}{2}}
N^{-\frac{1}{p}} \Big(\frac{p-1}{N-p}\Big)^{1-\frac{1}{p}}
\Big[\frac{\Gamma(1+\frac{N}{2})
\Gamma(N)}{\Gamma(\frac{N}{p})
\Gamma(N+1-\frac{N}{p})}\Big]^{1/N},
\end{equation}
 see, for instance, \cite{T}.  Fixing $q \in [1, p^*)$, again from
the Sobolev embedding theorem, there exists a positive constant
$c_q$ such that
\begin{equation}\label{eq2.8}
\|u\|_{L^q(\Omega)} \le c_q \|u\|, \quad \forall u \in X.
\end{equation}
Thus, the embedding  $X \hookrightarrow L^q(\Omega)$  is compact.
By \eqref{eq2.7}, as a simple consequence of H\"older's
inequality,  one has the  upper bound
\begin{equation}\label{eq2.9}
c_q \le  \pi^{-\frac{1}{2}} N^{-1/p}
\Big(\frac{p-1}{N-p}\Big)^{1-\frac{1}{p}}
\Big[\frac{\Gamma(1+\frac{N}{2})
\Gamma(N)}{\Gamma(\frac{N}{p})
\Gamma(N+1-\frac{N}{p})}\Big]^{1/N}
|\Omega|^{\frac{p^*-q}{p^* q}},
\end{equation}
 where $|\Omega|$ denotes the Lebesgue measure of the open set
$\Omega$.

Our main tools is an infinitely many critical points theorem
\cite{Bo}  which is recalled below.

\begin{theorem}\label{thm:2.1}
  Let $X$ be a reflexive real
Banach space; $\Phi, \Psi  : X \to \mathbf{R}$ be two
 Gateaux differentiable functionals such that $\Phi$ is  sequentially
 weakly lower semicontinuous,
 strongly continuous, and coercive and $\Psi$ is  sequentially weakly 
upper semicontinuous.
  For every $r > \inf_X \Phi$, let us put
\begin{gather*}
\varphi(r) = \inf _{u \in \Phi^{-1}(]-\infty, r[)}
\frac{\sup _{v \in \Phi^{-1}(]-\infty, r[)} \Psi(v) -
\Psi(u)}{r - \Phi(u)} ,\\
\gamma = \liminf _{r \to + \infty} \varphi(r), \quad
 \delta = \liminf _{r \to (\inf_X \Phi)^+} \varphi(r). 
\end{gather*}
 Then, one has
 \begin{itemize}
\item[(i)]   If $\gamma < + \infty$ then, for each $\lambda \in
]0, \frac{1}{\gamma}[$,
 the following alternative holds: either the functional $\Phi - \lambda \Psi$ 
has a global minimum,  or there exists a sequence $\{u_n\}$ of critical points 
(local minima) of $\Phi - \lambda \Psi$ such that
 $\lim_{n \to + \infty} \Phi(u_n) = + \infty$.

\item[(ii)]  If $\delta < + \infty$ then, for each 
$\lambda \in ]0, \frac{1}{\delta}[$,
 the following alternative holds: either there exists a global minimum of 
$\Phi$ which is a local minimum of
 $\Phi - \lambda \Psi$, or there exists a sequence $\{u_n\}$ of pairwise 
distinct critical points (local minima)
 of $\Phi - \lambda \Psi$, with  $\lim_{n \to + \infty} \Phi(u_n) = \inf_X \Phi$, 
which weakly converges to a global minimum of $\Phi$.
\end{itemize}
\end{theorem}

\section{Main results}

Pick $s > 0$ such that $B(0, s) \subset \Omega$, where $B(0, s)$
denotes the ball with center at $0$ and radius of $s$.  Let
\begin{equation}\label{eq3.1}
L = \frac{2 \pi^{N/2}}{\Gamma \left(\frac{N}{2}\right)}
  \int_{\frac{s}{2}}^s |\frac{12 (N+1)}{s^3} r
  - \frac{24 N}{s^2} + \frac{9(N - 1)}{s} \frac{1}{r}|^p
  r^{N-1} dr.
\end{equation}

\begin{theorem}\label{thm:3.1}
 Suppose that  {\rm (H1)} and $0 < a < m_0 H$ hold (with $H$ is as in \eqref{eq2.2}).
Also assume
\begin{itemize}
\item[(H2)]  $f \in C(\overline{\Omega} \times \mathbb{R})$, and
$F(x, t) \ge 0$ for every $(x, t) \in \Omega \times [0, +\infty[$;

\item[(H3)]   There exists $s > 0$ as considered in \eqref{eq3.1}
such that, if we put
 $$ 
\alpha := \liminf _{t \to + \infty}
\frac{ \sup_{\|\xi\|_{L^q(\Omega)} \le t}  \int_{\Omega} F(x, \xi)
dx}{t^p}, \quad  \beta := \limsup _{t \to + \infty}
\frac{\int_{B(0,s/2)} F(x, t/h) dx}{t^p},
$$
 one has
\begin{equation}\label{eq3.2}
 \alpha < R \beta,
 \end{equation}
where $R = \frac{(m_0 H - a) h^p}{m_1 H L c_q^p}$ 
(constants $h > 1$,  $c_q$ and $L$ are as in \eqref{eq2.8} and \eqref{eq3.1}, 
respectively).
\end{itemize}
Then, for every $\lambda \in \Lambda := \frac{m_0 H - a}{p H
c_q^p} \left]\frac{1}{R \beta}, \frac{1}{\alpha}\right[$ and for
every $g \in C(\overline{\Omega} \times \mathbb{R})$
 such that 
\begin{itemize}
\item[(H4)]
$G(t, u) \ge 0$, for all $(t,u) \in  \overline{\Omega} 
\times [0, + \infty[$, and
 $$ 
 G_{\infty} := \limsup_{t \to +\infty} 
\frac{ \sup_{\|\xi\|_{L^q(\Omega)} \le t}  \int_{\Omega}
 G(x, \xi) dx}{t^p},  
$$
\end{itemize}
if we put
\begin{equation}\label{eq3.3}
\mu^* =\begin{cases} 
\frac{m_0 H - a - p H c_q^p \alpha \lambda}{p H c_q^p G_{\infty}},
& G_{\infty} > 0,\\
+ \infty, & G_{\infty} = 0,
\end{cases}
\end{equation}
then  \eqref{eq1.1} possesses an unbounded sequence of weak
solutions in $X$ for every $\mu \in J:= [0, \mu_*[$.
\end{theorem}

\begin{proof}   
Our aim is to apply part (i) of Theorem \ref{thm:2.1}.
  Let $\Phi, \Psi$ be the functionals defined in
\eqref{eq2.3}.  From the above, we know that the Gateaux
derivative of  $\Phi$ and $\Psi$ are given by \eqref{eq2.5} and
\eqref{eq2.6}, respectively.  By \eqref{eq2.1}, it follows that
\begin{equation}\label{eq3.4}
 \frac{m_0 H - a}{p H} \|u\|^p \le
\Phi(u) \le \frac{m_1}{p} \|u\|^p, \quad u \in X,
\end{equation}
 which implies that $\Phi$ is coercive.    Moreover, from the
weakly lower semicontinuity of norm, and the monotonicity and
continuity of $\widehat{M}$,  we  known that $\Phi$ is
sequentially weakly lower semicontinuous.   The functional $\Psi$
has compact derivative, hence it is sequentially weakly upper
semicontinuous.

By \eqref{eq2.8} and \eqref{eq3.4}, we obtain
\begin{equation}
\begin{aligned}
\Phi^{-1}(]- \infty, r[) 
&= \{u \in X  : \Phi(u) < r\} \\
&\subset \big\{u \in X : \frac{m_0 H - a}{pH}
 \|u\|^p < r\big\} \\
& \subset \big\{u \in X : \|u\|_{L^q(\Omega)} 
<  c_q \big(\frac{pHr}{m_0H-a}\big)^{1/p}\big\}.
\end{aligned}\label{eq3.5}
\end{equation}

Note that $\Phi(0) = 0$ and $\Psi(0) = 0$.  For every $r > 0$, we
obtain by \eqref{eq3.5} that
\begin{align*}
 \varphi(r) 
&= \inf _{u \in \Phi^{-1}(]-\infty, r[)} \frac{\sup _{v \in
\Phi^{-1}(]-\infty, r[)} \Psi(v) - \Psi(u)}{r - \Phi(u)} \\
&  \le \frac{\sup _{v \in \Phi^{-1}(]-\infty, r[)}
 \Psi(v)}{r}  \\
& \le \frac{\sup_{\|\xi\|_{L^q(\Omega)} \le l} \int_{\Omega}  F(x, \xi) dx}{r}
  + \frac{\mu}{\lambda}
\frac{\sup_{\|\xi\|_{L^q(\Omega)} \le l} \int_{\Omega}  G(x,
\xi) dx}{r},
\end{align*}
 where $l = c_q \big(\frac{pHr}{m_0H-a}\big)^{1/p}$.

Let $\{\sigma_n\}$ be a  sequence of positive numbers such that
$\sigma_n \to + \infty$ and
\begin{equation}
\begin{aligned}
&\lim _{n \to + \infty}  \frac{ \sup_{\|\xi\|_{L^q(\Omega)}
\le \sigma_n}  \int_{\Omega} F(x, \xi) dx}{\sigma_n^p} \\
& =  \liminf _{t \to + \infty} \frac{
\sup_{\|\xi\|_{L^q(\Omega)} \le t}  \int_{\Omega} F(x, \xi)
dx}{t^p}.
\end{aligned}\label{eq3.6}
\end{equation}
 Let $r_n = \frac{m_0 H - a}{pH c_q^p} \sigma_n^p$ for all 
$n \in \mathbb{N}$.  From (H3), (H4) and \eqref{eq3.6},  we obtain 
\begin{equation}
\begin{aligned}
\gamma 
& = \liminf _{r \to + \infty} \varphi(r)
  \le \liminf _{n \to +
 \infty} \varphi(r_n) \\
& \le \frac{p H c_q^p}{m_0 H - a} \lim  _{n \to + \infty}
\frac{\sup_{\|\xi\|_{L^q(\Omega)} \le \sigma_n}  \int_{\Omega}
 F(x, \xi)dx}{\sigma_n^p} \\
 & \quad + \frac{\mu}{\lambda} \frac{p H c_q^p}{m_0 H - a} \limsup _{n \to + \infty}
\frac{\sup_{\|\xi\|_{L^q(\Omega)} \le \sigma_n}  \int_{\Omega}
 G(x, \xi) dx}{\sigma_n^p} \\
 &  \le  \frac{p H c_q^p}{m_0 H - a}
 \left(\alpha + \frac{\mu}{\lambda} G_{\infty}\right) < + \infty.
\end{aligned}\label{eq3.7}
\end{equation}
 By \eqref{eq3.3} and \eqref{eq3.7}, we easily check that

\begin{equation}\label{eq3.8}
 \gamma < \begin{cases}
\frac{p H c_q^p}{m_0 H - a}
 \big(\alpha + \frac{\mu^*}{\lambda} G_{\infty}\big) =
 \frac{1}{\lambda}, &  G_{\infty} > 0,\\
\frac{p H c_q^p}{m_0 H - a} \alpha <
 \frac{1}{\lambda}, &  G_{\infty} = 0
\end{cases}
\end{equation}
 From the definition of $\Lambda$ and \eqref{eq3.2}, we have that
$\Lambda \subset ]0, \frac{1}{\gamma}[$.


In the following, we  claim that the functional 
$\Phi - \lambda \Psi$  for $\lambda \in \Lambda$ is unbounded from below.
Indeed, since 
$\frac{1}{\lambda} < \frac{p H c_q^p}{m_0 H - a} R \beta 
= \frac{p h^p}{m_1 L} \beta $, there exists a
sequence $\{\tau_n\}$ of positive numbers and $\eta > 0$ such that
$\tau_n \to + \infty$ and
\begin{equation}\label{eq3.9}
\frac{1}{\lambda} < \eta < \frac{p h^p}{ m_1 L}
\frac{\int_{B(0,s/2)} F(x, \tau_n/h)
dx}{\tau_n^p},
\end{equation}
for $n$ large enough.

Let $h > 1$ be as in $R$ (\eqref{eq3.2}),  we consider a sequence
$\{w_n\}$ in $X$ defined by setting
\begin{equation}\label{eq3.10}
  w_n(x) =  \begin{cases}
0,  &  x \in \overline{\Omega} \setminus B(0, s),\\
 \frac{\tau_n}{h} \left(\frac{4}{s^3} \rho^3 - \frac{12}{s^2} \rho^2 
+ \frac{9}{s} \rho - 1\right), &
 x \in B(0, s) \setminus B (0, \frac{s}{2}),\\
\frac{\tau_n}{h}, & x \in B (0, \frac{s}{2})
\end{cases}
\end{equation}
 with $\rho = \operatorname{dist}(x, 0) = \sqrt{\sum_{i=1}^N x_i^2}$.
  Clearly $w_n \in X$.  A direct calculation shows
\[
\frac{\partial w_n(x)}{\partial x_i}  
= \begin{cases}
0,  &  x \in (\overline{\Omega} \setminus B(0, s)) \cap B (0, \frac{s}{2}),\\
 \frac{\tau_n}{h} \left(\frac{12 \rho x_i}{s^3} - \frac{24 x_i}{s^2} 
+ \frac{9 x_i}{s \rho}\right), &
 x \in B(0, s) \setminus B (0, \frac{s}{2})
\end{cases}
\]
 and
 \begin{equation}\label{eq3.11}
\frac{\partial^2 w_n(x)}{\partial x_i^2}  
= \begin{cases}
0,  &  x \in (\overline{\Omega} \setminus B(0, s)) \cap B 
(0,\frac{s}{2}),\\
 \frac{\tau_n}{h} \left(\frac{12(x_i^2 + \rho^2)}{s^3 \rho} 
- \frac{24}{s^2} + \frac{9(\rho^2 - x_i^2)}{s \rho^3}\right), &
 x \in B(0, s) \setminus B (0,\frac{s}{2}).
\end{cases}
\end{equation}
  By \eqref{eq3.11} and \eqref{eq3.1} we have
\begin{align*}
\sum _{i=1}^N \frac{\partial^2 w_n(x)}{\partial x_i^2}  =
\begin{cases}
0,  &  x \in (\overline{\Omega} \setminus B(0, s)) \cap B (0,\frac{s}{2}),\\
 \frac{\tau_n}{h} \left(\frac{12 \rho (N+1)}{s^3} - \frac{24 N}{s^2} + \frac{9(N - 1)}{s \rho}\right), &
 x \in B(0, s) \setminus B (0,\frac{s}{2}),
\end{cases}
\end{align*}
and
\begin{equation}
\begin{aligned}
&\int_{\Omega} |\Delta w_n(x)|^p dx \\
& =
 \left(\frac{\tau_n}{h}\right)^p \frac{2 \pi^{N/2}}{\Gamma \left(\frac{N}{2}\right)}
  \int_{\frac{s}{2}}^s |\frac{12 (N+1)}{s^3} r
  - \frac{24 N}{s^2} + \frac{9(N - 1)}{s} \frac{1}{r}|^p
  r^{N-1} dr 
  = \frac{L}{h^p}  \tau_n^p.
\end{aligned}\label{eq3.12}
\end{equation}
Thus,  we have by \eqref{eq2.4} and \eqref{eq3.12}
 that
\begin{equation}
\begin{aligned}
\Phi(w_n) &= \frac{1}{p} \widehat{M}(\|w_n\|^p) - \frac{a}{p}
\int_{\Omega} \frac{|w_n(x)|^p}{|x|^{2p}} dx \le \frac{1}{p}
\widehat{M} \Big( \int_{\Omega} |\Delta w_n(x)|^p dx\Big) \\
 & \le \frac{ m_1 L}{p h^p}\tau_n^p.
\end{aligned}\label{eq3.13}
\end{equation}
 On the other hand, by (H4), one has
\begin{equation}\label{eq3.14}
 \Psi(w_n) = \int_{\Omega} \left[F(x,
 w_n(x) + \frac{\mu}{\lambda} G(x, w_n(x))\right] dx 
\ge \int_{B (0,s/2)}  F(x, \tau_n/h) dx.
\end{equation}
 Hence, it follows from \eqref{eq3.13}, \eqref{eq3.14} and \eqref{eq3.9} that
\[
\Phi(w_n) - \lambda \Psi(w_n) 
\le \frac{m_1 L}{p h^p} \tau_n^p
- \lambda \int_{B (0,s/2)}  F(x, \tau_n) dx \\
  < \frac{m_1 L}{p h^p}
 (1-\lambda \eta)\tau_n^p
\]
 for every $n \in \mathbb{N}$ large enough, which leads
 to $\lim_{n \to +\infty} (\Phi(w_n) - \lambda \Psi(w_n)) = -
 \infty$.

The alternative of Theorem \ref{thm:2.1} case (i) assures the
existence of unbounded sequence $\{u_n\}$ of critical points of
the functional $\Phi - \lambda \Psi$.  This completes the proof in
view of the relation between the critical points of $\Phi -
\lambda \Psi$ and the weak solutions of problem \eqref{eq1.1}.
\end{proof}


\begin{remark} \label{rmk3.1} \rm
If $\alpha < \infty$, $\beta > 0$, and  $h > 1$ large enough, then
\eqref{eq3.2} holds.
\end{remark}


In the following, arguing in a similar way, but applying case (ii)
of Theorem \ref{thm:2.1}, we can establishes the  existence of
infinitely many solutions to \eqref{eq1.1} converging at
zero.


\begin{theorem}\label{thm:3.2}
  Suppose that {\rm (H1)} and $0 < a <
m_0 H$  hold (with $H$ is as in \eqref{eq2.2}). Also assume 
\begin{itemize}
\item[(H5)]  $f \in C(\overline{\Omega} \times \mathbb{R})$, and
there exists $c > 0$ such that $F(x, t) \ge 0$ for every $(x, t)
\in \Omega \times [0, c]$;

\item[(H6)]  There exists $s > 0$ as considered in \eqref{eq3.1}
such that, if we put
$$  
\alpha^0 := \liminf _{t \to
0^+} \frac{\sup_{\|\xi\|_{L^q(\Omega)} \le t} \int_{\Omega}  F(x,
\xi) dx}{t^p}, \quad  \beta^0 := \limsup _{t \to 0^+}
\frac{\int_{B(0,s/2)} F(x, t/h) dx}{t^p}, 
$$
 one has
\begin{equation}\label{eq3.15}
 \alpha^0 < R \beta^0,
\end{equation}
 where $R = \frac{(m_0 H - a) h^p}{m_1 H L
c_q^p}$ (constants $h > 1$,  $c_q$ and $L$ are as in
\eqref{eq2.8} and \eqref{eq3.1}, respectively). 
\end{itemize}
Then, for every
$\lambda \in \Lambda^0 := \frac{m_0 H - a}{p H c_q^p}
]\frac{1}{R \beta^0}, \frac{1}{\alpha^0}[$ and for
every $g \in C(\overline{\Omega} \times \mathbb{R}) $ such that
\begin{itemize}
\item[(H7)]  $G(t, u) \ge 0$,  for all $(t, u)
\in  \overline{\Omega} \times [0, c]$  and
 $$  
G_0 := \limsup_{t \to 0^+} \frac{
 \sup_{\|\xi\|_{L^q(\Omega)} \le t} \int_{\Omega} G(x, \xi)
dx}{t^p},  
$$
\end{itemize}
if we put
\begin{equation}\label{eq3.16}
\mu_* =\begin{cases}
\frac{m_0 H - a - p H c_q^p \alpha \lambda}{p
H c_q^p G_0}, &
G_0 > 0,\\
+ \infty, & G_0 = 0,
\end{cases}
\end{equation}
then  \eqref{eq1.1} admits a sequence $\{u_n\}$ of weak solutions
such that $u_n \to 0$ strongly in $X$ for every $\mu \in J:= [0,
\mu_*[$.
\end{theorem}

\begin{proof}   
We take $\Phi$ and $\Psi$ be as in \eqref{eq2.3}.
First, note that $\min _X \Phi = \Phi(0) = 0$.  Let
$\{\sigma_n\}$ be a sequence of positive numbers such that
$\sigma_n \to 0^+$,  and putting  $r_n = \frac{m_0 H - a}{pH
c_q^p} \sigma_n^p$.  Similarly as above,  we get
\begin{equation}
\begin{aligned}
\delta & := \liminf _{r \to 0^+} \varphi(r) \le \liminf_{n \to +
 \infty} \varphi(r_n) \\
&  \le \frac{p H c_q^p}{m_0 H - a}
 \left(\alpha^0 + \frac{\mu}{\lambda} G_0\right) < + \infty.
\end{aligned}\label{eq3.17}
\end{equation}
 From \eqref{eq3.16} and \eqref{eq3.17},  we have that $\Lambda^0
\subset ]0, \frac{1}{\delta}[$.  Now, for 
$\lambda \in \Lambda^0$, we claim that $\Phi - \lambda \Psi$ does not have a
local minimum at zero. Indeed,  let $\{\tau_n\}$ be a sequence of
positive numbers in $]0, \tau[$ and $\eta > 0$ such that 
$\tau_n \to 0^+$ and
 $$ 
\frac{1}{\lambda} < \eta  < \frac{p h^p}{m_1 L}
\frac{\int_{B(0,s/2)} F(x, \tau_n/h)dx}{\tau_n^p},
 $$
 for $n$ large enough. Let $\{w_n\}$ be the
sequence in $X$ defined in \eqref{eq3.10}.  
By (H7), one has that \eqref{eq3.14} holds.  
Thus, from \eqref{eq3.13}, \eqref{eq3.14}
and \eqref{eq3.9} we obtain that
 $$ 
\Phi(w_n) - \lambda \Psi(w_n) < \frac{m_1 L }{p h^p} (1-\lambda
\eta)\tau_n^p < 0 = \Phi(0) - \lambda \Psi(0) 
$$
 for every $n \in \mathbb{N}$ large enough. This
 together with the fact that $\|w_n\| \to 0$  shows that 
$\Phi - \lambda \Psi$ has not a local minimum at zero.  The conclusion
 follows from the alternative of Theorem \ref{thm:2.1} case (ii).
\end{proof}

\subsection*{Acknowledgments}
The authors want to than the referees for their valuable and helpful
suggestions and comments that improved this article.
 This work is supported by the Natural Science Foundation of Jiangsu Province 
(BK2011407),  and by the Natural Science Foundation of China (11571136 and 11271364).


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\end{document}