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\begin{document}

\title
{MULTIPLICITY RESULTS FOR KIRCHHOFF TYPE ELLIPTIC PROBLEMS WITH
HARDY POTENTIAL}
\author[M. Bagheri] {Mahnaz Bagheri}

\author[G.A. Afrouzi]
{Ghasem A. Afrouzi}
\address{Department of Mathematics, Faculty of
Mathematical Sciences, University of Mazandaran, Babolsar, Iran}
\email{ afrouzi@umz.ac.ir}\email{m.bagheri\_umz@yahoo.com}
\keywords{$p$-biharmonic type operators; Navier
condition; Hardy potential; Variational methods; Critical point theory.\\
\hspace*{.3cm} {\it 2010 Mathematics Subject Classifications}:
35J40, 58E05.}
\begin{abstract}
In this paper, we are concerned with the existence of solutions
for fourth-order Kirchhoff type elliptic problems with Hardy
potential. In fact, employing a consequence of the local minimum
theorem due to Bonanno and mountain pass theorem we look into the
existence results for the problem under algebraic conditions with
the classical Ambrosetti-Rabinowitz (AR) condition on the
nonlinear term. Furthermore, by combining two algebraic conditions
on the nonlinear term using two consequences of the local minimum
theorem due to Bonanno we ensure the existence of two solutions,
applying the mountain pass theorem given by Pucci and Serrin we
establish the existence of third solution for our problem.
\end{abstract}

\maketitle

\numberwithin{equation}{section}

\section{Introduction}
Consider the following $p$-biharmonic equation with Hardy
potential of fourth-order Kirchhoff-type elliptic problem
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
\displaystyle 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) &\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N} (N\geq 3)$
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 the $p$-biharmonic operator of fourth order,
$\lambda$ is nonnegative parameter, $M:[0,+\infty)\rightarrow
\mathbb{R}$ is continuous function and $f:\Omega \times \mathbb{R}
\rightarrow \mathbb{R}$ is an $L^2$-Carath\'{e}odory function.

The problem \eqref{e1.1} is related to the stationary problem
  \begin{equation}
  \label{2}
  \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}for $0 < x < L$, $t\geq0$, where $u = u(x, t)$ is the lateral displacement at the space coordinate
$x$ and the time $t$, $E$ the Young modulus, $\rho$ the mass
density, $h$ the cross-section area, $L$ the length and $\rho_0$
the initial axial tension, proposed by Kirchhoff \cite{K} as an
extension of the classical D'Alembert's wave equation for free
vibrations of elastic strings. Kirchhoff model can also be used
for describing the dynamics of an axially moving string. In recent
years, axially moving string-like continua such as wires, belts,
chains, band-saws have been subjects of the study of researchers
(see \cite{SP}). Similar nonlocal problems also model several
physical and biological systems where $u$ describes a process that
depends on the average of itself, for example, the population
density. Problems of Kirchhoff-type have been widely investigated,
we refer the reader to papers
\cite{new4,new3,new2,GHK,HZ,MolRad1,MolRad2,Rb,S} and the
references therein.

Fourth-order equations can describe the static form change of beam
or the sport of rigid body. In \cite{LM}, Lazer and McKenna have
pointed out that this type of nonlinearity furnishes a model to
study travelling waves in suspension bridges. Since then more
nonlinear biharmonic equations and $p$-biharmonic equations have
been studied. Existence and multiplicity of solutions of nonlinear
fourth order differential equations have been deserved a great
deal of interest, for instance see
\cite{CLL,CL,CM,HL,LT2,LT3,LS,MR,LC}.

 Recently, combined problems of
Kirchhoff-type with $p$-biharmonic operator have been widely
investigated such that many researchers have discussed the
existence of at least one solution, or multiple solutions, or even
many solutions for such problems with different method. we refer
the reader to the papers \cite{FKH,HKS,MHTT,XZCL} and references
therein. For example, in \cite{MHTT} employing variational methods
and critical point theory, Massar et al. ensured the existence of
infinitely many solutions the following perturbed $p$-biharmonic
Kirchhoff-type problem
\begin{equation*}
\left\{\begin{array}{ll}
\displaystyle \Delta(|\Delta u|^{p-2}\Delta u)-\left[ M(\int_\Omega
|\nabla u|^pdx)\right] ^{p-1}\Delta _pu+\rho|u|^{p-2}u=
\lambda f(x,u) &\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega,
\end{array}\right.
\end{equation*}
where $p>\max \{1,\frac{N}{2}\}$ , $\lambda>0$ is a real number,
$\Omega \subset\mathbb{R}^N(N\geq1)$ is a bounded smooth domain,
$\rho>0$ and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
 is an continues function and $M:[0,+\infty)
 \rightarrow \mathbb{R}$ is continuous function, while in \cite{FKH} using variational methods and critical point theory, multiplicity results of
nontrivial and nonnegative solutions for the same problem were
established. Xiu et al. in \cite{XZCL} by employing variational
method studied multiplicity of solutions for the following
$p$-biharmonic equation
\begin{equation*}
\left\{\begin{array}{ll}
\displaystyle(a+b\int_{\Omega}\left( |\Delta u|^{p}+|u|^p\right) dx)(\Delta _{p}^{2}u+
|u|^{p-2}u)=h(x)|u|^{r-2}u+H(x)|u|^{q-2}u+g(x), &\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega,
\end{array}\right.
\end{equation*}
where $1<p<\frac{N}{2}$, $\Delta_{p}^{2}u=\Delta (|\Delta u|^{p-2}
\Delta u)$ is the $p$-biharmonic operator of fourth order,
$\Omega\subset \mathbb{R}^N$ is an unbounded domain, and $h(x),
H(x)$ and $g(x)$ are nonnegative
functions with sufficient conditions.

On the other hand, singular boundary value problems arise in the
context of chemical heterogeneous catalysts and chemical catalyst
kinetics, in the theory of heat conduction in eletirically
conducting materials, singular minimal surfaces, as well as in the
study of non-Newtonian fluids and boundary layer phenomena for
viscous fluids. Furthermore, nonlinear singular elliptic equations
are also encountered in glocial advance, in transport of coal
slurries down conveyor belts and in several other geophysical and
industrial contents. For the use of singular problem in the
mathematical literature, see \cite{HL,L,PP}. In recent years, some
interesting results for singular $p$-biharmonic equation of
Kirchhoff-type were obtained. For instance, Xu and Bai in
\cite{XB} by using critical point theory, discussed the existence
of infinitely many weak solutions for similar problem to
\eqref{e1.1}.

In the present article, we establish the existence of two
solutions for the problem \eqref{e1.1} using a consequence of the
local minimum theorem due to Bonanno and mountain pass theorem
under some algebraic conditions with the classical
Ambrosetti-Rabinowitz (AR) condition on the nonlinear term.
Moreover, by combining two algebraic conditions on the nonlinear
term employing two consequences of the local minimum theorem due
to Bonanno we guarantee the existence of two solutions, applying
the mountain pass theorem given by Pucci and Serrin (\cite{PS}) we
establish the existence of third solution for the problem
\eqref{e1.1}.

 For a through on the subject, we also refer the reader to
\cite{D, HFK, HFSC}.
\section{Preliminaries}
For a given nonempty set $X$, and two functionals $\Phi,\Psi:
X\rightarrow\mathbb{R}$, we define the following functions
$$\beta(r_1,r_2)=\inf_{v\in\Phi^{-1}(r_1,r_2)}\frac{\sup_{u\in
\Phi^{-1}(r_1,r_2)}\Psi(u)-\Psi(v)}{r_2-\Phi(v)},
$$$$
\rho_1(r_1,r_2)=\sup_{v\in\Phi^{-1}(r_1,r_2)}\frac{\Psi(v)
-\sup_{u\in\Phi^{-1}(-\infty,r_1]}\Psi(u)}{\Phi(v)-r_1}$$ for all
$r_1, r_2\in\mathbb{R}, r_1<r_2$, and
$$\rho_2(r)=\sup_{v\in\Phi^{-1}(r,+\infty)}\frac{\Psi(v)-
\sup_{u\in\Phi^{-1}(-\infty,r]}\Psi(u)}{\Phi(v)-r}
$$for all $r\in\mathbb{R}$.

\begin{theorem} {\cite[Theorem 5.1]{B}}\label{the2.1}
Let $X$ be a real Banach space; $\Phi:X\rightarrow\mathbb{R}$ be a
sequentially weakly lower semicontinuous, coercive and
continuously G\^{a}teaux differentiable function whose G\^{a}teaux
derivative admits a continuous inverse on $X^*$,
$\Psi:X\rightarrow \mathbb{R}$ be a continuously G\^{a}teaux
differentiable function
whose G\^{a}teaux derivative is compact. Assume that there are
$r_1, r_2\in\mathbb{R}, r_1<r_2$, such that
$$\beta(r_1,r_2)<\rho_1(r_1,r_2).$$
Then, setting $I_\lambda:=\Phi-\lambda\Psi$, for each $\lambda\in
(\frac{1}{\rho_1(r_1,r_2)},\frac{1}{\beta(r_1,r_2)})$ there is
$u_{0,\lambda}\in\Phi^{-1}(r_1,r_2)$ such that $I_{\lambda}
(u_{0,\lambda})\leq I_{\lambda}(u)$ for all
$u\in\Phi^{-1}(r_1,r_2)$ and $I^{'}_{\lambda}(u_{0,\lambda})=0.$
\end{theorem}

\begin{theorem} {\cite[Theorem 5.3]{B}}\label{the2.2}
Let $X$ be a real Banach space; $\Phi:X\rightarrow\mathbb{R}$
be a continuously G\^{a}teaux differentiable function whose
G\^{a}teaux derivative admits a continuous inverse on $X^*$,
$\Psi:X\rightarrow \mathbb{R}$  be a continuously G\^{a}teaux
differentiable function whose G\^{a}teaux derivative is compact.
Fix $\inf _X\Phi<r<\sup_X\Phi$ and assume that
$$\rho_2(r)>0,$$and for each $\lambda>\frac{1}{\rho_2(r)}$, the functional $I_\lambda
:=\Phi-\lambda\Psi$ is coercive. Then for each $\lambda\in]\frac{1}
{\rho_2(r)},+\infty[$ there is $u_{0,\lambda}\in\Phi^{-1}(r,+\infty)$
such that $I_{\lambda}(u_{0,\lambda})\leq I_{\lambda}(u)$ for all
$u\in\Phi^{-1}(r,+\infty)$ and $I^{'}_{\lambda}(u_{0,\lambda})=0.$
\end{theorem}
Let $X$ denote the space $W^{2,p}(\Omega) \cap
W_{0}^{1,p}(\Omega)$ endowed with the norm$$ \|u\|=\left(
\int_{\Omega}|\Delta u|^{p}dx\right) ^\frac{1}{p} .$$ We recall
the following Rellich inequality \cite{DH}, which says that, for
each $u\in X$,
\begin{equation}\label{Rel}
\int_{\Omega}\frac{|u(x)|^{p}}{|x|^{2p}}dx\leq
\frac{1}{H}\int_{\Omega}|\Delta u|^{p}
\end{equation}
where the best constant is
\begin{equation}\label{e2.2}
H=\Big(\frac{(p-1)N(N-2p)}{p^{2}}\Big)^{p}.
\end{equation}
Now, let $M:\mathbb{R}^{+} \rightarrow\mathbb{R}^{+}$ be a continuous function such that
there exists two positive constants $m_0$ and $m_1$ such that
$$m_0\leq M(t)\leq m_1,$$for all $t\in\mathbb{R}^{+}$ and $f:\Omega\times \mathbb{R}
\rightarrow\mathbb{R}$ be an $L^2$-Carath\'{e}odory function,
namely, $x\mapsto f(x,t)$ is continuous for almost every $x\in
\Omega$, and for every $s>0$ there exists a function $l_s\in
L^2(\Omega)$ such that
$$\sup_{|t|\leq s}|f(x,t)|\leq l_s(x)$$
for almost every $x\in \Omega$.
 Set $p^*=\frac{pN}{N-p}$. By the
Sobolev embedding theorem there exist a positive constant $c$ such
that
$$\|u\|_{L^{p^*}(\Omega)}\leq c\|u\|,\quad\forall u\in X,$$ where
\begin{equation*}
c:=\pi^{-\frac{1}{2}}N^{-\frac{1}{p}}\left( \frac{p-1}{N-p}\right)^{1-\frac{1}{p}}
\left[\frac{\Gamma(1+\frac{N}{2})\Gamma(N)}{\Gamma(\frac{N}{p})
\Gamma(N+1-\frac{N}{p})} \right]^{\frac{1}{N}},
\end{equation*}
see, \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{e2.3}\|u\|_{L^q(\Omega)}\leq c_q\|u\|,\quad\quad\forall u\in X.
\end{equation}Thus the embedding $X\hookrightarrow L^q(\Omega)$ is compact. By Holder inequality, one has the upper bound
$$c_q\leq\pi^{-\frac{1}{2}}N^{-\frac{1}{p}}\left( \frac{p-1}{N-p}\right)^{1-\frac{1}{p}}
\left[\frac{\Gamma(1+\frac{N}{2})\Gamma(N)}{\Gamma(\frac{N}{p})
\Gamma(N+1-\frac{N}{p})} \right]^{\frac{1}{N}}|\Omega|^{\frac{p^*-q}{p^*q}},
$$ where $|\Omega|$ denote the Lebesgue measure of the open set $\Omega$.

Fixing the real parameter $\lambda,$ a function
$u \in W^{1,p}(\Omega)$ is said to be a weak solution of
$\eqref{e1.1}$ if for all $v \in {W^{1,p}(\Omega)},$
\begin{align*}
&M\Big(\int_{\Omega}|\Delta u| ^{p}dx\Big)\int_{\Omega}|\Delta u|
^{p-2}\Delta u(x)\Delta v(x)dx-a\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^{2p}}u(x)v(x)dx
\\&=\lambda\int_{\Omega}f(x,u(x))v(x)dx.
\end{align*}

By assumption $m_0>\frac{a}{H}$, we state the following
proposition which we need in the proofs of our main result.
\begin{proposition}\label{pro2.3}
Let $T:X\rightarrow X$ be the operator defined by
$$
T(u)h=M\Big(\int_{\Omega}|\Delta u| ^{p}dx\Big)\int_{\Omega}|\Delta u|
^{p-2}\Delta u(x)\Delta h(x)dx-a\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^{2p}}u(x)h(x)dx
$$ for every $u,v\in X$. Then, $T$ admits a continuous inverse on $X^*$.
\end{proposition}
\begin{proof}
Since
\begin{eqnarray*}
T(u)h&=&M\Big(\int_{\Omega}|\Delta u| ^{p}dx\Big)\int_{\Omega}|\Delta u|
^{p-2}\Delta u(x)\Delta h(x)dx-a\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^{2p}}u(x)h(x)dx\\
&\geq& m_0\|u\|^p-\frac{a}{H}\|u\|^p\\
&= &\left( m_0-\frac{a}{H}\right) \|u\|^p,
\end{eqnarray*}
and since $,m_0>\frac{a}{H}$, this follows that $T$ is coercive.
Taking into account (2.2) of \cite{SJ} for $p>1$ there exists a positive constant
$C_p$ such that if $p\geq 2$, then$$\left\langle |x|^{p-2}x-|y|^{p-2}y,x-y\right\rangle\geq C_p|x-y|^p, $$
 if $1<p<2$, then$$\left\langle |x|^{p-2}x-|y|^{p-2}y,x-y\right\rangle\geq C_p\frac{|x-y|^2}{(|x|+|y|)^{2-p}} $$
 where $\left\langle .,.\right\rangle $ denotes the usual inner product in $\mathbb{R}^N$.Then, we observe that
 $$\left\langle T(u)-T(v),u-v\right\rangle \geq C\|u-v\|^p>0$$for some $C>0$ for every
$u,v\in X$, which means that $T$ is strictly monotone. Furthermore, since $X$ is reflexive,
for $u_n\rightarrow u$ strongly in $X$ as $n\rightarrow+\infty$, one has $T(u_n)\rightarrow T(u)$
weakly in $X^*$ as $n\rightarrow+\infty$. Hence, $T$ is demicontinuous, so by {\cite[Theorem 26.A(d)]{Z}},
 the inverse operator $T^{-1}$ of $T$ exists. $T^{-1}$ is continuous. Indeed, let $(\nu_n)$
 be a sequence of $X^*$ such that $\nu_n\rightarrow \nu$ strongly in $X^*$ as $n\rightarrow
 +\infty$. Let $u_n$ and $u$ in $X$ such that $T^{-1}(\nu_n)=u_n$ and $T^{-1}(\nu)=u$.
 Taking in to account that $T$ is coercive, one has that the sequence $u_n$ is bounded
 in the reflexive space $X$. For a suitable subsequence, we have $u_n\rightarrow\widehat{u}$
  weakly in $X$ as $n\rightarrow +\infty$, which concludes
  $$ \lim_{n\rightarrow +\infty}\langle T(u_n)-T(u),u_n-\widehat{u}
  \rangle=\langle\nu_n-\nu,u_n-\widehat{u}\rangle=0. $$
 Note that if $u_n\rightarrow \widehat{u}$ weakly in $X$ as $n\rightarrow +\infty$ and
 $T(u_n)\rightarrow T(\widehat{u})$ strongly in $X^*$ as $n\rightarrow +\infty$,
 one has $u_n\rightarrow \widehat{u}$ strongly in $X$ as $n\rightarrow +\infty$,
 and since $T$ is continuous, we have $u_n\rightarrow\widehat{u}$ weakly in
 $X$ as $n\rightarrow +\infty$ and $T(u_n) \rightarrow T(\widehat{u})=T(u)$
 strongly in $X^*$ as $n\rightarrow +\infty$. Hence, taking into account that
 $T$ is an injective, we have $u=\widehat{u}$.
\end{proof}
\section{Main results}
In this section, we formulate our main results as follow.

Put $$\widehat{M}(t)=\int_{0}^{t} M(s)ds, \qquad t\geq0
$$and$$F(x,t)=\int_{0}^{t}f(x,\xi)d\xi, \qquad (x,t)\in\Omega\times\mathbb{R}.$$

Choose $s>0$ such that $B(0,s)\subset\Omega$, where $B(0,s)$
denotes the open ball in $\mathbb{R}^N$ of radius s with center at
$0$. Put$$L=\frac{2\pi^{\frac{N}{2}}}{\Gamma
(\frac{N}{2})}\int_{\frac{s}{2}}^s\Big{|}\frac{12r(N+1)}{s^3}-\frac{24N}{s^2}+\frac{9(N-1)}{sr}\Big{|}^p
r^{N-1}dr.$$ For a nonnegative constant $\eta$ and a positive
constant $\delta$ with$$ (m_0H-a)\eta^p\neq m_1HL(c_q\delta)^p
$$we set
$$a_\eta(\delta):=p\frac{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx-
\int_{B(0,\frac{s}{2})}F(x,\delta)dx}{(m_0H-a)\eta^p-m_1HL(c_q\delta)^p}.$$

We now present our main result as follows.
\begin{theorem}\label{the3.1}
Suppose that $0<a<m_0H$ (with H is as in \eqref{e2.2}). Moreover, assume
that there exist a nonnegative constant $\eta_1$ and two positive
constants $\eta_2$ and $\delta$ with
\begin{equation}\label{e3.1}
\frac{1}{c_qL^{\frac{1}{p}}}\eta_1< \delta< \left(
\frac{m_0H-a}{m_1LH} \right)
^{\frac{1}{p}}\frac{\eta_2}{c_q}\end{equation} such that
\begin{enumerate}\item[(A1)] $F(x,t)\geq0$ for each $(x,t)\in(B(0,s)\setminus
B(0,\frac{s}{2}))\times\mathbb{R}$; \item[(A2)]
$a_{\eta_1}(\delta)<a_{\eta_2}(\delta)$; \item[(A3)] there exist
two constants $\xi >p$ and $R>0$ such that
\begin{equation}\label{e3.2}
0<\xi F(x,t)\leq tf(x,t),
\end{equation}
for all $|t|\geq R$ and for all $x\in \Omega$.
\end{enumerate}
Then for each $\lambda\in \left(
\frac{1}{Hc_q^p}\frac{1}{a_{\eta_1}(\delta)},
\frac{1}{Hc_q^p}\frac{1}{a_{\eta_2}(\delta)}\right) $, the problem
$\eqref{e1.1}$ admits at least two nontrivial weak solutions $u_1$
and $u_2$ in $x$, such that
$$\frac{m_0H-a}{m_1Hc_q^p}\eta_1^p<\|u_1\|^p<m_0\eta_2^p$$.
\end{theorem}
\begin{proof}
Our aim is to apply Theorem \ref{the2.1} to the problem \eqref{e1.1}.
Let $\Phi$ and $\Psi$ be the the functionals defined by
\begin{equation}\label{e3.3}
\Phi (u)=\frac{1}{p}\widehat{M}(\|u\| ^{p})-\frac{a}{p}
\int_{\Omega} \frac{|u(x)|^{p}}{|x|^{2p}}dx ,
\end{equation}and\begin{equation}\label{e3.4}
\Psi (u)=\int_{\Omega} F(x,u(x))dx.\end{equation} Put
$I_\lambda(u)=\Phi(u)-\lambda\Psi(u)$ for all $u\in X$. It is easy
to show that the functionals $\Phi$ and $\Psi$ are well define and
continuously G\^{a}teaux differentiable. Moreover, we introduce
the functional $I_{\lambda}:W^{1,p} (\Omega)\to\mathbb{R}$
associated with problem $\eqref{e1.1}$,
\begin{equation*}
I_{\lambda}(u):=\frac{1}{p}\widehat{M}(\|u\| ^{p})-\frac{a}{p}
\int_{\Omega}
\frac{|u(x)|^{p}}{|x|^{2p}}dx-\lambda\int_{\Omega}F(x,u(x))dx.
\end{equation*}Clearly
$\Phi$ and $\Psi$ are continuously G\^{a}teaux differentiable and
\begin{equation*}
\Phi ^{'}(u)(v)=M\Big(\int_{\Omega}|\Delta u| ^{p}dx\Big)\int_{\Omega}|\Delta u|
^{p-2}\Delta u(x)\Delta v(x)dx-a\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^{2p}}u(x)v(x)dx,
\end{equation*}and
\begin{equation*}\Psi ^{'}(u)(v)=\int_{\Omega}f(x,u(x))v(x)dx
\end{equation*}for every $v \in X.$ Proposition \ref{pro2.3} follows that $\Phi'$ admits a continuous inverse
on $X^*$. Moreover, $\Psi'$ is a compact operator. Note that the
critical points of $I_{\lambda}$ are exactly the weak solutions of
problem $\eqref{e1.1}$. Since $m_0\leq M(t)\leq m_1$ for all
$t\in\mathbb{R}^+$, we see that
\begin{equation}\label{e3.5}
\frac{m_0H-a}{pH}\|u\|^p\leq \Phi (u)\leq\frac{m_1}{p}\|u\|^p.
\end{equation}Put\begin{equation}\label{e3.6}
r_1=\frac{m_0H-a}{pHc_q^p}\eta_1^p\ \textrm{and}\ \quad
r_2=\frac{m_0H-a}{pHc_q^p}\eta_2^p,
\end{equation}
and define $w_\delta\in X$ by
\begin{equation}\label{e3.7}
w_\delta(x):=\left\{\begin{array}{ll}
\displaystyle 0,&x\in \overline{\Omega}\setminus B(0,s) ,\\
\displaystyle\delta(\frac{4}{s^3}l^3-\frac{12}{s^2}l^2+\frac{9}{s}l-1),&x\in B(0,s)\setminus B(0,\frac{s}{2}) ,\\
\displaystyle \delta,&x\in B(0,\frac{s}{2}) ,
\end{array}\right.
\end{equation}with $l=dist(x,0)=\sqrt{\sum_{i=1}^{N}x_{i}^{2}}$. Then
\begin{equation*}
\frac{\partial w_\delta(x)}{\partial x_i}:=\left\{\begin{array}{ll}
\displaystyle 0,&x\in \overline{\Omega}\setminus B(0,s)\cap B(0,\frac{s}{2}) ,\\
\displaystyle\delta(\frac{12lx_{i}}{s^3}-\frac{24x_i}{s^2}+
\frac{9x_i}{sl}),&x\in B(0,s)\setminus B(0,\frac{s}{2}) ,
\end{array}\right.
\end{equation*}
and\begin{equation*} \frac{\partial^2w_\delta(x)}{\partial
x_i^2}:=\left\{\begin{array}{ll}
\displaystyle 0,&x\in \overline{\Omega}\setminus B(0,s)\cap B(0,\frac{s}{2}) ,\\
\displaystyle\delta(\frac{12(x_{i}^2+l^2)}{s^3l}-\frac{24}{s^2}+
\frac{9(l^2-x_i^2)}{sl^3}),&x\in B(0,s)\setminus B(0,\frac{s}{2}).
\end{array}\right.
\end{equation*}
Therefore, we have
\begin{equation*}
\sum_{i=1}^{N}\frac{\partial^2w_\delta(x)}{\partial x_i^2}:=\left\{\begin{array}{ll}
\displaystyle 0,&x\in \overline{\Omega}\setminus B(0,s)\cap B(0,\frac{s}{2}) ,\\
\displaystyle\delta(\frac{12l(N+1)}{s^3}-\frac{24N}{s^2}+
\frac{9(N-1)}{sl}),&x\in B(0,s)\setminus B(0,\frac{s}{2}).
\end{array}\right.
\end{equation*}
and
\begin{equation}\label{e3.8}
\int_{\Omega}\|\Delta w_\delta (x)\|^pdx=\delta^p\frac{2 \pi^{\frac{N}{2}}}
{\Gamma( \frac{N}{2})}\int_{\frac{s}{2}}^s\Big{|}(\frac{12r(N+1)}{s^3}-\frac{24N}{s^2}+
\frac{9(N-1)}{sr})\Big{|}^pr^{(N-1)}dr=L\delta^p.
\end{equation}
So, from \eqref{e3.5}, we have
\begin{equation}\label{e3.9}
\frac{m_0H-a}{pH}\delta L^p\leq \Phi(w_\delta)\leq\frac{m_1}{p}\delta L^p.
\end{equation}
From the condition \eqref{e3.1}, we obtain $r_1<\Phi(u)<r_2$. Then, for all
$u\in X$, we see that
\begin{eqnarray*}
\Phi^{-1}(-\infty,r_2)&=&\lbrace u\in X, \Phi(u)\leq r_2\rbrace\\
&\subseteq & \lbrace u \in X, \|u\|_{L^q(\Omega)}\leq\eta_2 \rbrace
\end{eqnarray*}
and it follows that
$$\sup_{u\in \Phi^{-1}(-\infty,r_2)}\Psi(u)\leq \int_\Omega
\sup_{\|t\|_{L^q(\Omega)}\leq\eta_2}F(x,t)dx.$$ Therefore, by (A1) one has
\begin{eqnarray*}
\beta(r_1,r_2)&\leq&\frac{\sup_{u\in \Phi^{-1}(-\infty,r_2)}\Psi(u)
-\Psi(w_\delta)}{r_2-\Phi(w_\delta)}\\
&\leq & \frac{\int_{\Omega}\sup_{\|t\|_{L^q(\Omega)}\leq\eta_2}F(x,t)dx-\int_\Omega
F(x,w_\delta(x))dx}{\frac{m_0H-a}{pc_q^pH}\eta_2^p-\frac{m_1}{p}L\delta^p}\\
&\leq& pHk^p \frac{\int_{\Omega}\sup_{\|t\|_{L^q(\Omega)}\leq\eta_2}F(x,t)dx-\int_{B(0,\frac{s}{2})}
F(x,\delta)dx}{(m_0H-a)\eta_2^p-m_1HL(c_q\delta)^p}\\
&=&Hc_q^pa_{\eta_2}(\delta).
\end{eqnarray*}
On the other hand, arguing as before, one has
\begin{eqnarray*}
\rho_2(r_1,r_2)&\geq&\frac{\Psi(w_\delta)-\sup_{u\in \Phi^{-1}
(-\infty,r_1]}\Psi(u)}{\Phi(w_\delta)-r_1}\\
&\geq & \frac{\int_\Omega F(x,w_\delta(x))dx-\int_{\Omega}\sup_
{\|t\|_{L^q(\Omega)}\leq\eta_2}F(x,t)dx}{\frac{m_1}{p}L\delta^p-\frac{m_0H-a}{pc_q^pH}\eta_1^p}\\
&\geq& pHk^p \frac{\int_{B(0,\frac{s}{2})} F(x,\delta)dx-\int_
{\Omega}\sup_{\|t\|_{L^q(\Omega)}\leq\eta_2}F(x,t)dx}{m_1HL(c_q\delta)^p-(m_0H-a)\eta_1^p}\\
&=&Hc_q^pa_{\eta_1}(\delta).
\end{eqnarray*}
Hence, from assumption (A2), one has $\beta(r_1,r_2)<\rho_2(r_1,r_2)$.
Therefore, from Theorem 2.1, for each $\lambda\in\left
( \frac{1}{c_q^pH}\frac{1}{a_{\eta_1(\delta)}},\frac{1}
{c_q^pH}\frac{1}{a_{\eta_2}(\delta)}\right), $ the functional $I_\lambda$
admits at least one nontrivial critical point $u_1$ such that
$$r_1<\Phi(u_1)<r_2,
$$that is,$$\frac{m_0H-a}{m_1Hc_q^p}\eta_1^p<\|u_1\|^p<m_0\eta_2^p.
$$Now, we prove the existence of the second local minimum distinct
from the first one. To this purpose, we verify the hypotheses of the
mountain-pass theorem for the functional $\Phi-\lambda\Psi$.
Clearly, the functional $\Phi-\lambda\Psi$ is of class $C^1$ and $(\Phi-
\lambda\Psi)(0)=0$. The first part of proof guarantees that $u_1\in X$
is a local nontrivial local minimum for $\Phi-\lambda\Psi$ in $X$.
Now, we can assume that $u_1$ is a strict local minimum of
$\Phi-\lambda\Psi$ on $X$. Therefore, there is $s>0$ such that
$$\inf_{\|u-u_1\|=s}(\Phi-\lambda\Psi)(u)>(\Phi-\lambda\Psi)(u_1).
$$So the condition {\cite[$(I_1)$, Theorem 2.2]{R}} is verified. From $(A3)$
there is a positive constant $C_1, C_2$ such that
$$F(x,t)\geq C_1|t|^\xi+C_2.$$Fixed $u\in X\setminus \{0\}$, for each $t>1$ one has
\begin{eqnarray*}
(\Phi-\lambda\Psi)(tu) &=&\frac{1}{p}\widehat{M}\left(\int_\Omega |\Delta tu|^p dx\right)
-\frac{a}{p}\int_\Omega\frac{|tu|^p}{|x|^{2p}}dx - \lambda
\int_\Omega F(x,tu)dx\\
&\leq &\frac{m_1}{p}t^{p} \left( \int_\Omega |\Delta u|^p
dx\right) - \lambda Ct^\xi\int_\Omega|u|^\rho dx+\lambda C_2.
\end{eqnarray*}
Since $\xi>p,$ this condition guarantees that $I_{\lambda}$ is
unbounded from below. So the condition {\cite[$(I_2)$, Theorem 2.2]{R}} is fulfilled.
Now we prove that $I_{\lambda}:=\Phi-\lambda\Psi$
satisfies $\rm(PS)$-condition for every $\lambda>0$. Namely, we will
prove that any sequence $\{u_{n}\}\subset X$ satisfying
\begin{equation*}\label{PS}
h:=\sup_{n}I_{\lambda}(u_{n})<+\infty,\quad
\lim_{n\rightarrow+\infty}\|I'_{\lambda}(u_{n})\|=0.
\end{equation*} From above, we can actually assume that
$$|\frac{1}{\xi}\langle I'_{\lambda}(u_{n}),u_{n}\rangle|\leq
\|u_{n}\|.$$ For $n$ large enough, we have
$$h\geq I_{\lambda}(u_{n})=\frac{1}{p}\widehat{M}\left(\int_\Omega |\Delta u_{n}(x)|^p dx\right)
-\frac{a}{p}\int_\Omega\frac{|u_{n}(x)|^p}{|x|^{2p}}dx - \lambda
\int_\Omega F(x,u_{n}(x))dx,$$ then\begin{eqnarray*}
I_{\lambda}(u_n)-\frac {1}{\xi}\langle I'_{\lambda}(u_n),u_n\rangle &=&\frac{1}{p}\widehat{M}(\int_\Omega
|\Delta u_n(x)|^p dx)
-\frac{a}{p}\int_\Omega\frac{|u_n(x)|^p}{|x|^{2p}}dx - \lambda
\int_\Omega F(x,u_n(x))dx\\
&-&\frac{1}{\xi} M(\int_\Omega
|\Delta u_{n}(x)|^p dx)\int_\Omega|\Delta u_{n}(x)|^p dx\\
&- &\frac{a}{\xi}\int_\Omega \frac{|\Delta u_n(x)|^p}{|x|^{2p}}dx+\frac{\lambda}{\xi} \int_\Omega f(x,u_n(x))u_n(x)dx\\
&\geq &m_0\left(\frac{1}{p} - \frac{1}{\xi} \right) \int_\Omega
|\Delta u_n(x)|^p dx
-\frac{a}{H}\left( \frac{1}{p}-\frac{1}{\xi}\right) \int_\Omega |\Delta u_{n}(x)|^p dx\\
&=&\left(\frac{1}{p} - \frac{1}{\xi} \right) \left( \frac{m_0H-a}{H}\right) \|u_{n}\|^{p}.
\end{eqnarray*}
Thus,$$h+\|u_{n}\|\geq I_{\lambda}(u_{n})-\frac{1}{\xi}\langle
I'_{\lambda}(u_{n}),u_{n}\rangle\geq\left(\frac{1}{p}
-\frac{1}{\xi} \right) \left( \frac{m_0H-a}{H}\right)
\|u_{n}\|^{p}.$$ Consequently, $\{\|u_{n}\|\}$ is bounded. By the
Eberlian-Smulyan theorem, without loss of generality, we assume
that $u_{n}\rightharpoonup u.$ Then $\Psi'(u_{n})\rightarrow
\Psi'(u)$. Since $I'_{\lambda}(u_{n})=\Phi'(u_{n})-\lambda
\Psi'(u_{n})\rightarrow0,$ then $\Phi'(u_{n})\rightarrow \lambda
\Psi'(u)$. Since $\Phi'$ has a continuous inverse,
$u_{n}\rightarrow u$ and so $I_{\lambda}$ satisfies
$\rm(PS)$-condition. Hence, the classical theorem of Amberosetti
and Rabinowitz gives a critical point $u_2$ of $\Phi-\lambda\Psi$
such that $(\Phi-\lambda\Psi)(u_2)> (\Phi-\lambda\Psi)(u_1)$. So
$u_1$ and $u_2$ are distinct weak solutions of the problem
(1.1).Hence, the proof is complete.
\end{proof}
\begin{theorem}\label{th3.2}
Suppose that $f(x,0)\neq 0$ for all $x\in\Omega$ and there exist two
positive constants $\delta$ and $\eta$, with
$$\delta<\left(
\frac{m_0H-a}{m_1LH}\right)^\frac{1}{p}\frac{\eta}{c_q}$$ such that
the assumptions (A1) and (A3) in Theorem 3.1 hold. Furthermore,
assume that
\begin{equation}\label{e3.10}
\frac{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx}{(m_0H-a)\eta^p}<
\frac{\int_{B(0,\frac{s}{2})}F(x,\delta)dx}{m_1HLc_q^p\delta^p}.
\end{equation}
Then, for each
$$\lambda\in \left(\frac{1}{p}\frac{m_1L\delta^p}{\int_{B(0,\frac{s}
{2})}F(x,\delta)dx},\frac{1}{pHc_q^p}\frac{(m_0H-a)\eta^p}
{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx}  \right)
$$the problem \eqref{e1.1} admits at least two nontrivial weak solutions
$u_1$ and $u_2$ in $X$ such that
$$\|u_1\|^p<m_0\eta^p.$$
\begin{proof}
Our aim is to employ Theorem \ref{the3.1}, by choosing
$\eta_1=0$ and $\eta_2=\eta$. Therefore, owing to the inequality
\eqref{e3.10} and (A1), we see that
\begin{eqnarray*}
a_\eta(\delta)&=&p\frac{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx-\int_{B(0,\frac{s}{2})}
F(x,\delta)dx}{(m_0H-a)\eta^p-m_1HLc_q^p\delta^p}\\
&<&p\frac{\left( 1-\frac{m_1HLk^p\delta^p}{(m_0H-a)\eta^p} \right) \int_\Omega
\sup_{|t|\leq \eta}F(x,t)dx}{(m_0H-a)\eta^p-m_1HLc_q^p\delta^p}\\
&<&p\frac{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx}{(m_0H-a)\eta^p}\\
&<&p\frac{\int_{B(0,\frac{s}{2})}F(x,\delta)dx}{m_1HLc_q^p\delta^p}\\
&=&a_0(\delta).
\end{eqnarray*}
In particular, one has
$$a_\eta(\delta)<p\frac{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx}{(m_0H-a)\eta^p}
$$which follows
$$\frac{1}{pHc_q^p}\frac{(m_0H-a)\eta^p}{\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\eta}F(x,t)dx}
<\frac{1}{Hc_q^p}\frac{1}{a_\eta(\delta)}.$$ Hence, Theorem 3.1
yields the desired conclusion.\end{proof}
Now, we present an application of Theorem 2.2 which will be used later to obtain multiple
solutions for the problem\eqref{e1.1}.\begin{theorem}\label{th3.3}
Suppose that there exist two positive constants $\overline{\eta}$ and $\overline{\delta}$ with
\begin{equation*}
\overline{\delta}>\left( \frac{m_0H-a}{m_1LH}\right)^{\frac{1}{p}}\frac{\overline{\eta}}{c_q}
\end{equation*}
such that assumption (A1) in Theorem 3.1 holds. Moreover, assume that
\begin{equation*}
\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\overline{\eta}}F(x,t)dx<\int_{B(0,
\frac{s}{2})}F(x,\overline{\delta})dx
\end{equation*}
and\begin{equation}\label{e3.11}
\limsup_{|\xi|\rightarrow+\infty}\frac{F(x,\xi)}{|\xi|^p}\leq0
\quad \text{uniformly in }\mathbb{R}.\end{equation} Then, for each
$\lambda>\widehat{\lambda}$, where
\begin{equation*}\widehat{\lambda}:=\frac{m_1HLc_q^p\overline{\delta}^p-(m_0H-a)
\overline{\eta}^p}{pHc_q^p( \int_{B(0,\frac{s}{2})}F(x,\overline
{\delta})dx-\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\overline{\eta}}F(x,t)dx)}
\end{equation*}the problem \eqref{e1.1} admits at least one nontrivial weak solution $\overline{u_1}\in X$ such that
\begin{equation*}
\|\overline{u_1}\|^p>\frac{m_0H-a}{m_1Hc_q^p}\overline{\eta}^p.\end{equation*}\end{theorem}\begin{proof}
Our goal is apply Theorem 2.2 to the functional $I_\lambda=\Phi- \lambda\Psi$ where $\Phi$ and $\Psi$ are given as
in \eqref{e3.3} and \eqref{e3.4}, respectively. We observe that the all assumptions of Theorem 2.2 on
$\Phi$ and $\Psi$ are satisfied. Moreover, for $\lambda>0$, the functional $I_\lambda$ is
coercive. Indeed, fix $0<\epsilon<\frac{m_0H-a}{p\lambda
\text{meas}(\Omega)Hc_q^p}$. From \eqref{e3.11} there is a function
$\varrho_\varepsilon\in L^1(\Omega)$ such that
$$F(x,t)\leq \varepsilon t^p+\varrho_\varepsilon(x),
$$for every $x\in \Omega$ and $t\in \mathbb{R}$. Therefore, for each
$u\in X$ with $\|u\|\geq1$, we see that
\begin{eqnarray*}
\Phi(u)-\lambda\Psi(u)&\geq &\frac{m_oH-a}{pH} \|u\|^p-\lambda\varepsilon
\int_\Omega u^p(x)dx-\lambda\|\varrho_\varepsilon\|_{L^1}\\
&\geq &\left( \frac{m_0H-a}{pH}-\lambda \text{meas}(\Omega)c_q^p\varepsilon\right)\|u\|^p-
\lambda\|\varrho_\varepsilon\|_{L1}
\end{eqnarray*}and thus$$\lim_{\|u\|\rightarrow+\infty}(\Phi(u)-\lambda\Psi(u))=+\infty
$$which means the functional $I_\lambda=\Phi-\lambda\Psi$ is coercive. Put
$$\overline{r}=\frac{m_0H-a}{pHc_q^p}\overline{\eta}^p,$$and choose\begin{equation*}
\overline{w}(x):=\left\{\begin{array}{ll}
\displaystyle 0,&x\in \overline{\Omega}\setminus B(0,s) ,\\
\displaystyle\overline{\delta}(\frac{4}{s^3}l^3-\frac{12}{s^2}l^2+\frac
{9}{s}l-1),&x\in B(0,s)\setminus B(0,\frac{s}{2}) ,\\
\displaystyle \overline{\delta},&x\in B(0,\frac{s}{2}).
\end{array}\right.\end{equation*}
Using the condition (A1) and arguing as in the proof of Theorem 3.1, we obtain that
\begin{equation*}
\rho_2(\overline{r})\geq pHc_q^p\frac{ \int_{B(0,\frac{s}{2})}F(x,\overline
{\delta})dx-\int_\Omega\sup_{\|t\|_{L^q(\Omega)}\leq\overline{\eta}}F(x,t)dx}{m_1HLc_q^p
\overline{\delta}^p-(m_0H-a)\overline{\eta}^p}.
\end{equation*}Thus, it follows that $\rho(\overline{r})>0$.
Hence, from Theorem 2.2 for each $\lambda>\widehat{\lambda}$,
the functional $I_\lambda$ admits at least one local minimum $\overline{u_1}$ such that
$$\|\overline{u_1}\|^p>\frac{m_0H-a}{m_1Hc_q^p}\overline{\eta}^p
$$the desired conclusion is achieved.\end{proof}
Now, we point out a special situation of our main result when the function $f$ has
separated variables. To be precise, let $\alpha:\Omega\rightarrow\mathbb{R}$ be a nonnegative and
nonzero function such that $\alpha\in L^1(\Omega)$ and let
$g:\mathbb{R} \rightarrow\mathbb{R}$ be a nonnegative continuous
function. Consider the following problem
\begin{equation}\label{e3.12}
\left\{\begin{array}{ll}
\displaystyle M\left( \int_{\Omega}|\Delta u|^{p} dx\right) \Delta _{p}^{2}u-
\frac{a}{|x|^{2p}}|u|^{p-2}u=\lambda \alpha(x)g(u)&\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega.
\end{array}\right.
\end{equation}
 Put $G(t)=\int_0^tg(\xi)d\xi\quad \textrm{for all }t\in
\mathbb{R}$ and set $f(x,t)=\alpha(x)g(t)$ for every $(x,t)\in
\Omega\times\mathbb{R}$. The following existence results are
consequences of Theorems 3.1-3.3, respectively.
\begin{theorem}\label{th3.4}
Suppose that $g(0)\neq0$ and there exist a nonnegative constant $\eta_1$ and
two positive constants $\eta_2$ and $\delta$, with
$$\frac{1}{c_qL^{\frac{1}{p}}}\eta_1< \delta< \left( \frac{m_0H-a}{m_1LH}
\right) ^{\frac{1}{p}}\frac{\eta_2}{c_q}
$$such that$$\frac{\|\alpha\|_{L^1(\Omega)}G(\eta_2)-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta)}
{(m_0H-a)\eta_2^p-m_1HLc_q^p\delta^p}
<\frac{\|\alpha\|_{L^1(\Omega)}G(\eta_1)-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta)}
{(m_0H-a)\eta_1^p-m_1HLc_q^p\delta^p}.
$$Moreover, assume that there exist constants $v>p$ and $R>0$ such that for all
$|\xi|\geq R$ and for all $x\in \Omega$
\begin{equation}\label{e3.13}
0<vG(\xi)\leq \xi g(\xi).
\end{equation}
Then, for each $\lambda\in]\lambda_1,\lambda_2[$, where
$$\lambda_1:=\frac{1}{Hc_q^p}\frac{(m_0H-a)\eta_1^p-m_1HLc_q^p\delta^p}
{\|\alpha\|_{L^1(\Omega)}G(\eta_1)-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta)}
$$and$$\lambda_2:=\frac{1}{Hc_q^p}\frac{(m_0H-a)\eta_2^p-m_1HLc_q^p\delta^p}
{\|\alpha\|_{L^1(\Omega)}G(\eta_2)-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta)}$$
the problem \eqref{e3.12} admits at least two nontrivial weak
solutions $u_1$ and $u_2$ such that
$$\|u_1\|^p<m_0\eta^p.$$
\end{theorem}\begin{theorem}\label{th3.5}
Suppose that $g(0)\neq 0$ and there exist two positive constant $\delta$ and $\eta$, with
$$\delta< \left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\eta}{c_q}
$$such that
\begin{equation}\label{e3.14}\frac{\|\alpha\|_{L^1(\Omega)}G(\eta)}{\eta^p}
<\frac{m_0H-a}{m_1HLc_q^p\delta^p}\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta).
\end{equation}Moreover, assume that the assumption \eqref{e3.11} holds. Then, for every
$$\lambda\in \left( \frac{m_1L\delta^p}{\|\alpha\|_{L^1(B(0,\frac{s}{2}))}
G(\delta)},\frac{(m_0H-a)\eta^p}{Hc_q^p\|\alpha\|_{L^1(\Omega)}G(\eta)}\right)
$$the problem \eqref{e3.12} admits at least two nontrivial weak solutions $u_1$ and $u_2$
in $X$ such that
$$\|u_1\|^p<m_0\eta^p.$$\end{theorem}
\begin{theorem}\label{th3.6}
Suppose that there exist two positive constant $\overline{\eta}$ and $\overline{\delta}$ with
$$\overline{\delta}> \left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\overline{\eta}}{c_q}
$$such that
\begin{equation}\label{e3.15}
G(\overline{\eta})<\frac{\|\alpha\|_{L^1(B(0,\frac{s}{2}))}}{\|\alpha\|_{L^1(\Omega)}}G(\overline{\delta})
\end{equation}
and$$\limsup_{\xi \rightarrow
+\infty}\frac{g(\xi)}{|\xi|^{p-1}}\leq 0.$$Then, for each
$\lambda>\overline{\lambda}$, where
$$\overline{\lambda}:=\frac{1}{Hc_q^p}\frac{(m_0H-a)\overline{\eta}^p-m_1HLc_q^p\overline{\delta}^p}
{\|\alpha\|_{L^1(\Omega)}G(\overline{\eta})-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\overline{\delta})}
$$the problem \eqref{e3.12} admits at least one nontrivial weak solution $\overline{u_1}$
such
that$$\|\overline{u_1}\|^p>\frac{m_0H-a}{m_1Hc_q^p}\overline{\eta}^p.
$$\end{theorem}
A further consequence of Theorem 3.1 is the following existence
result.\begin{theorem}\label{th3.7} Suppose that $g(0)\neq 0$ and
\begin{equation}\label{e3.16}
\lim_{\xi\rightarrow 0^+}\frac{g(\xi)}{\xi^{p-1}}=+\infty.
\end{equation}
Moreover, assume that the assumption \eqref{e3.13} holds. Then, for
every $\lambda\in(0,\lambda_\eta^*)$, where
$$\lambda_\eta^*:=\frac{m_0H-a}{pHc_q^p\|\alpha\|_{L^1(\Omega)}}
\sup_{\eta>0}\frac{\eta^p}{G(\eta)}$$the problem \eqref{e3.12}
admits at least two nontrivial weak solutions in $X$.
\end{theorem}
\begin{proof}
Fix $\lambda\in ]0,\lambda_\eta^*[$. then there is $\eta>0$ such that
$\lambda<\frac{m_0H-a}{pHc_q^p\|\alpha\|_{L^1(\Omega)}}
\frac{\eta^p}{G(\eta)}$. From \eqref{e3.16} there exists a positive constant $\delta$ with
$$\delta< \left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\eta}{c_q}
$$such that$$\lambda<\frac{m_1L\delta^p}{p\|\alpha\|_{L^1(B(0,\frac{s}{2}))}}.
$$Therefore, the conclusion follows from Theorem 3.2.
\end{proof}
Now, we present the following example to illustrate Theorem 3.7.
\begin{example}Let $N=3,\ p=\frac{5}{4},\ \Omega=\{(x_1,x_2,x_3)\in \mathbb{R}^3;
x_1^2+x_2^2+x_3^2<1\} , M(t)=3+\cos x \text{ for all } t\in
[0,+\infty]$, $\alpha(x_1,x_2,x_3)=\frac{1} {\sqrt{x_1^2+x_2^2+x_3^2}}\text{
for all } (x_1,x_2,x_3)\in \mathbb{R}^3$ and $g(t) =\frac{1}{6}+t^2|t|$.
thus$$\|\alpha\|_{L^1(\Omega)}=\int_0^{2\pi} \int_0^{\pi}
\int_0^1 r\sin\phi dr d\phi d\theta =2\pi
$$and$$\lim_{\xi\rightarrow
0^+}\frac{g(\xi)}{\xi^{p-1}}=\lim_{\xi\rightarrow
0^+}(\frac{1}{6\xi^{\frac{1}{4}}}+\xi^{\frac{7}{4}})=+\infty
$$Now, by choosing $\nu =3$ and $R=2$, we see that the assumption \eqref{e3.13} is satisfied.
Hence, by applying theorem 3.7, for every $\lambda\in(0,\lambda^*)$ where
\begin{eqnarray*}\lambda^{*}&=&\frac{m_0H-a}{pHc_q^p\|\alpha\|_{L^1(\Omega)}}
\sup_{\eta>0}\frac{\eta^p}{G(\eta)}\\ &=&\left( \frac{4\sqrt[4]{6}-\sqrt{5}}{5\sqrt[4]{6}c_q^\frac{5}{4}\pi}\right)
\sup_{\eta>0}\frac{\eta^{\frac{1}{4}}}{2+3\eta^3}\\&\geq& \left( \frac{4\sqrt[4]{6}-\sqrt{5}}
{5\sqrt[4]{6} c_q^\frac{5}{4}\pi}\right) \frac{\eta^{\frac{1}{4}}}
{2+3\eta^3}\mid_{\eta=1}\\
&=& \frac{48\sqrt[4]{6}-12\sqrt{5}}{25\sqrt[4]{6} c_q^\frac{5}{4}\pi}
\end{eqnarray*}the problem
\begin{equation*}
\left\{\begin{array}{ll}
\displaystyle \Big(3+\cos(\int_{\Omega}|\Delta u|^{p} dx)\Big)\Delta _{p}^{2}u-
\frac{6}{50|x|^{2p}}|u|^{p-2}u=\frac{\lambda}{\sqrt{x_1^2+x_2^2+x_3^2}}g(u(x_1,x_2,x_3))  &\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega,
\end{array}\right.
\end{equation*}
has at least two nontrivial weak solutions.
\end{example}
 Next, by applying Theorems 3.5 and 3.6, we obtain the following theorem of existence of
three solutions for the problem\eqref{e1.1}.
\begin{theorem}\label{th3.8}
Suppose that $g(0)\neq 0$ and
\begin{equation}\label{e3.17}
\limsup_{|\xi|\rightarrow+\infty}\frac{G(\xi)}{|\xi|^p}\leq0.
\end{equation}
Furthermore, assume that there exist four positive constants $\eta, \delta,
\overline{\eta}$ and $\overline{\delta}$ with
$$\delta< \left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\eta}{c_q}
\leq \left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\overline{\eta}}
{c_q}<\overline{\delta}
$$such that \eqref{e3.14} and \eqref{e3.15} hold, and
\begin{equation}\label{e3.18}
\frac{\|\alpha\|_{L^1(\Omega)}G(\eta)}{(m_0H-a)\eta^p}<
\frac{\|\alpha\|_{L^1(\Omega)}G(\overline{\eta})-\|\alpha\|_{L^1(B(0,\frac{s}{2}))}
G(\overline{\delta})}{(m_0H-a)\overline{\eta}^p-m_1HLc_q^p\overline{\delta}^p}
\end{equation}
is satisfied. Then, for each
\begin{equation*}
\lambda\in \Lambda=\left(\max\{\overline{\lambda},\frac{m_1L\delta^p}
{\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\delta)}\},\frac{(m_0H-a)\eta^p}
{Hc_q^p\|\alpha\|_{L^1(\Omega)}G(\eta)}\right)
\end{equation*}
the problem \eqref{e3.10} admits at least three nontrivial weak
solutions $u_1$, $\overline{u_1}$ and $u_3$ such
that\begin{equation*} \|u_1\|^p<m_0\eta^p,
\quad\|\overline{u_1}\|^p>\frac{m_0H-a}{m_1Hc_q^p}\overline{\eta}^p.
\end{equation*}
\end{theorem}
\begin{proof}It is easy to see that $\Lambda\neq \emptyset$ from \eqref{e3.18}. Fix
$\lambda\in \Lambda$. Using Theorem 3.5, there is a nontrivial weak
solution $u_1$ such that
$$\|u_1\|^p<m_0\eta^p$$
which is a local minimum for the associated functional $I_\lambda$, and Theorem 3.6
guarantees the second a nontrivial weak solution $\overline{u_1}$ such that
$$\|\overline{u_1}\|^p>\frac{m_0H-a}{m_1Hc_q^p}\overline{\eta}^p$$which
is local minimum for $I_\lambda$. Now, by employing the proof of Theorem 3.3,
from the condition \eqref{e3.17} we see that the functional $I_\lambda$
satisfies the (PS) condition. Hence, the conclusion follows from the mountain pass
theorem as given by Pucci and Serrin (see {\cite{PS}}).
\end{proof}
We now point out the following consequence of Theorem 3.8.
\begin{theorem}\label{th3.9}
Suppose that $g(0)\neq 0$,
\begin{equation}\label{e3.19}
\limsup_{\xi\rightarrow 0^+}\frac{G(\xi)}{\xi^p}=+\infty,
\end{equation}
and\begin{equation}\label{e3.20} \limsup_{\xi\rightarrow
+\infty}\frac{G(\xi)}{\xi^p}=0.
\end{equation}
Moreover, assume that there exist two positive constants $\overline{\eta}$ and
$\overline{\delta}$ with
\begin{equation}\label{e3.21}
\left( \frac{m_0H-a}{m_1LH}\right) ^{\frac{1}{p}}\frac{\overline{\eta}}{c_q}<\overline{\delta}
\end{equation}
such that\begin{equation}\label{e3.22}
\frac{\|\alpha\|_{L^1(\Omega)}G(\overline{\eta})}{(m_0H-a)\overline{\eta}^p}<
\frac{\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\overline{\delta})}{m_1HLc_q^p\overline{\delta}^p}
\end{equation}
Then, for each$$\lambda\in \left(\frac{m_1L\overline{\delta}^p}
{\|\alpha\|_{L^1(B(0,\frac{s}{2}))}G(\overline{\delta})},\frac{(m_0H-a)\overline{\eta}^p}
{Hc_q^p\|\alpha\|_{L^1(\Omega)}G(\overline{\eta})}\right)
$$the problem \eqref{e3.12} admits at least three nontrivial weak solutions.
\end{theorem}
\begin{proof}We easily observe form \eqref{e3.20} that the condition \eqref{e3.17} is satisfied. Moreover, by choosing $\delta$ small
enough and $\eta=\overline{\eta}$, one can drive the condition
\eqref{e3.14} from \eqref{e3.19} as well as the conditions
\eqref{e3.15} and \eqref{e3.18} from \eqref{e3.22}. Hence, the
conclusion follows from Theorem 3.8.
\end{proof}
Finally, we illustrate Theorem 3.9 by presenting the following example.
\begin{example}Consider the problem
\begin{equation}\label{e3.23}
\left\{\begin{array}{ll}
\displaystyle (3+\sin(\int_{\Omega}|\Delta u|^{p} dx))\Delta _{p}^{2}u-
\frac{1}{4\sqrt[5]{4}|x|^{2p}}|u|^{p-2}u=\lambda u(x) &\textrm{ in } \Omega,\\
\displaystyle u=\Delta u=0, & \text{ on }\partial\Omega,
\end{array}\right.
\end{equation}
where $N=3, p=\frac{6}{5}, \Omega=\{x=(x_1,x_2,x_3)\in \mathbb{R}^3;
|x_1|+|x_2|+|x_3|<2\} , M(t)=3+\sin t \text{ for all } t\in
[0,+\infty)$, $\alpha(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2 \text{
for all } (x_1,x_2,x_3)\in \mathbb{R}^3, s=2$.
Thus$$\|\alpha\|_{L^1(\Omega)}=\frac{64}{5}, \quad \quad\|\alpha\|_{L^1(B(0,1))}=\frac{4\pi}{5},
$$Let $g(t)=1+e^{-t}(1-t)$ for all $t\in \mathbb{R}$.
Thus $g$ is nonnegative continuous, $g(0)\neq 0$ and $$G(t)=t(1+e^{-t}).$$
Since  $$\lim_{t\rightarrow0^+}\frac{t(1+e^{-t})}{t^{\frac{6}{5}}}=+\infty, \quad\quad \lim_{t\rightarrow
+\infty}\frac{t(1+e^{-t})}{t^{\frac{6}{5}}}=0,$$we see that conditions \eqref{e3.19} and \eqref{e3.20} hold true.Moreover,
by $\overline{\eta}=4480c_q$ and $\overline{\delta}=\frac{1}{c_q},$
we see that  the conditions \eqref{e3.19} and \eqref{e3.20} hold true. Then, by applying Theorem 3.9, for every
$$\lambda\in \left(\frac{302}{c_q^{\frac{1}{5}}(1+e^{-c_q^{-1}})},\frac{350}{c_q^{\frac{1}{5}}(1+e^{-4480c_q})} \right) $$
the problem \eqref{e3.23} has at least three nontrivial weak solutions.\\

\end{example}
\end{theorem}
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\end{document}
