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\fancyhead[CE]{M.Khodabakhshi and M.R.  Heidari Tavani} 
\fancyhead[CO]{Existence of three weak solutions }



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\begin{document}
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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}



{\Large \bf 
Existence of three weak solutions for some \\singular elliptic
problems with Hardy potential\\}


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}


{\bf Mehdi Khodabakhshi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Amir Kabir University of Technology} \vspace{2mm}


{\bf  Mohammad Reza Heidari Tavani$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Islamic Azad University of Ramhormoz} \vspace{2mm}



\end{center}
 
\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.}~In this paper,under growth conditions on the nonlinearity, we obtain
the existence of at least three weak solutions for some singular
elliptic Dirichlet problems involving the $p$-Laplacian.~The approach is based on variational methods and critical point theory. 
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 34B15; 35J20

\noindent{\bf Keywords and Phrases:} Singular problem, $p$-Laplace operator, Variational methods, Critical point.
\end{quotation}}


\section{Introduction}

In this paper, we want to investigate the following problem
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
\displaystyle-\Delta_{p}u + \frac{|u|^{p-2}u}{|x|^p} =\lambda f(x,u), &\textrm{ in } \Omega,\\
\displaystyle u=0, & \text{ on }
\partial\Omega,
\end{array}\right.
\end{equation}
where $\lambda$ is positive parameter and $\Delta_{p}u$:=div$(|\nabla u|^{p-2}\nabla u)$ denotes the
$p$-Laplace operator, $\Omega$ is a bounded domain
in $\mathbb{R}^{N}(N\geq2)$ containing the origin and with smooth
boundary $\partial\Omega$, $1<p<N$, and
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory
function such that
\begin{enumerate}
\item[$\rm(f_1)$]\hspace{1cm} $|f(x,t)|\leq a_1+a_2|t|^{q-1},\qquad\forall (x,t)\in\Omega\times\mathbb{R}$,
\end{enumerate}
for some non-negative constants $a_1,a_2$ and $q\in]1,p^*[,$ where
$$
p^*:=\frac{pN}{N-p}.
$$
Several results are known concerning the existence of 
solutions for singular elliptic problems, and we mention the works
\cite{CPV,GB,k2,GR,k3}. For example in \cite{CPV}, the authors 
 obtained the
existence of one solution for the problem

\begin{equation}\label{e1.4m}
\left\{\begin{array}{ll}
\displaystyle-\Delta_{p}u = \frac{|u|^{p-2}u}{|x|^p} +\lambda f(x,u), &\textrm{ in } \Omega,\\
\displaystyle u|_{\partial\Omega}=0, 
\end{array}\right.
\end{equation}
based on variational methods and critical point theory.




 Nonlinear singular elliptic equations are encountered in
glacial advance, in transport of coal slurries down conveyor belts
and in several other geophysical and industrial contents( see \cite{dfk}).\\
In this work, our goal is to obtain the existence of at least three
weak solutions for the problem \eqref{e1.1} , by using
variational methods. \\
Recall that a function
$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is said to be a
Carath\'{e}odory function , if\\($C_{1}$) the function $x\rightarrow
f(x,t)$ is measurable for every $t\in\mathbb{R}$;\\
$(C_{2})$ the function $t\rightarrow f(x,t)$ is continuous for a.e.
$x\in\Omega$.


%----------------------------------------------------------------------------------------------------------

\section{Preliminaries and Basic Definitions}

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}(N\geq2)$
containing the origin and with smooth boundary $\partial\Omega$.
Further, denote by $X$ the space $W_{0}^{1,p}$ $(\Omega)$ endowed
with the norm
$$\|u\|:=\Bigg(\int_\Omega |\nabla u(x)|^p\,dx\Bigg)^{1/p}.$$
Let $1<p<N,$  we recall classical Hardy's inequality,  which says
that\begin{equation}\label{c12}\int_\Omega \frac{|u(x)|^p}{|x|^p}dx\leq
\frac{1}{H}\int_\Omega |\nabla u(x)|^pdx,\qquad(\forall u\in
X)\end{equation}where $H:=(\frac{N-p}{p})^p$;( see, for instance, the
paper \cite{GB}).
 By the compact embedding $X\hookrightarrow L^q(\Omega)$ for each
$q\in [1,p^*[$, there exists a positive constant $c_{q}$ such that
\begin{equation}\label{130}
\|u\|_{L^q(\Omega)}\leq c_q\|u\|,\qquad(\forall u\in X)
\end{equation}
where $c_{q}$ is the best constant.\\Let us define
$F(x,\xi):=\int_{0}^{\xi} f(x,t)dt$, for every $(x,\xi)$ in
$\Omega\times\mathbb{R}.$  Moreover, we introduce the functional
$I_{\lambda}:X\to\mathbb{R}$ associated with
$\eqref{e1.1}$,$$I_{\lambda}(u):=\Phi(u)-\lambda\Psi(u),\qquad
(\forall u\in X)$$ where
\begin{equation*}\Phi(u):=\frac{1}{p}\left(\int_{\Omega}|\nabla u(x)|^pdx
+\int_{\Omega}\frac{|u(x)|^p}{|x|^p }dx\right),\,\,\Psi(u):=\int_{\Omega}F(x,u(x))dx.
\end{equation*}It is known that $\Phi ,\Psi\in
C^1(W_{0}^{1,p}(\Omega),\mathbb{R}),$ and
$$\Phi^{'}(u)(v)=\int_{\Omega}|\nabla u|^{p-2}\nabla u.\nabla v dx + \int_{\Omega}\frac{|u|^{p-2}}{|x|^p}uvdx$$
and $$\Psi^{'}(u)(v)=\int_{\Omega}f(x,u(x))v(x)dx$$ for each $u,v
\in W_{0}^{1,p}(\Omega).$\\\\
Now we present one proposition that will be needed to prove the main theorem of this paper.
 \begin{proposition}
 Let $T:X\to X^*$ be the operator defined by
\begin{equation*}
T(u)(v):=\int_\Omega |\nabla u(x) |^{p-2}\nabla u(x) \nabla v(x) dx
+ \int_\Omega \frac{|u(x)|^{p-2}}{|x|^p}u(x)v(x)dx,
\end{equation*}
 for every $u , v\in X$. Then $T$ is strictly monotone.
 \end{proposition}
\begin{proof}
Clearly $T$ is coercive .Taking into account $\rm(2.2)$ of \cite{JS} for
$p>1$ there exists a positive constant $C_{p}$ such that if
$p\geq2,$ then
\begin{equation*}
\langle |x|^{p-2}x-|y|^{p-2}y,x-y\rangle \geq C_{p} |x-y|^p,
\end{equation*}\\ if $1<p<2,$ then
\begin{equation*}
\langle |x|^{p-2}x-|y|^{p-2}y,x-y\rangle \geq C_{p}\frac
{|x-y|^2}{(|x|+|y|)^{2-p}},
\end{equation*}\\ where $\langle.,.\rangle$
denotes the usual inner product in $\mathbb{R}^N.$  Thus, it is easy
to see that, if $p\geq2,$  then, for any $u , v \in X, with\quad
u\neq v,$
\begin{equation*}
\langle Tu-Tv,u-v\rangle \geq C_{p} \int_\Omega |\nabla u(x)-\nabla
v(x) |^p dx=C_{p} \|(u-v)\|^p >0,
\end{equation*}\\ and if $1<p<2$ then,
\begin{equation*}
\langle Tu-Tv,u-v\rangle \geq C_{p} \int_\Omega \frac{|\nabla
u(x)-\nabla v(x)|^2}{(|\nabla u| +|\nabla v|)^{2-p}} dx >0,
\end{equation*}\\ for every $u,v \in X,$ which means that $T$ is strictly
monotone.
\end{proof}
 Moreover, by Theorem 3.1 of \cite{KLV} and proposition 2.1 ,$\Phi$ is weakly lower
semicontinuous and $\Phi^{'}:W_{0}^{1,p}(\Omega)\rightarrow
(W_{0}^{1,p}(\Omega))^*$ is a homeomorphism. Condition ($f_{1}$) and
compact embedding $W_{0}^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$
imply that the functional $\Psi$ has compact derivative.
 From the Hardy's inequality (see \eqref{c12}), it follows
that\begin{equation}
\frac{\|u\|^p}{p}\leq\Phi(u)\leq\left(\frac{H+1}{pH}\right)\|u\|^p,
\end{equation}
 for every $u\in X.$

Fixing the real parameter $\lambda,$ a function $
u:\Omega\to\mathbb{R}$ is said to be a weak solution of
\eqref{e1.1} if $u \in X$ and
\begin{equation*}
\int_{\Omega}|\nabla u(x)|^{p-2}\nabla u(x)\nabla v(x)dx +
\int_{\Omega}\frac{|u(x)|^{p-2}}{|x|^p}u(x)v(x) dx\end{equation*}
\begin{equation*}-\lambda\int_{\Omega}f(x,u(x))v(x)dx =0,\end{equation*}
for every $v\in X$. Hence, the critical points of $I_{\lambda}$ are
exactly the weak solutions of \eqref{e1.1}.

Our main tools are the following critical point theorems.

\begin{theorem}[\cite{bmk}, Theorem 3.6]\label{the2.1}
 Let $X$ be a reflexive real Banach space, $\Phi:X\rightarrow \mathbb{R}$ be
 a coercive, continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional
 whose G\^{a}teaux derivative admits a continuous inverse on $X^*$,
$\Psi:X\rightarrow \mathbb{R}$ be a continuously G\^{a}teaux
differentiable functional whose G\^{a}teaux derivative is compact
such that$$\inf_{x\in X}\Phi(x)=\Phi(0)=\Psi(0)=0.$$ Assume that
there exist $r>0$ and $\bar{x}\in X,$ with $r<\Phi(\bar{x}),$ such
that:\\ $(a_1) \frac{\sup_{\Phi(x)\leq
r}\Psi(x)}{r}<\frac{\Psi(\bar{x})}{\Phi(\bar{x})};$\\
$(a_2)$ for each $\lambda\in
\Lambda_r:=\left]\frac{\Phi(\bar{x})}{\Psi(\bar{x})} ,
\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}\right[$ the functional
 $\Phi-\lambda\Psi$ is coercive.\\\\Then, for each $\lambda\in\Lambda_r,$ the functional $\Phi-\lambda\Psi$ has
 at least three distinct critical points in $X.$
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{theorem}[\cite{BC}, Corollary 3.1]\label{the2.2} Let $X$ be a reflexive real Banach space,
$\Phi:X \longrightarrow \mathbb{R}$ be a convex, coercive and
continuously G\^ateaux differentiable functional whose derivative
admits a continuous inverse on $X^\ast$, $\Psi:X \longrightarrow
\mathbb{R}$ be a continuously G\^ateaux differentiable functional
whose derivative is compact, such that

 1. $\inf_{X}\Phi=\Phi(0)=\Psi(0)=0;$

 2. for each $\lambda>0$ and for every $u_1,u_2\in X$ which are local
 minima for the functional $\Phi-\lambda\Psi$ and such that $\Psi(u_1)\geq
 0$ and $\Psi(u_2)\geq
 0$, one has
  $$\inf_{s\in[0,1]}\Psi(su_1+(1-s)u_2)\geq 0.$$
Assume that there are two positive constants $r_1,r_2$ and
$\overline{v}\in X,$ with $2r_1<\Phi(\overline{v})<\frac{r_2}{2},$ such that \\

 $(b_1)$ $\displaystyle\frac{\sup_{u\in\Phi^{-1}(]-\infty,r_1[)}\Psi(u)}{r_1}<
 \frac{2}{3}\frac{\Psi(\overline{v})}{\Phi(\overline{v})};$

 $(b_2)$ $\displaystyle\frac{\sup_{u\in\Phi^{-1}(]-\infty,r_2[)}\Psi(u)}{r_2}<
 \frac{1}{3}\frac{\Psi(\overline{v})}{\Phi(\overline{v})}.$\\\\
 Then, for each
\begin{equation*}\lambda\in\displaystyle\left]\frac{3}{2}\frac{\Phi(\overline{v})}{\Psi(\overline{v})},\
\min\left\{
\frac{r_1}{\sup_{u\in\Phi^{-1}(]-\infty,r_1[)}\Psi(u)},\
\frac{\frac{r_2}{2}}{\sup_{u\in\Phi^{-1}(]-\infty,r_2[)}\Psi(u)}\right\}\right[\end{equation*},
the functional $\Phi-\lambda \Psi$ has at least three distinct
critical points which lie in $\Phi^{-1}(]-\infty,r_2[)$.
\end{theorem}
%----------------------------------------------------------------------------------------------------------

\section{Main results}

In this section we establish the main  results of this paper. Now, fix $x_0\in\Omega$ and pick $D>0$ such that $B(x_0,D)\subset\Omega$ 
not containing origin, where $B(x_0,D)$ denotes the ball with center $x_0$ and radious $D.$

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.1}
Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a
Carath\'{e}odory function such that condition $\rm(f_1)$ holds.
Moreover,  assume that
\begin{enumerate}
\item[$\rm(f_2)$] there exist $\alpha\in[0,+\infty[$ and $1<\gamma<p$ such that
$$ F(x,t)\leq\alpha(1+|t|^\gamma),$$
\end{enumerate}for each $(x,t)\in \Omega\times\mathbb{R};$ \begin{enumerate}
\item[$\rm(f_3)$] $F(x,t)\geq0$ for each $(x,t)\in\Omega\times\mathbb{R^+};$\end{enumerate}
\begin{enumerate}
\item[$\rm(f_4)$] there exist $r>0$ and $\delta>0$ with $r<\frac{1}{p}(\frac{2\delta}{D})^p m(D^N-(\frac{D}{2})^N)$
such that
$$\bar{\omega_r}:=\frac{1}{r}\left(a_{1}c_{1}(pr)^{\frac{1}{p}}+
\frac{a_2}{q}(c_q)^q(pr)^{\frac{q}{p}}\right)<\frac{p\inf_{x\in\Omega
}F(x,\delta)}{\left(\frac{H+1}{H}\right)(\frac{2\delta}{D})^p(2^N-1)}$$ where $c_1$  and $ c_q$ are the best constants in \eqref{130}.
\end{enumerate} Then, for each $\lambda \in\Lambda_{r,\delta}=\left]\frac{\left(\frac{H+1}{H}\right)(\frac{2\delta}{D})^p(2^N-1)}
{p\inf_{x\in\Omega}F(x,\delta)},\frac{1}{\bar{\omega_r}}\right[,$
the problem \eqref{e1.1} admits at least three weak solutions.
\end{theorem}

\begin{proof}
Our aim is to apply Theorem 2.2 to problem \rm\eqref{e1.1}.
To this end let $X:=W_{0}^{1,p}(\Omega)$ with the norm
$$ \|u\|:=\left(\int_\Omega |\nabla u(x)|^p dx\right)^{1/p},$$   and the functionals
$\Phi,\Psi:X\rightarrow\mathbb{R}$ be defined by
$$
\Phi(u):=\frac{1}{p}\left(\int_\Omega |\nabla u(x)|^pdx +
\int_\Omega\frac {|u(x)|^p}{|x|^p} dx\right),$$ and $$\Psi
(u):=\int_\Omega F(x,u(x))dx,$$ for all $u\in X.$ \\As seen before,
the functionals $\Phi$ and $\Psi$ satisfy the regularity assumptions
requested in Theorem 2.2. Now, let $\bar{v}\in X$ be defined
by\begin{equation*}\bar{v}(x)=\left\{\begin{array}{lll}
\displaystyle0  &\textrm x\in \Omega\setminus B(x_{0},D)\\
\displaystyle \frac{2\delta}{D}(D-|x-x_{0}|)  &\text x\in
B(x_{0},D)\setminus B(x_{0},\frac{D}{2}), \\
\displaystyle \delta  &
\text x\in B(x_{0},\frac{D}{2})
\end{array}\right.
\end{equation*}\\
where $|.|$ denotes the Euclidean norm on $\mathbb{R^N}.$ We have
\begin{equation*}
\frac{1}{p}\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)\leq\Phi(\bar{v})
 \end{equation*}\begin{equation*}\leq\left(\frac{H+1}{pH}\right)\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)
\end{equation*} where $m:=\frac{\pi^{\frac{N}{2}}}{\frac{N}{2}\Gamma(\frac{N}{2})}$
is the measure of unit ball of $\mathbb{R^N}$ and $\Gamma$ is the
Gamma function. Thanks to $(f_3),$
$$\Psi(\bar{v})\geq\int_{B(x_0,\frac{D}{2})}F(x,\bar{v}(x))dx
\geq\inf_{x\in \Omega}F(x,\delta)m\left(\frac{D}{2}\right)^N$$ and
so \begin{equation}
\frac{\Psi(\bar{v})}{\Phi(\bar{v})}\geq\frac{p\inf_{x\in\Omega}F(x,\delta)}{\left(\frac{H+1}{H}\right)\left(\frac{2\delta}{D}\right)^p\left(2^N-1\right)}.
\end{equation}From $r<\frac{1}{p}\left(\frac{2\delta}{D}\right)^pm\left(D^N-\left(\frac{D}{2}\right)^N\right),$
one has $r<\Phi(\bar{v}).$ Bearing
in mind define the functional  $\Phi$, we see that
\begin{eqnarray*}
 \Phi^{-1}(]-\infty,r])&=&\{u\in X;\ \Phi(u)\leq r\}
 \\&\subseteq&\left\{u\in X;\ \frac{\|u\|^{p}}{p}\leq r\right\} .\end{eqnarray*} \\So, the compact embedding
$X\hookrightarrow L^q(\Omega)$  and $(f_1)$ imply that, for each
$u\in\Phi^{-1}(]-\infty,r]),$ we have
$$\Psi(u)\leq
a_1\int_{\Omega}|u(x)|dx+\frac{a_2}{q}\int_{\Omega}|u(x)|^qdx\leq
a_1c_1\|u\|+\frac{a_2}{q}\left(c_q\|u\|\right)^q$$$$\leq
a_1c_1(pr)^{\frac{1}{p}}+\frac{a_2}{q}(c_q)^q(pr)^{\frac{q}{p}}$$

and so \begin{equation}\frac{1}{r}\sup_{\Phi(u)\leq
r}\Psi(u)\leq\frac{1}{r}\left(a_1c_1(pr)^{\frac{1}{p}}+\frac{a_2}{q}(c_q)^q(pr)^{\frac{q}{p}}\right)\end{equation}
and so condition $(a_1)$ of Theorem 2.2 is verified. Now, let us
introduce the integral functional related to problem \eqref{e1.1}
$$I_\lambda(.):=\Phi(.)-\lambda\Psi(.)$$ and we prove that, for each $\lambda>0,$ $I_\lambda$ is
coercive. By arguments similar to those used before, we obtain
$$\int_\Omega |u(x)|^\gamma dx\leq (c_\gamma \|u\|)^\gamma $$
and so, for each $u\in X$ with
$\|u\|\geq\max\{1,\frac{1}{c_\gamma}\},$ from $(f_2)$ one has
$$\Psi(u)=\int_\Omega F(x,u(x))dx\leq\int_\Omega \alpha(1+|u(x)|^\gamma)dx\leq\alpha\left(|\Omega|+(c_\gamma\|u\|)^\gamma\right).$$
This leads to
$$I_\lambda(u)\geq\frac{1}{p}\|u\|^p-\lambda\alpha\left(|\Omega|+(c_\gamma\|u\|)^\gamma\right)$$
and, since $\gamma<p,$ coercivity of $I_\lambda$ is obtained. Taking
into account that
$$\Lambda_{r,\delta}\subseteq\left]\frac{\Phi(\bar{v})}{\Psi(\bar{v})},\frac{r}{\sup_{\Phi(u)\leq r}\Psi(u)}\right[,$$
Theorem 2.2 ensures that, for each $\lambda\in\Lambda_{r,\delta},$
the functional $I_\lambda$ admits at least three critical points in
$X$ that are weak solutions of the problem $\rm\eqref{e1.1}.$
\end{proof}
%................................................................................................................

\begin{remark}
In Theorem $\rm3.1,$ if we consider $f(x,0)\neq0$, then we
obtain the existence of at least three non-zero weak solutions.
\end{remark}

\begin{remark}
According to the Sobolev
embedding theorem there is a positive constant $c$ such that
\begin{equation}\label{121}
 \|u\|_{L^{p^*}(\Omega)}\leq c\|u\|,\qquad (\forall u\in
 X).
 \end{equation}
 The best approximation for constant $c$ in
\eqref{121} is \begin{equation}
c:=\frac{1}{N\sqrt{\pi}}\left(\frac{N!\Gamma(\frac{N}{2})}{2\Gamma(\frac{N}{p})
\Gamma(N+1-\frac{N}{p})}\right)^{1/N}\eta^{1-\frac{1}{p}},
\end{equation} where
$$\eta:=\frac{N(p-1)}{N-p},$$ (see, for instance, \cite{Dp}). The consequences of using the $\rm H\ddot{o}lder$'s
inequality, in \eqref{130},
 is as follows
$$c_{q}\leq \frac{{\rm meas}(\Omega)^{\frac{p^*-q}{p^*q}}}{N\sqrt{\pi}}
\left(\frac{N!\Gamma(\frac{N}{2})}{2\Gamma(\frac{N}{p})\Gamma(N+1-N/p)}\right)^{1/N}\eta^{1-1/p},$$\\
where $\rm meas$ $(\Omega)$ denotes the Lebesgue measure of the set
$\Omega$.
\end{remark}

%----------------------------------------------------------------------------------------------------------
Another  the main result of this section is as follows.
\begin{theorem}\label{the3.1}
Let  $f:\Omega\times \mathbb{R}\to
\mathbb{R}$ be  Carath\'{e}odory function  with assumption $(f_1)$  that satisfies the condition $f(x,t)\geq 0$ for every
$(x,t)\in \Omega\times\mathbb{R}$.
Moreover,~assume that
there exist three positive constants $r_1,\ r_2$ and
$\delta$ with\\$r_1<\frac{1}{2p}\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)$ and $2\left(\frac{H+1}{pH}\right)\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)<r_2$.~Furthermore,~suppose that\\\\
$(B_1)$ $\bar{\omega}_{r_{1}}:=\frac{1}{r_1}\left(a_1c_1(pr_1)^{\frac{1}{p}}+\frac{a_2}{q}(c_q)^q(pr_1)^{\frac{q}{p}}\right)<\frac{2}{3}\frac{p\inf_{x\in\Omega}F(x,\delta)}{\left(\frac{H+1}{H}\right)\left(\frac{2\delta}{D})\right)^p\left(2^N-1)\right)}$;\\\\$(B_2)$
$\bar{\omega}_{r_{2}}:=\frac{1}{r_2}\left(a_1c_1(pr_2)^{\frac{1}{p}}+\frac{a_2}{q}(c_q)^q(pr_2)^{\frac{q}{p}}\right)<\frac{1}{3}\frac{p\inf_{x\in\Omega}F(x,\delta)}{\left(\frac{H+1}{H}\right)\left(\frac{2\delta}{D})\right)^p\left(2^N-1)\right)}$.\\\\Then, for each $\lambda \in\left]\frac{3}{2}\frac{\left(\frac{H+1}{H}\right)(\frac{2\delta}{D})^p(2^N-1)}
{p\inf_{x\in\Omega}F(x,\delta)},\min\{\frac{1}{\bar{\omega}_{r_{1}}},\frac{1}{2\,\bar{\omega}_{r_{2}}}\}\right[,$
the problem \eqref{e1.1} admits at least three weak solutions $u_i$ for $i=1,2,3$,
such that $\| u_i\|<(p\,r_2)^{\frac{1}{p}}$.\end{theorem}\begin{proof}  Take $\Phi$
and $\Psi$ as in the proof of Theorem $3.1$.~Our aim is to verify $(b_1)$ and
$(b_2)$  in Theorem $2.3$. To this end, choose $\bar{v}$ as given in Theorem $3.1$. Using \begin{equation*}
\frac{1}{p}\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)\leq\Phi(\bar{v})
\leq \end{equation*}$$\left(\frac{H+1}{pH}\right)\left(\frac{2\delta}{D}\right)^p
m\left(D^N-\left(\frac{D}{2}\right)^N\right)
$$ and theorem data it is clear that we have $2r_1<\Phi(\overline{v})<\frac{r_2}{2}$ . Now we have \begin{eqnarray*}\frac{\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_1[)}\Psi(u)}{r_1}
&=&\frac{\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_1[)}\int_{\Omega}F(x,u(x))dx}{r_1}
\\ &\leq  & \bar{\omega}_{r_{1}}<\frac{1}{\lambda}
 <\frac{2}{3}\frac{\Psi(\bar{v})}{\Phi(\bar{v})}\end{eqnarray*}and\\ \begin{eqnarray*}\frac{2\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_2[)}\Psi(u)}{r_2}
&=&\frac{2\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_2[)}\int_{\Omega}F(x,u(x))dx}{r_2}
\\ &\leq  &2\,\bar{\omega}_{r_{2}}<\frac{1}{\lambda}
 <\frac{2}{3}\frac{\Psi(\bar{v})}{\Phi(\bar{v})}\,\,.
 \end{eqnarray*}Therefore, $(b_1)$ and $(b_2)$ of Theorem $2.3$ are established.
 Finally, we verify that $\Phi-\lambda\Psi$ satisfies the assumption 
 \textit{2.} of Theorem $2.3$. Let
$u_1$ and $u_2$ be two local minima for $\Phi-\lambda\Psi$.~Then
$u_1$ and $u_2$ are critical points for $\Phi-\lambda\Psi$, and
so, they are weak solutions for the problem $(1)$.~We will show that they are nonnegative.\par
Let $\bar{u}$ be a weak solution of problem $(1)$.~Using the argument of  contradiction, assume that the set
 $T=\big\{x \in\Omega : \bar{u}(x)<0\big\}$ is non-empty and of positive measure. Put $u^{*}(x)=\min\{0, \bar{u}(x)\}$
 for all $x \in\Omega$.~It is clear that, $u^{*}\in X$ and hence we have
\begin{equation*}
\int_{\Omega}|\nabla \bar{u}(x)|^{p-2}\nabla \bar{u}(x)\nabla u^{*}(x)dx +
\int_{\Omega}\frac{|\bar{u}(x)|^{p-2}}{|x|^p}\bar{u}(x)u^{*}(x) dx \end{equation*}
$$-\lambda\int_{\Omega}f(x,\bar{u}(x))u^{*}(x)dx =0.$$\par Thus,~from our sign assumptions on the
data,~we have
$$
0\leq\int_{T}|\nabla \bar{u}(x)|^{p}dx +
\int_{T}\frac{|\bar{u}(x)|^{p}}{|x|^p}dx\leq 0.
$$
Hence,  $\bar{u}=0$  in $T$ and this is antithesis.~Hence, $u_{1}(x)\geq 0$
and $u_{2}(x)\geq 0$ for every $x\in\Omega $.~Thus, it follows that
$su_{1}+(1-s)u_{2}\geq 0$ for all $s\in [0,1]$, and that $$
f(x,su_1+(1-s)u_2)\geq 0,$$ and consequently,
$\Psi(su_1+(1-s)u_2)\geq 0$, for every $s\in [0,1]$.\par
 By using Theorem
$2.3$, for every
$$\lambda \in\left]\frac{3}{2}\frac{\left(\frac{H+1}{H}\right)(\frac{2\delta}{D})^p(2^N-1)}
{p\inf_{x\in\Omega}F(x,\delta)},\min\{\frac{1}{\bar{\omega}_{r_{1}}},\frac{1}{2\,\bar{\omega}_{r_{2}}}\}\right[ \subseteq$$$$
\left]\frac{3}{2}\frac{\Phi(w)}{\Psi(w)},\ \min\left\{
\frac{r_1}{\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_1[)}\Psi(u)},\
\frac{{r_2}/{2}}{\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r_2[)}\Psi(u)}\right\}\right[,
$$
the functional $\Phi-\lambda\Psi$ has at least three distinct
critical points which are the weak solutions of the problem
\eqref{e1.1} and the proof  is complete.
 \end{proof}\\To illustrate the Theorem $3.2$,~we provide an example.\begin{example}Let  $f: \mathbb{R}\to
\mathbb{R}$ be a non-negative continuous function with $F(t)=\int_0^t f(\xi)d\xi$ and $f(t)\leq t^{2}$ for every $t\in\mathbb{R}$.~Also suppose that there exist positive constants $r_1$ , $r_2$ and $\delta$ such that the following inequalities hold.\\$(i_1)$ $r_{1}<\min\{\delta^{\frac{3}{2}}\pi,\frac{4}{3(c_3)^3}\}$    ,  $4\left(\frac{H+1}{H}\right)\delta^{\frac{3}{2}}\pi<r_2<\frac{2}{3(c_3)^3}$ \\$(i_2)$ $3\left(\frac{H+1}{H}\right)\delta^{\frac{3}{2}}<\int_0^\delta f(\xi)d\xi$ \\ where $c_3$ is best constant in $(3)$.In this case problem $$\left\{\begin{array}{ll}
\displaystyle-\Delta_{\frac{3}{2}}u + \frac{|u|^{\frac{-1}{2}}u}{|x|^\frac{3}{2}} = f(u), &\textrm{ in } \Omega,\\
\displaystyle u=0, & \text{ on }
\partial\Omega,
\end{array}\right. $$ where  $\Omega$ be a bounded domain in $\mathbb{R}^{2}$
containing the origin and containing the ball with radious 2 not containing origin and  with smooth boundary $\partial\Omega$, has at least three weak solutions. To this end according to conditions $f(t)\leq |t|^{2}$ and $(f_1)$ we can consider $a_1=0$ , $a_2=1$ and $q=3$. also consider $p=\frac{3}{2}$ and $ D=2$. On the other hand, from $N=2$ we have $m=\pi$. Hence according to Theorem $3.2$, it is enough to show that $$\lambda=1 \in\left]\frac{3}{2}\frac{\left(\frac{H+1}{H}\right)(\frac{2\delta}{D})^p(2^N-1)}
{p\inf_{x\in\Omega}F(x,\delta)},\min\{\frac{1}{\bar{\omega}_{r_{1}}},\frac{1}{2\,\bar{\omega}_{r_{2}}}\}\right[$$ and this, according to the inequalities $(i_1)$ and $(i_2)$ can be easily researched. \end{example}
\section*{Acknowledgements}The authors express their gratitude to the referees
 who reviewed this paper. 
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{\small

\noindent{\bf Mehdi Khodabakhshi}

\noindent Department of Mathematics and computer sciences



\noindent Amir Kabir University of Technology

\noindent Tehran, Iran

\noindent E-mail:  m.khodabakhshi11@gmail.com}\\



{\small

\noindent{\bf Mohammad Reza Heidari Tavani}\\
\noindent Assistant Professor of Mathematics\\
\noindent Department of Mathematics



\noindent Ramhormoz Branch, Islamic Azad University

\noindent Ramhormoz, Iran

\noindent E-mail: m.reza.h56@gmail.com}

\end{document}