\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 32, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/32\hfil Elliptic equations with a concave term]
{Singular elliptic equations involving a concave term and
critical Caffarelli-Kohn-Nirenberg exponent with sign-changing
weight functions}
\author[M. Bouchekif, A. Matallah\hfil EJDE-2010/32\hfilneg]
{Mohammed Bouchekif, Atika Matallah} % in alphabetical order
\address{Mohammed Bouchekif \newline
Universit\'{e} Aboubekr Belkaid,
D\'{e}partement de Math\'{e}matiques,
BP 119 (13000) Tlemcen - Alg\'{e}rie}
\email{m\_bouchekif@yahoo.fr}
\address{Atika Matallah \newline
Universit\'{e} Aboubekr Belkaid,
D\'{e}partement de Math\'{e}matiques,
BP 119 (13000) Tlemcen - Alg\'{e}rie.}
\email{atika\_matallah@yahoo.fr}
\thanks{Submitted September 28, 2009. Published March 3, 2010.}
\subjclass[2000]{35A15, 35B25, 35B33, 35J60}
\keywords{Variational methods; critical
Caffarelli-Kohn-Nirenberg exponent; \hfill\break\indent
concave term; singular and sign-changing
weights; Palais-Smale condition}
\begin{abstract}
In this article we establish the existence of at least two distinct
solutions to singular elliptic equations involving a concave term
and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing
weight functions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
This article shows the existence of at least two
solutions to the problem
\begin{equation} \label{p-lambda-mu}
\begin{gathered}
-\mathop{\rm div}\big(\frac{\nabla u}{|x|^{2a}}\big) -\mu
\frac{u}{|x|^{2(a+1)}}=\lambda h(x) \frac{| u|
^{q-2}u}{|x|^{c}}+k(x)\frac{| u|^{2_{\ast }-2}u}{|x| ^{2_{\ast
}b}} \quad \text{in }\Omega \backslash\{0\}\\
u=0 \quad \text{on}\ \partial \ \Omega
\end{gathered}
\end{equation}
where $\Omega \subset\mathbb{R}^{N}$ is an open bounded domain,
$N\geq 3$, $0\in \Omega $, $a<( N-2) /2$,
$a\leq b0$ such that $(\mathcal{P}_{\lambda ,0})$ for $\lambda $
fixed in $(0,\Lambda _{0})$ has at least two positive solutions by
using sub-super method and the Mountain Pass Theorem, problem \eqref{p-lambda-mu} for $\lambda =\Lambda _{0}$ has also
a positive solution and no positive solution for
$\lambda>\Lambda _{0}$. When $\mu >0$, $a=b=c=0$, Chen \cite{C}
studied the
asymptotic behavior of solutions to problem \eqref{p-lambda-mu}
by using the Moser's iteration. By applying the Ekeland
Variational Principle he obtained a first positive solution, and
by the Mountain Pass Theorem he proved the existence of a second
positive solution. Recently, Bouchekif and Matallah
\cite{BA} extended the results of \cite{C} to problem
$(\mathcal{P}_{\lambda ,\mu })\ $with $a=c=0$, $0\leq b<1$, they
established the existence of two positive solutions under some
sufficient conditions for $\lambda $ and $\mu $. Lin \cite{Lin}
considered a more general problem \eqref{p-lambda-mu}
with $0\leq a<(N-2)/2$, $a\leq b0$.
For the case $h\not\equiv 1$ or $k\not\equiv 1$, we refer the
reader to \cite{BA2,HH,T,W} and the
references therein. Tarantello \cite{T} studied the problem
\eqref{p-lambda-mu} for $\mu=0$, $a=b=c=0$, $q=\lambda =1$,
$k\equiv 1$ and $h$ not necessarily equals to $1$, satisfying some
conditions.
Recently, problem \eqref{p-lambda-mu} in $\Omega =
\mathbb{R}^{N}$ with $q=1$ has considered in \cite{BA2}.
Wu \cite{W} showed the existence of multiple positive solutions
for problem \eqref{p-lambda-mu} with $a=b=c=0$,
$10$, the function
\begin{equation} \label{e1.4}
u_{\varepsilon }(x)=C_{0}\varepsilon ^{\frac{2}{2_{\ast }-2}
}\Big(\varepsilon ^{\frac{2\sqrt{\bar{\mu }_{a}-\mu }}{\sqrt{
\bar{\mu }_{a}-\mu }-b}}|x|^{\frac{2_{\ast }-2}{2}
(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu }_{a}-\mu
})}+|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu }
_{a}}+\sqrt{\bar{\mu }_{a}-\mu })}\Big)^{-\frac{2}{2_{\ast }-2 }}
\end{equation}
with a suitable positive constant $C_{0}$, is a weak solution
of
\[
-\mathop{\rm div}\big(|x|^{-2a}\nabla u\big)
-\mu | x| ^{-2(a+1)}u=|x| ^{-2_{\ast}b}|u|^{2_{\ast }-2}u\quad
\text{in }\mathbb{R}^{N}\backslash \{0\}.
\]
Furthermore,
\begin{equation}
\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla u_{\varepsilon }|^{2}dx
-\mu \int_{\mathbb{R}^{N}}|x|^{-2(a+1)}u_{\varepsilon }^{2}dx
=\int_{\mathbb{R}^{N}}|x|^{-2_{\ast }b}|u_{\varepsilon }|^{2_{\ast }}dx
=A_{a,b,\mu}, \label{0}
\end{equation}
where $A_{a,b,\mu }$ is the best constant,
\begin{equation} \label{e1.6}
A_{a,b,\mu }=\inf_{u\in H_{\mu }\backslash \{0\}}
E_{a,b,\mu }(u)=E_{a,b,\mu }(u_{\varepsilon }),
\end{equation}
with
\[
E_{a,b,\mu }(u):=\frac{\int_{\mathbb{R}^{N}}
|x|^{-2a}|\nabla u|^{2}dx-\mu \int_{\mathbb{R}^{N}}
|x|^{-2(a+1)}u^{2}dx}{( \int_{\mathbb{R}^{N}}
|x|^{-2_{\ast }b}|u|^{2_{\ast }}dx)^{2/2_{\ast }}}.
\]
Also in \cite{K} and \cite{Lin}, they proved that for
$0\leq a<(N-2)/2$, $a\leq b0$ as
\begin{equation}
v_{\varepsilon }(x)=(2.2_{\ast }\varepsilon ^{2}( \bar{\mu
}_{a}-\mu ))^{\frac{1}{2_{\ast }-2}}\Big(\varepsilon ^{2}| x|^{\frac{(
2_{\ast }-2)(\sqrt{\bar{\mu }_{a}}-\sqrt{\bar{\mu
}_{a}-\mu })}{2} }+| x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu
} _{a}}+\sqrt{\bar{\mu }_{a}-\mu })}\Big)^{-\frac{2}{2_{\ast }-2
}}
\end{equation}
is a weak solution of
\[
-\mathop{\rm div}(|x|^{-2a}\nabla u)-\mu | x| ^{-2(a+1)}u
=|x| ^{-2_{\ast}b}|u|^{2_{\ast }-2}u\quad \text{in }
\mathbb{R}^{N}\backslash \{0\},
\]
and satisfies
\begin{equation}
\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla v_{\varepsilon }|^{2}dx
-\mu \int_{\mathbb{R}^{N}}|x|^{-2(a+1)}v_{\varepsilon }^{2}dx
=\int_{\mathbb{R}^{N}}|x|^{-2_{\ast }b}|v_{\varepsilon }|^{2_{\ast }}dx
=B_{a,b,\mu},
\end{equation}
where $B_{a,b,\mu }$ is the best constant,
\begin{equation}
B_{a,b,\mu }:=\inf_{u\in H_{\mu }\backslash \{0\}}
E_{a,b,\mu }(u)=E_{a,b,\mu }(v_{\varepsilon }).
\end{equation}
A natural question that arises in concert applications is to see what
happens if these elliptic (degenerate or non-degenerate) problems are
affected by a certain singular perturbations. In our work we prove the
existence of at least two distinct nonnegative critical points of energy
functional associated to problem \eqref{p-lambda-mu} by
splitting the Nehari manifold (see for example Tarantello \cite{T}
or Brown and Zhang \cite{BZ}).
In this work we consider the following assumptions:
\begin{itemize}
\item[(H)] $h$ is a continuous function defined in
$\bar{\Omega}$ and there exist $h_{0}$ and $\rho_{0}$
positive such that
$h(x)\geq h_{0}$ for all $x\in B(0,2\rho _{0})$,
where $B(a,r)$ is a ball centered at $a$ with radius $r$;
\item[(K)] $k$ is a continuous function defined in
$\bar{\Omega}$ and satisfies
$k(0)=\max_{x\in \bar{\Omega} }k(x)>0$,
$k(x)=k(0)+o (x^{\beta })$ for $x\in B(0,2\rho _{0})$ with
$\beta>2_{\ast }\sqrt{\bar{\mu }_{a}-\mu }$;
\end{itemize}
and one of the following two assumptions
\begin{itemize}
\item[(A1)] $N >2(|b|+1)$ and
\[
(a,\mu )\in ]-1,0[ \times ] 0, \bar{\mu }_{a}-b^{2}[ \cup
[ 0,\tfrac{N-2}{2}[ \times] a(a-N+2),\bar{\mu }_{a}-b^{2}[ ,
\]
\item[(A2)] $N \geq 3$, $(a,\mu )\in [0,\frac{N-2}{2}[ \times
[ 0,\bar{\mu }_{a}[$.
\end{itemize}
Following the method introduced in \cite{T,HH}, we obtain the
following existence result.
\begin{theorem} \label{thm1}
Suppose that $a<(N-2)/2$, $a\leq b0$ such
that for $\lambda \in ( 0,\Lambda ^{\ast })$
problem \eqref{p-lambda-mu} has at least two nonnegative solutions in
$H_{\mu }$.
\end{theorem}
This paper is organized as follows.
In section 2 we give some preliminaries.
Section 3 is devoted to the proof of Theorem \ref{thm1}.
\section{Preliminary results}
We start by giving the following definitions.
Let $E$ be a Banach space and a functional $I\in
\mathcal{C}^{1}(E,\mathbb{R) }$. We say that $(u_{n})$ is a Palais
Smale sequence at level $l$ ($(PS)_l$ in short)
if $I(u_{n})\to l$ \ and $I'(u_{n})\to 0$ in $E'$
(dual of $E$) as $n\to \infty $. We say also that $I$
satisfies the Palais Smale condition at level $l$ if any
$(PS)_l$ sequence has a subsequence converging strongly in $E$.
Define
\begin{equation}
w_{\varepsilon }:= \begin{cases}
u_{\varepsilon } &\text{if }(a,\mu )\in ] -1,0[ \times
] 0,\bar{\mu }_{a}-b^{2}[ \cup [0,\tfrac{N-2}{2} [ \times ] a(a-N+2) ,
\bar{\mu}_{a}-b^{2}[, \\
v_{\epsilon } &\text{if }(a,\mu )\in [ 0,\tfrac{N-2}{2}
[ \times [ 0,\bar{\mu }_{a}[ ,
\end{cases}
\label{e2.1} %u
\end{equation}
and
\begin{equation}
\begin{aligned}
&S_{a,b,\mu }:=\\
&\begin{cases}
A_{a,b,\mu }&\text{if }(a,\mu )\in ] -1,0[ \times
] 0,\bar{\mu }_{a}-b^{2}[ \cup [0,\tfrac{N-2}{2}[ \times ] a(a-N+2) ,
\bar{\mu}_{a}-b^{2}[, \\
B_{a,b,\mu }&\text{if }(a,\mu )\in [ 0,\tfrac{N-2}{2}[
\times [ 0,\bar{\mu }_{a}[ ,.
\end{cases} \label{s}
\end{aligned}
\end{equation}
Since our approach is variational, we define the functional
$I_{\lambda ,\mu }$ as
\[
I_{\lambda ,\mu }(u)=\frac{1}{2}\| u\|_{\mu ,a}^{2}-\frac{\lambda
}{q}\int_{\Omega }h(x)| x| ^{-c}|u|^{q}dx-\frac{1}{2_{\ast }}
\int_{\Omega }k(x)|x| ^{-2_{\ast }b}|u|^{2_{\ast }}dx,
\]
for $u\in H_{\mu }$.
By \eqref{ee} and \eqref{e1.3} we can guarantee that $I_{\lambda ,\mu }$
is well defined in $H_{\mu }$ and
$I_{\lambda ,\mu }\in C^{1}(H_{\mu }, \mathbb{R})$.
$u\in H_{\mu }$ is said to be a weak solution of \eqref{p-lambda-mu}
if it satisfies
\[
\int_{\Omega }(|x|^{-2a}\nabla u\nabla v-\mu | x|
^{-2(a+1)}uv-\lambda h( x) |x|^{-c}|u|^{q-2}uv-k(x) |x| ^{-2_{\ast
}b}|u|^{2_{\ast }-2}uv)dx=0
\]
for all $v\in H_{\mu }$. By the standard elliptic regularity
argument, we have that $u\in C^{2}(\Omega \backslash \{ 0\} )$.
In many problems as \eqref{p-lambda-mu}, $I_{\lambda ,\mu }$ is
not bounded below on $H_{\mu }$ but is bounded below on an appropriate
subset of $H_{\mu }$ and a minimizer in this set (if it exists) may give
rise to solutions of the corresponding differential equation.
A good candidate for an appropriate subset of $H_{\mu }$ is the so called
Nehari manifold
\[
\mathcal{N}_{\lambda }=\{u\in H_{\mu }\backslash \{0\},
\langle I_{\lambda ,\mu }'( u),u\rangle =0\}.
\]
It is useful to understand $\mathcal{N}_{\lambda }$ in terms of the
stationary points of mappings of the form
\[
\Psi _{u}(t)=I_{\lambda ,\mu }(tu),\text{ \ }t>0,
\]
and so
\[
\Psi _{u}'(t)=\langle I_{\lambda ,\mu }'(tu),u\rangle
=\frac{1}{t}\langle I_{\lambda ,\mu }'(tu),tu\rangle .
\]
An immediate consequence is the following proposition.
\begin{proposition} \label{prop1}
Let $u\in H_{\mu }\backslash \{0\}$ and $t>0$. Then $tu\in
\mathcal{N}_{\lambda }$ if and only if $\Psi _{u}'(t)=0$.
\end{proposition}
Let $u$ be a local minimizer of $I_{\lambda ,\mu }$, then
$\Psi _{u}$ has a
local minimum at $t=1$. So it is natural to split $\mathcal{N}_{\lambda }$
into three subsets $\mathcal{N}_{\lambda }^{+}$, $\mathcal{N}_{\lambda }^{-}$
and $\mathcal{N}_{\lambda }^{0}$ corresponding respectively to local
minimums, local maximums and points of inflexion.
We define
\begin{align*}
\mathcal{N}_{\lambda }^{+}
&=\big\{u\in \mathcal{N}_{\lambda }:
(2-q)\| u\|_{\mu ,a}^{2}-(2_{\ast }-q)\int_{\Omega }k(x)\frac{|u|
^{2_{\ast }}}{|x|^{2_{\ast }b}}dx>0\big\}\\
&=\{u\in \mathcal{N}_{\lambda }:(2-2_{\ast
})\| u\|_{\mu ,a}^{2}+( 2_{\ast }-q)\lambda \int_{\Omega
}h(x)\frac{|u|^{q}}{ |x| ^{c}}dx>0\}.
\end{align*}
Note that $\mathcal{N}_{\lambda }^{-}$ and
$\mathcal{N}_{\lambda }^{0}$ similarly by
replacing $>$ by $<$ and $=$ respectively.
\begin{equation}
c_{\lambda }:=\inf_{u\in \mathcal{N}_{\lambda }}I_{\lambda ,\mu }(
u);\text{ }c_{\lambda }^{+}:=\inf_{u\in \mathcal{N}_{\lambda
}^{+}}I_{\lambda ,\mu }(u);\quad
c_{\lambda }^{-}:=\inf_{u\in
\mathcal{N}_{\lambda }^{-}}I_{\lambda ,\mu }(u).
\end{equation}
The following lemma shows that minimizers on $\mathcal{N}_{\lambda }$ are
critical points for $I_{\lambda ,\mu }$.
\begin{lemma} \label{lem1}
Assume that $u$ is a local minimizer for $I_{\lambda ,\mu }$ on
$\mathcal{N} _{\lambda }$ and that $u\notin \mathcal{N}_{\lambda
}^{0}$. Then $I_{\lambda ,\mu }'(u)=0$.
\end{lemma}
The proof of the above lemma is essentially the same as that of
\cite[Theorem 2.3]{BZ}.
\begin{lemma} \label{lem2}
Let
\[
\Lambda _1:=\Big(\frac{2-q}{2_{\ast }-q}\Big)^{\frac{2-q}{2_{\ast }-q}
}\Big(\frac{2_{\ast }-2}{(2_{\ast }-q)C_1}\Big)| h^{+}| _{\infty
}^{-1}|k^{+}|_{\infty }(S_{a,b,\mu })^{\frac{N( 2-q)}{4(a+1-b)}},
\]
where $\eta ^{+}(x)=\max (\eta (x),0)$, and
$|\eta ^{+}|_{\infty}=\sup_{x\in \Omega }ess|\eta ^{+}(x)|$. Then
$\mathcal{N}_{\lambda }^{0}=\emptyset$ for all
$\lambda \in( 0,\Lambda _1)$.
\end{lemma}
\begin{proof}
Suppose that $\mathcal{N}_{\lambda }^{0}\neq \emptyset $.
Then for $u\in \mathcal{N}_{\lambda }^{0}$, we have
\begin{gather*}
\| u\|_{\mu ,a}^{2}=\frac{2_{\ast }-q}{2-q}\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx, \\
\| u\|_{\mu ,a}^{2}=\lambda \frac{2_{\ast }-q}{2_{\ast }-2}
\int_{\Omega }h(x)\frac{|u|^{q}}{ |x| ^{c}}dx.
\end{gather*}
Moreover by (H), (K), Caffarelli-Kohn-Nirenberg and H\"{o}lder
inequalities, we obtain
\begin{gather*}
\| u\|_{\mu ,a}^{2}\geq \Big(\frac{2-q}{( 2_{\ast }-2)|k^{+}|_{\infty
}}( S_{a,b,\mu }) ^{2_{\ast }/2}\Big)^{2/(2_{\ast }-2)}, \\
\| u\|_{\mu ,a}^{2}\leq \Big(\lambda \frac{2_{\ast }-q}{ 2_{\ast
}-2}(S_{a,b,\mu })^{-q/2}C_1| h^{+}| _{\infty }\Big)^{2/(2-q)}.
\end{gather*}
Thus
$\lambda \geq \Lambda _1$.
From this, we can conclude that $\mathcal{N}_{\lambda
}^{0}=\emptyset $ if $\lambda \in (0,\Lambda _1)$.
\end{proof}
Thus we conclude that $\mathcal{N}_{\lambda }=\mathcal{N}_{\lambda
}^{+}\cup \mathcal{N}_{\lambda }^{-}$ for all $\lambda \in
(0,\Lambda _1)$.
\begin{lemma} \label{lem3}
Let $c_{\lambda }^{+}$, $c_{\lambda }^{-}$ defined in \eqref{e2.1}.
Then there exists $\delta _{0}>0$ such that
\[
c_{\lambda }^{+}<0\; \forall \lambda \in (0,\Lambda
_1)\quad \text{and}\quad
c_{\lambda }^{-}>\delta _{0}\;\forall
\lambda \in (0,\frac{q}{2}\Lambda _1).
\]
\end{lemma}
\begin{proof}
Let $u\in \mathcal{N}_{\lambda }^{+}$. Then
\[
\int_{\Omega }k(x)\frac{|u|^{2_{\ast }}}{ | x| ^{2_{\ast
}b}}dx<\frac{2-q}{2_{\ast }-q}\| u\|_{\mu ,a}^{2},
\]
which implies
\begin{align*}
c_{\lambda }^{+} &\leq I_{\lambda ,\mu }(u)\\
&= \big(\frac{1}{2}-\frac{1}{q}\big)\| u\|_{\mu
,a}^{2}+\big(\frac{1}{q}-\frac{1}{2_{\ast }}\big)\int_{\Omega }k(x)
\frac{|u|^{2_{\ast }}}{|
x|^{2_{\ast }b}}dx \\
&< -\frac{(2-q)(2_{\ast }-2)}{2.2_{\ast }q}
\| u\|_{\mu ,a}^{2}
< 0.
\end{align*}
Let $u\in \mathcal{N}_{\lambda }^{-}$. Then
\[
\frac{2-q}{2_{\ast }-q}\| u\|_{\mu ,a}^{2}<\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x| ^{2_{\ast }b}}dx.
\]
Moreover by (H), (K) and Caffarelli-Kohn-Nirenberg inequality,
we have
\[
\int_{\Omega }k(x)\frac{|u|^{2_{\ast }}}{ | x| ^{2_{\ast
}b}}dx\leq ( S_{a,b,\mu }) ^{-2_{\ast }/2}\| u\|_{\mu ,a}^{2_{\ast
}}| k^{+}|_{\infty }.
\]
This implies
\[
\| u\|_{\mu ,a}>\big(\frac{2-q}{(2_{\ast }-2) |k^{+}|_{\infty
}}\big)^{1/(2_{\ast }-2) }(S_{a,b,\mu })^{2_{\ast }/(2(2_{\ast }-2))}.
\]
On the other hand,
\[
I_{\lambda ,\mu }(u)\geq \| u\|_{\mu
,a}^{q}\Big(\big(\frac{1}{2}-\frac{1}{2_{\ast }}\big)\| u\|_{\mu
,a}^{2-q}-\lambda \frac{2_{\ast }-q}{2_{\ast }q}(S_{a,b,\mu
})^{-q/2}C_1|h^{+}|_{\infty }\Big)
\]
Thus, if $\lambda \in (0,\frac{q}{2}\Lambda _1)$ we get
$I_{\lambda ,\mu }(u)\geq \delta _{0}$,
where
\begin{align*}
\delta _{0}
&:= \Big(\tfrac{2-q}{(2_{\ast }-2)| k^{+}| _{\infty
}}\Big)^{\frac{q}{2_{\ast }-2}}( S_{a,b,\mu })^{\frac{2_{\ast
}q}{2(2_{\ast }-2) }}
\Big(\big(\frac{1}{2}-\frac{1}{2_{\ast }}\big)(S_{a,b,\mu
})^{ \frac{2_{\ast }(2-q)}{2(2_{\ast }-2)}}\big( \tfrac{2-q}{(2_{\ast
}-q) |k^{+}|_{\infty }}
\big)^{\frac{2-q}{2_{\ast }-2}}\\
&\quad -\lambda \tfrac{2_{\ast }-q}{2_{\ast }-2}(S_{a,b,\mu
})^{-q/2}C_1|h^{+}|_{\infty }\Big).
\end{align*}
\end{proof}
As in \cite[Proposition 9]{W}, we have the following result.
\begin{lemma} \label{lem4}
\begin{itemize}
\item[(i)] If $\lambda \in (0,\Lambda _1)$, then there exists a
$(PS)_{c_{\lambda }}$ sequence $( u_{n})\subset
\mathcal{N}_{\lambda }$ for $I_{\lambda ,\mu }$.
\item[(ii)] If $\lambda \in (0,\frac{q}{2}\Lambda _1)$, then there
exists a $(PS)_{c_{\lambda }^{-}}$ sequence $(u_{n})\subset
\mathcal{N}_{\lambda }^{-}$ for $I_{\lambda ,\mu }$.
\end{itemize}
\end{lemma}
We define
\begin{gather*}
K^{+}:=\big\{u\in \mathcal{N}_{\lambda }:\int_{\Omega }k(
x)\frac{|u|^{2_{\ast }}}{|x|^{2_{\ast }b}}dx>0\big\},
\quad
K_{0}^{-}:=\big\{u\in \mathcal{N} _{\lambda }:\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{|x|^{2_{\ast }b}}dx\leq 0\big\},
\\
H^{+}:=\{u\in \mathcal{N}_{\lambda }:\int_{\Omega }h(
x)\frac{|u|^{q}}{|x|^{c}} dx>0\},
\quad
H_{0}^{-}:=\{u\in \mathcal{N}_{\lambda }:
\int_{\Omega }h(x)\frac{|u|^{q}}{ |x|^{c}}dx\leq 0\},
\end{gather*}
and
\[
t_{\rm max}=t_{\rm max}(u):=\big(\frac{2-q}{2_{\ast }-2}\big)^{1/( 2_{\ast
}-2)}\| u\|_{\mu ,a}^{2/( 2_{\ast }-2) }
\Big(\int_{\Omega
}k(x)\frac{|u|^{2_{\ast }}}{| x| ^{2_{\ast }b}}dx
\Big)^{-1/(2_{\ast}-2)},
\]
for $u\in K^{+}$.
Then we have the following result.
\begin{proposition} \label{prop2}
For $\lambda \in (0,\Lambda _1)$ we have
\begin{itemize}
\item[(1)] If $u\in K^{+}\cap H_{0}^{-}$ then there exists unique
$t^{+}>t_{\rm max}$ such that $t^{+}u\in \mathcal{N}_{\lambda
}^{-}$and
\[
I_{\lambda ,\mu }(t^{+}u)\geq I_{\lambda ,\mu }( tu) \quad
\text{for } t\geq t_{\rm max};
\]
\item[(2)] If $u\in K^{+}\cap H^{+}$, then there exist unique $t^{-}$,
$t^{+}$ such that $00$ such
that $tu\in \mathcal{N}_{\lambda }$.
\item[(4)] If $u\in K_{0}^{-}\cap H^{+}$, then there exists unique
$00$, contradiction.
\end{proof}
\subsection*{Existence of a local minimum for $I_{\lambda ,
\mu }$ on $\mathcal{N}_{\lambda }^{-}$}
To prove the existence of a second nonnegative solution we need the
following results.
\begin{lemma} \label{lem5}
Let $(u_{n})$ is a $(PS)_l$ sequence with
$u_{n}\rightharpoonup u$ in $H_{\mu }$. Then there exists positive
constant $\tilde{C}:=C(a,b,N,q,|h^{+}|_{\infty },S_{a,b,\mu })$
such that
\[
I_{\lambda ,\mu }'(u)=0\quad \text{and}\quad
I_{\lambda ,\mu }(u)\geq -\tilde{C}\lambda ^{2/(2-q)}.
\]
\end{lemma}
\begin{proof}
It is easy to prove that $I_{\lambda ,\mu }'(u) =0$,
which implies that $\langle I_{\lambda ,\mu }^{'}(u),u\rangle =0$,
and
\[
I_{\lambda ,\mu }(u)-\frac{1}{2_{\ast }}\langle I_{\lambda
,\mu }^{'}(u),u\rangle
=( \frac{1}{2}-\frac{1}{2_{\ast }})\| u\|_{\mu ,a}^{2}
-\lambda (\frac{1}{q}-\frac{1}{2_{\ast }})\int_{\Omega }h(x)
\frac{|u|^{q}}{|x|^{c}}dx.
\]
By Caffarelli-Kohn-Nirenberg, H\"{o}lder and Young inequalities
we find that
\[
I_{\lambda ,\mu }(u)\geq ( \frac{1}{2}-\frac{1}{2_{\ast }} ) \|
u\|_{\mu ,a}^{2}-\lambda \frac{2_{\ast }-q}{ 2_{\ast
}q}(S_{a,b,\mu })^{-q/2}C_1|h^{+}|_{\infty }\| u\| _{\mu
,a}^{q}.
\]
There exists $\tilde{C}$ such that
\[
(\frac{1}{2}-\frac{1}{2_{\ast }})t^{2}-\lambda \frac{2_{\ast }-q
}{2_{\ast }q}(S_{a,b,\mu })^{-q/2}C_1|h^{+}|_{\infty }t^{q}\geq
-\tilde{C}\text{ }\lambda ^{2/(2-q)} \quad \text{for all
}t\geq 0.
\]
Then we conclude that
$I_{\lambda ,\mu }(u)\geq -\tilde{C}\text{ }\lambda ^{2/(2-q)}$.
\end{proof}
\begin{lemma} \label{lem6}
Let $(u_{n})$ in $H_{\mu }$ be such that
\begin{gather}
I_{\lambda ,\mu }(u_{n})\to l0$ such that for all $\lambda \in (0,\Lambda _{4})$ we have
$l^{\ast }>0$ and $\underset{t\geq 0}{\text{ }\sup }I_{\lambda
,\mu }(t\tilde{v}_{\varepsilon })0$ be such that
\[
(\frac{1}{2}-\frac{1}{2_{\ast }})|k^{+}|_{\infty }(S_{a,b,\mu
})^{2_{\ast }/( 2_{\ast }-2)}- \tilde{C}\lambda ^{2/(2-q)}>0\quad
\text{for all }\lambda \in (0,\Lambda _{2}).
\]
Then
\[
f(0)=0<(\frac{1}{2}-\frac{1}{2_{\ast }})| k^{+}| _{\infty
}(S_{a,b,\mu })^{2_{\ast }/( 2_{\ast }-2)}-\tilde{C}\lambda
^{2/(2-q)}\quad \text{for all } \lambda \in (0,\Lambda _{2}).
\]
By the continuity of $f(t)$, there exists $t_1>0$ small enough such that
\[
f(t)<(\frac{1}{2}-\frac{1}{2_{\ast }})|k^{+}|_{\infty }(S_{a,b,\mu
})^{2_{\ast }/( 2_{\ast }-2) }-\tilde{C}\lambda ^{2/(2-q)}\quad
\text{for all }t\in (0,t_1).
\]
On the other hand,
\[
\max_{t\geq 0}\tilde{f}(t)=(\frac{1}{2}-\frac{1}{2_{\ast }})
|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\varepsilon ^{\frac{N-2( a+1-b)}{ 2(a+1-b)}}).
\]
Then
\begin{align*}
\sup_{t\geq 0} I_{\lambda ,\mu }(t\tilde{v}_{\varepsilon })
&< (\frac{1}{2}-\frac{1}{2_{\ast
}})|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\varepsilon ^{\frac{N-2(a+1-b)}{2(a+1-b)}})\\
&\quad -\lambda \frac{t_1^{q}}{q}h_{0}\int_{B(0,\rho _{0})
}\frac{ |\tilde{v}_{\varepsilon }|^{q}}{|x|^{c}}dx.
\end{align*}
Let $0<\varepsilon <\rho _{0}^{(2_{\ast }-2) \sqrt{\bar{\mu
}_{a}-\mu }}$ then
\begin{align*}
&\int_{B(0,\rho _{0})}\frac{|\tilde{v}_{\varepsilon }|
^{q}}{|x|^{c}}dx \\
&= \int_{B(0,\rho _{0})}| x| ^{-c}(\varepsilon
^{\frac{2\sqrt{ \bar{\mu }_{a}-\mu }}{\sqrt{\bar{\mu
}_{a}-\mu }-b}}|x|^{\frac{2_{\ast }-2}{2}(\sqrt{\bar{\mu
}_{a}}-\sqrt{ \bar{\mu }_{a}-\mu })}+|x| ^{\frac{2_{\ast
}-2}{2}(\sqrt{\bar{\mu }_{a}}+\sqrt{\bar{\mu }_{a}-\mu }
)})^{-\frac{2q}{2_{\ast }-2}}dx \\
&\geq C_{2}.
\end{align*}
Now, taking $\varepsilon =\lambda ^{\frac{2(2_{\ast }-2)}{ 2_{\ast
}-q}}$ we get $\lambda <\rho _{0}^{(2-q)\sqrt{\bar{\mu}_{a}-\mu }}$ and
\[
\sup_{t\geq 0} I_{\lambda ,\mu
}(t\tilde{v}_{\varepsilon })<(\frac{1}{2}-\frac{1}{2_{\ast
}})|k^{+}|_{\infty }(S_{a,b,\mu }) ^{2_{\ast }/(2_{\ast
}-2)}+O(\lambda ^{2/( 2-q)})-\lambda
\frac{t_1^{q}}{q}h_{0}C_{2}.
\]
Choosing $\Lambda _{3}>0$ such that
\[
O(\lambda ^{2/(2-q)})-\lambda \frac{t_1^{q}}{q}h_{0}C_{2}<-
\tilde{C}\lambda ^{2/( 2-q)}\quad \text{for all }\lambda \in (0,\Lambda
_{3}).
\]
Then if we take $\Lambda _{4}=\min \{\Lambda _{2},\Lambda
_{3},\rho _{0}^{(2-q)\sqrt{\bar{\mu }_{a}-\mu }}\}$ we deduce
that
\[
\sup_{t\geq 0} J_{\lambda }(t\tilde{v}_{\varepsilon
})0$. Similarly as the proof of Proposition
\ref{prop3}, we conclude that $I_{\lambda ,\mu }$ has a
minimizer $v_{\lambda }$
in $\mathcal{N}_{\lambda }^{-}$ for all $\lambda \in ( 0,\Lambda
^{\ast })$ such that $I_{\lambda ,\mu }(v_{\lambda }) =c_{\lambda
}^{-}>0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1}]
By Propositions \ref{prop2} and \ref{prop4},
there exists $\Lambda ^{\ast }>0$
such that \eqref{p-lambda-mu} has two
nonnegative solutions $u_{\lambda }\in \mathcal{N}_{\lambda }^{+}$
and $v_{\lambda }\in \mathcal{ N}_{\lambda }^{-}$ since
$\mathcal{N}_{\lambda }^{+}\cap \mathcal{N} _{\lambda
}^{-}=\emptyset$.
\end{proof}
\begin{thebibliography}{00}
\bibitem{AB} A. Ambrosetti, H. Br\'{e}zis, G. Cerami:
Combined effects of
concave and convex nonlinearities in some elliptic problems, J. Funct. Anal.
122 (1994), 519--543.
\bibitem{BA} M. Bouchekif, A. Matallah: Multiple positive solutions for
elliptic equations involving a concave term and critical Sobolev-Hardy
exponent, Appl. Math. Lett. 22 (2009), 268-275.
\bibitem{BA2} M. Bouchekif, A. Matallah: On singular nonhomogeneous elliptic
equations involving critical Caffarelli-Kohn-Nirenberg exponent, Ricerche
math. doi 10. 1007/s 11587-009-0056-y.
\bibitem{BL} H. Br\'{e}zis, E. Lieb: A relation between pointwise
convergence of functions and convergence of functionals, Proc. Amer. Math.
Soc. 88 (1983), 486-490.
\bibitem{BZ} K.J. Brown, Y. Zhang: The Nehari manifold for a semilinear
elliptic equation with a sign-changing weight function, J. Differential
Equations 193 (2003), 481-499.
\bibitem{CK} L. Caffarelli, R. Kohn, L. Nirenberg: First order interpolation
inequality with weights, Compos. Math. 53 (1984), 259--275.
\bibitem{CW} F. Catrina, Z. Wang: On the Caffarelli-Kohn-Nirenberg
inequalities: sharp constants, existence (and nonexistence), and symmetry of
extremal functions, Comm. Pure Appl. Math. 54 (2001), 229-257.
\bibitem{C} J. Chen: Multiple positive solutions for a class of nonlinear
elliptic equations, J. Math. Anal. Appl. 295 (2004), 341-354.
\bibitem{CC} K.S. Chou, C.W. Chu: On the best constant for a weighted
Sobolev-Hardy Inequality, J. London Math. Soc. 2 (1993), 137-151.
\bibitem{E} L.C. Evans: Partial differential equations, in: graduate studies
in mathematics 19, Amer. Math. Soc. Providence, Rhode Island, (1998).
\bibitem{FG} A. Ferrero, F. Gazzola: Existence of solutions for singular
critical growth semilinear elliptic equations. J. Differential Equations 177
(2001), 494-522.
\bibitem{HH} T.S. Hsu, H.L. Lin: Multiple positive solutions for singular
elliptic equations with concave-convex nonlinearities and sign-changing
weights. Boundary Value Problems, doi: 10 1155/2009/584203.
\bibitem{K} D. Kang, G. Li, S. Peng: Positive solutions and critical
dimensions for the elliptic problem involving the Caffarelli-Kohn-Nirenberg
inequalities, Preprint.
\bibitem{Lin} M. Lin: Some further results for a class of weighted nonlinear
elliptic equations, J. Math. Anal. Appl. 337(2008), 537-546.
\bibitem{X.} B. J. Xuan: The solvability of quasilinear
Br\'{e}zis-Nirenberg-type problems with singular weights, Nonlinear Anal.
62 (4) (2005), 703-725.
\bibitem{X} B. J. Xuan, S. Su, Y. Yan: Existence results for
Br\'{e}zis-Nirenberg problems with Hardy potential and singular
coefficients, Nonlinear Anal. 67 (2007), 2091-2106.
\bibitem{T} G. Tarantello: On nonhomogeneous elliptic equations involving
critical Sobolev exponent, Ann. Inst. H. Poincar\'{e} Anal. Non.
Lin\'{e}aire 9 (1992), 281-304.
\bibitem{W} T. F. Wu: On semilinear elliptic equations involving
concave-convex nonlinearities and sign-changing weight function, J. Math.
Anal. Appl. 318 (2006), 253-270.
\end{thebibliography}
\end{document}