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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 115, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/115\hfil On the existence of weak solutions]
{On the existence of weak solutions for $p,q$-laplacian systems
with weights}

\author[O. H. Miyagaki, R. S. Rodrigues\hfil EJDE-2008/115\hfilneg]
{Olimpio H. Miyagaki, Rodrigo S. Rodrigues}  % in alphabetical order

\address{Olimpio H. Miyagaki \newline
Departamento de Matem\'atica,
Universidade Federal de Vi\c{c}osa,
36571-000, Vi\c{c}osa, MG, Brazil}
\email{olimpio@ufv.br}

\address{Rodrigo S. Rodrigues \newline
Departamento de Matem\'atica, 
Universidade Federal de S\~ao Carlos,  13565-905, S\~ao Carlos, SP, Brazil}
\email{rodrigosrodrigues@ig.com.br}

\thanks{Submitted May 19, 2008. Published August 20, 2008.}
\thanks{The first author was supported  by  the
CNPq-Brazil and AGIMB--Millenium
Institute \hfill\break\indent
MCT/Brazil. The second author was supported  by Capes-Brazil}

\subjclass[2000]{35B25, 35B33, 35D05, 35J55, 35J70}
\keywords{Degenerate quasilinear equations;  elliptic system;
\hfill\break\indent critical exponent; singular perturbation}

\begin{abstract}
 This paper studies degenerate quasilinear elliptic
 systems involving $p,q$-superlinear and critical nonlinearities
 with singularities.
 Existence results are obtained by using properties of the best
 Hardy-Sobolev constant together with an approach
 developed by Brezis and Nirenberg.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\allowdisplaybreaks

\section{Introduction}

 In a  well-known paper,  Brezis and Nirenberg
\cite{BrezisNirenberg} proved that, under certain
conditions, the elliptic problem with Dirichlet boundary condition
 \begin{equation} \label{PBN}
 \begin{gathered}
  -\Delta u =\lambda u^q+u^{2^*-1} \quad \quad\text{in }\Omega,  \\
  u>0 \quad\text{in }\Omega,  \\
  u=0 \quad\text{on } \partial\Omega
 \end{gathered}
 \end{equation}
possesses at least a solution, for all $\lambda > 0$, where $1<q<
2^*={2N}/({N-2})$, $N \geq 3$, $2^*$ is said to be the critical
Sobolev exponent, and $\Omega \subset \mathbb{R}^N\ (N \geq 3)$ is
a bounded smooth domain. In general, the main difficulty in this
type of problem is the lack of compactness of the injection
$H_0^1(\Omega) \hookrightarrow L^{2^*}(\Omega)$.

 We recall that the perturbation $\lambda u^q$ is essential
in this kind of the problem. By Pohozaev identity \cite{Poho},
problem \eqref{PBN} does not possess any solution when $\lambda
\leq 0$.


  Garc\'ia and Peral in \cite{Garcia} studied the existence
of  nontrivial solution  for a class of problems involving the
p-laplacian  operator, namely,
 \begin{gather*}
   -\Delta_p u\equiv - \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)
   = \lambda |u|^{q-2}u
   +\mu |u|^{p^*-2}u \quad\text{in }\Omega,   \\
   u\geq 0   \quad\text{in }\Omega,   \\
   u=0 \quad\text{on } \partial\Omega,
 \end{gather*}
in a bounded smooth domain $\Omega \subset \mathbb{R}^N (N > p)$,
with $1< p\leq q <p^* = {Np}/({N-p})$. When $p<q<p^*$, we say that
the above problem is $p$-superlinear. These type of problems,
which are related to the Brezis and Nirenberg problem
\cite{BrezisNirenberg} (problem \eqref{PBN} with $p=2$), have
been widely treated by several authors and we would like to
mention some of them, e.g., \cite{ChenLi, Ghoussoub,
GoncalvesAlves} for $1 < p < N $ and  \cite{Miyagaki, Nicolaescu,
Pan} for $p=2$, see also references cited there.


  Caffarelli, Kohn and Nirenberg in \cite{Caffarelli-Kohn-Nirenberg}
proved that if $1 <p <N$, $-\infty <a<(N-p)/p$, $a\leq c_1\leq a+1$,
$d_1=1+a-c_1$, and $p^*=p^*(a,c_1,p):=Np/(N-d_1p)$, there exists
$C_{a,p}>0$ such that the following Hardy-Sobolev type inequality
with weights is satisfied
 \begin{equation*}  \label{Caffarelli-Kohn-Nirenberg}
   \Big(\int_{\mathbb{R}^N}
   |x|^{-c_1p^*}|u|^{p^*}dx\Big)^{p/p^*}
   \leq
   C_{a,p}\Big({\int_{\mathbb{R}^N}
   |x|^{-ap}|\nabla u|^p }dx\Big), \quad \forall
   u\in C_0^{\infty}(\mathbb{R}^N).
 \end{equation*}
Note that several papers have been appeared on this subject,
mainly, the works  about the existence of solution for a
class of quasilinear elliptic problems of the type
 \begin{equation*}
   -Lu_{ap}=g(x,u) + |x|^{-e_1p^*}|u|^{q-2}u \quad\text{in } \Omega,
 \end{equation*}
where $ Lu_{ap}= \mathop{\rm div}(|x|^{-ap} |\nabla u|^{p-2} \nabla u)$,
under certain suppositions on the exponents $1<p<N$,
$-\infty<a<(N-p)/p$, $a\leq e_1<a+1$, $d=1+a-e_1$, and
$p^*=Np/(N-dp)$, and on the function
$g:\Omega\times\mathbb{R}\to \mathbb{R}$. See, for
instance, \cite{Carriao, Assuncao, Catrina-Wang, Djairo, Wang,
Xuan1} and references therein. The lack of compactness is overcame
proving that all the  Palais Smale sequence at the level $c$, (
$(PS)_c$-sequence, in short), with $ c < (d/N)(C_{a,p}^*)^{N/dp}$, is
relatively compact. $(d/N)(C_{a,p}^*)^{N/dp}$ is so called the
critical level and  $C_{a,p}^*$ is the best Hardy-Sobolev constant
and it is characterized by
 \begin{equation*}
   C_{a,p}^*=C_{a,p}^*(\Omega):=
   {\inf_{u\in W_0^{1,p}(\Omega,|x|^{-ap})
   \setminus\{0\}}} \Big\{ \frac{\int_{\Omega}
   |x|^{-ap}|\nabla u|^p dx}{\big(\int_{\Omega}|x|^{-e_1p^*}
   |u|^{p^*} dx\big)^{p/p^*}} \Big\}\,.
 \end{equation*}



 Besides the great number of the applications known for the
scalar case, for instance, in fluid mechanics, in newtonian
fluids, in flow through porous media, reaction-diffusion problems,
nonlinear elasticity, petroleum extraction, astronomy, glaciology,
etc, see \cite{Radu}, the above  systems can involve another
phenomena, such as competition model in population dynamics, see
\cite{Fleckinger} and reference therein. For the systems case we
would like to mention the
 papers \cite{Abdelaziz, Stavrakakis} and a survey paper
 \cite{Djairo(System)} as well as in the
references therein.


  In our work, we will use a version of the well-known mountain pass
theorem \cite{AR} to establish conditions for the existence of a
nontrivial solution for a quasilinear elliptic system involving the
above operator and a $p,q$-superlinear nonlinear perturbation
 \begin{equation}\label{sistemaperturbado1}
 \begin{gathered}
   -Lu_{ap}=\lambda    \theta|x|^{-\beta_1}|u|^{\theta-2}|v|^{\delta}u + \mu\alpha
   |x|^{-\beta_2}|u|^{\alpha-2}|v|^{\gamma}u
   \quad \text{in }\Omega,    \\
   -Lv_{bq}=\lambda    \delta |x|^{-\beta_1} |u|^{\theta}|v|^{\delta-2}v
   + \mu \gamma    |x|^{-\beta_2}|u|^{\alpha}|v|^{\gamma-2}v
   \quad \text{in } \Omega,    \\
   u=v=0\quad  \text{on }  \partial \Omega,
 \end{gathered}
 \end{equation}
where
 \begin{equation} \label{HOmega}
\text{$\Omega$ is a bounded smooth
domain of $\mathbb{R}^N$ with $0\in \Omega$,}
\end{equation}
the parameters $\lambda,\mu$ are
positive real numbers and the exponents satisfy
\begin{equation}
\begin{gathered}
   1 <p,\quad q <N,\quad -\infty< a<(N-p)/p,\quad   -\infty< b<(N-q)/q,   \\
 a\leq c_1<a+1,\quad b\leq c_2<b+1,\quad d_1=1+a-c_1,  \quad d_2=1+b-c_2,   \\
p^*=Np/(N-d_1p),\quad q^*=Nq/(N-d_2q),    \\
\alpha, \gamma, \theta, \delta >1, \quad \beta_1,\beta_2  \in \mathbb{R},
 \end{gathered} \label{Hexp}
\end{equation}
with one of the following two sets of conditions satisfied:
 \begin{equation}\label{H1}
 \begin{gathered}
      \frac{\theta}{p}+\frac{\delta}{q},\,
   \frac{\alpha}{p}+\frac{\gamma}{q} >  1 \quad  \text{($p,q$-superlinear)} \\ \\
  \frac{\theta}{p^*}+\frac{\delta}{q^*},\,
   \frac{\alpha}{p^*}+\frac{\gamma}{q^*} < 1 \quad
   \text{($p,q$-subcritical),}
 \end{gathered}
 \end{equation}
or
  \begin{equation}\label{HH1}
   \frac{\theta}{p^*}+\frac{\delta}{q^*}
  < 1 <   \frac{\theta}{p}+\frac{\delta}{q}
  \text{ and }
  \frac{\alpha}{p^*}+\frac{\gamma}{q^*}=1
\quad\text{$p,q$-superlinear/critical case)}
\end{equation}


  However, the variational systems behave, in a certain
sense, like in the scalar case,  there exist some  additional
difficulties mainly coming from the mutual actions of the
variables $u$ and $v$, see  e. g. \cite{Han, stavrakakis1}.
Another difficulty, even in the  regular case, are the  systems
involving $p$-laplacian and $q$-laplacian operators and their
respective critical exponents. In this situation, it is hard to
find a well appropriated critical level, mainly, when $p\neq q$.
This open question  was pointed out in Adriouch and Hamidi
\cite{Adriouch1}. But, recently  Silva and Xavier in \cite{Elves}
were able to prove, in a certain context and in the regular case,
the existence of weak solution for a system involving
$p$-laplacian and $q$-laplacian operators with $p\neq q$. Still in
the regular case and $p=q$, we would like to  mention  the papers
\cite{Abdelaziz,AlvesMoraisSouto, MoraisSouto, Stavrakakis,
YangJianfu(system)}, also  a survey paper
\cite{Djairo(System)}. In particular, Morais and Souto in
\cite{MoraisSouto} defined the following critical level number
$S_H/p$, where
\begin{equation*}
   S_H={ \inf_{W\setminus\{0\}}}
   \Big\{\frac{\int_{\Omega}
   |\nabla u|^p+|\nabla v|^pdx}{\big(
   \int_{\Omega}H(u,v)dx\big)^{p/p^*}}\Big\},
 \end{equation*}
 $W=W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ and
$H$ is homogeneous nonlinearity of degree $p^*$.
In this work, we will improve the critical level by proving that all the
Palais Smale sequences at the level $c$ are relatively compact provided that
$$
c<(\frac{1}{p}-\frac{1}{p^*})(\mu p^*)^{\frac{-p}{p^*-p}}
\tilde{S}^{\frac{p^*}{p^*-p}}
+\lambda(\frac{1}{p}-\frac{1}{p_1})M,
$$
 where $\tilde{S}$ depends of $C_{a,p}^*$ and
$M=M(u_n,v_n)\geq 0$ depends of Palais Smale sequence.

  Our first result deals with $p,q$-superlinear and subcritical
nonlinear perturbation.

 \begin{theorem}\label{thm1}
 In addition to \eqref{HOmega}, \eqref{Hexp}, and \eqref{H1}, assume
that $p_i\in (p,p^*)$, $q_i\in (q,q^*)$, $i=1,2$, with
$\theta/p_1+\delta/q_1=\alpha/p_2+ \gamma/q_2=1$ and
 \begin{equation}\label{hipotese555}
   \beta_i< \min\big\{(a+1)p_i+ N\big(1-\frac{p_i}{p}\big),
   (b+1)q_i+ N\big(1-\frac{q_i}{q}\big)\big\}, \quad i=1,2.
 \end{equation}
Then system \eqref{sistemaperturbado1} possesses  a weak
solution, where each component is nontrivial and nonnegative, for
each $\lambda\geq 0$  and $\mu >0$.
 \end{theorem}

 The next result treats the $p,q$-superlinear and critical case.

\begin{theorem}\label{edr}
 Assume \eqref{HOmega}, \eqref{Hexp} and \eqref{HH1}, with $p=q$
and $a=b\geq 0$. Suppose also $p_1=q_1\in(p,p^*)$, with
$\theta/p_1+\delta/q_1=1$, $p^*=q^*$, $\beta_2=c_1p^*$, and
$\beta_1=(a+1)p_1-c$ with
 \begin{equation*}
   -N\left[1-({p_1}/{p})\right]<c
   <\frac{(p_1-p+1)N-(a+1)p_1}{p-1}
   -\frac{(N-p-ap)(p_1-p)}{p(p-1)}\cdot
 \end{equation*}
 Then, system \eqref{sistemaperturbado1}
possesses a weak solutions, where each component is  nontrivial and
nonnegative, for each $\lambda, \mu > 0$.
\end{theorem}

The  $p,q$-superlinear and critical case with $p \neq q$ is studied
in the following result.

\begin{theorem} \label{dif}
In addition to \eqref{HOmega}, \eqref{Hexp},
and \eqref{HH1}, assume that $p_1\in(p,p^*)$, $q_1\in(q,q^*)$,
with $\theta/p_1+\delta/q_1=1$, $\beta_2=c_1p^*=c_2q^*$, and
$\beta_1$ as in \eqref{hipotese555}. Then there exists $\mu_0$
sufficiently small such that system \eqref{sistemaperturbado1}
posesses a weak solution, where each component is nontrivial and
nonnegative, for each $\lambda > 0$  and $0<\mu<\mu_0$.
\end{theorem}

\section{Preliminaries}

 We will set some spaces and their norms.
If $\alpha \in \mathbb{R}$ and $l\geq 1$, we define $L^l(\Omega,
|x|^{\alpha})$ as being the subspace of $L^l(\Omega)$ of the
Lebesgue measurable functions $u:\Omega \to \mathbb{R}$
satisfying
   \begin{equation*}
\|u\|_{L^{l}(\Omega,|x|^{\alpha})} :=  {
      \Big(\int_{\Omega}|x|^\alpha |u|^ldx\Big)^{1/l}}< \infty.
   \end{equation*}
If $1 < p < N$ and $-\infty<a<(N-p)/p$, we define
$W_0^{1,p}(\Omega,|x|^{-ap})$ as
being the completion of $C_0^{\infty}(\Omega)$
with respect to the norm $\| \cdot \|$
defined by
 \begin{equation*}
   \|u\|:=\Big({\int_{\Omega}|x|^{-ap}|
   \nabla u|^p}dx\Big)^{1/p}.
 \end{equation*}


  First of all, from  the Caffarelli, Kohn and Nirenberg
inequality (see \cite{Caffarelli-Kohn-Nirenberg}) and by the
boundedness of $\Omega$, it is easy to see that there exists $C>0$
such that
 \begin{equation*}
   {\Big( \int_{\Omega}|x|^{-\delta}
   |u|^r dx\Big)^{p/r}}\leq C
   \Big({\int_{\Omega}|x|^{-ap}
   |\nabla u|^{p}dx}\Big),\quad
   \forall u\in W_0^{1,p}(\Omega,|x|^{-ap}),
 \end{equation*}
where $1\leq r \leq Np/(N-p)$ and $\delta \leq (a+1)r+N[1-({r}/{p})]$.

\begin{lemma}\label{1.121}
  Suppose that $\Omega$ is a bounded smooth domain of
$\mathbb{R}^N$ with $0\in\Omega$, $1 <p<N$,
$-\infty< a<(N-p)/p$, $a\leq e_1< a+1$,
$d_1=1+a-e_1$, $p^*=Np/(N-d_1p)$, and $\alpha+\gamma=p^*$, then
 \begin{equation*}
   \tilde{S} := {\inf_{(u,v)\in \tilde{W}}}
   \Big\{ \frac{\int_{\Omega}
   |x|^{-ap}(|\nabla u|^p
   +|\nabla v|^p) dx}{\big(
   \int_{\Omega}|x|^{-e_1p^*}|u|^{\alpha}
   |v|^{\gamma}dx \big)^{p/p^*}}\Big\},
 \end{equation*}
where
 \begin{equation*}
   \tilde{W}=\big\{(u,v)\in \big(W_0^{1,p}(\Omega,|x|^{-ap})\big)^2:
   |u\|v|\not\equiv 0\big\},
 \end{equation*}
satisfies
 \begin{equation*}
   \tilde{S}=\big[({\alpha}/{\gamma})^
   {{\gamma}/{p^*}} + ({\alpha}/
   {\gamma})^{{-\alpha}/{p^*}}\big]C^*_{a,p}.
 \end{equation*}
 \end{lemma}

The proof of the above lemma  is  similar
to the proof of  \cite[Theorem 5]{AlvesMoraisSouto} (see also
\cite[Lemma 3]{MoraisSouto} for $p\neq 2$).

    Let us consider $\Omega$ a smooth domain of $\mathbb{R}^N$
(not necessarily bounded), $0\in \Omega$, $1<p<N$, $0\leq a <
(N-p)/{p}$, $a\leq c_1<a+1$, $d_1=1+a-c_1$, and $p^*=Np/(N-d_1p)$.
We define the space
 \begin{equation*}
   W_{a,c_1}^{1,p}(\Omega)=\big\{u\in L^{p^*}
   (\Omega,|x|^{-c_1p^*}):
   |\nabla u|\in L^p(\Omega,|x|^{-ap})\big\},
 \end{equation*}
equipped with the norm
 \begin{equation*}
   \|u\|_{W_{a,c_1}^{1,p}(\Omega)}= \|u\|_{L^{p^*}
   (\Omega,|x|^{-c_1p^*})}
   +\|\nabla u\|_{L^p(\Omega,|x|^{-ap})}.
 \end{equation*}
We consider the best Hardy-Sobolev constant given by
 \begin{equation*}
   \tilde{S}_{a,p}
   = {\inf_{W_{a,c_1}^{1,p}(\mathbb{R}^N)
   \setminus\{0\}}} \Big\{\frac{\int_{\mathbb{R}^N}
   |x|^{-ap}|\nabla u|^pdx}{\big(\int_{\mathbb{R}^N}
   |x|^{-c_1p^*} |u|^{p^*}dx\big)^{p/p^*}}
   \Big\}.
 \end{equation*}
Also, we define
 \begin{equation*}
   R^{1,p}_{a,c_1}(\Omega)=\big\{u\in W_{a,c_1}^{1,p}
   (\Omega): u(x)=u(|x|)\big\},
 \end{equation*}
endowed with the norm
 \begin{equation*}
   \|u\|_{R_{a,c_1}^{1,p}(\Omega)}=\|u\|_{W_{a,c_1}^{1,p}(\Omega)}.
 \end{equation*}
Actually,  Horiuchi in \cite{Horiuchi} proved that
 \begin{equation*}
   \tilde{S}_{a,p,R} =  {\inf_{R_{a,c_1}^{1,p}(\mathbb{R}^N)
   \setminus\{0\}}} \Big\{\frac{\int_{\mathbb{R}^N}
   |x|^{-ap}|\nabla u|^pdx}{\big(\int_{\mathbb{R}^N}
   |x|^{-c_1p^*} |u|^{p^*}dx\big)^{p/p^*}}
   \Big\} =\tilde{S}_{a,p}
 \end{equation*}
and it is achieved by functions of the form
 \begin{equation*}
   y_{\epsilon}(x):= k_{a,p}(\epsilon)U_{a,p,\epsilon}(x),
   \quad \forall \epsilon>0,
 \end{equation*}
where
 \begin{equation*}
   k_{a,p}(\epsilon)=c_0\epsilon^{(N-d_1p)/d_1p^2}
   \quad\text{and}\quad
   U_{a,p,\epsilon}(x)
   = {\Big(\epsilon+|x|^{\frac{d_1p(N-p-ap)}{(p-1)(N-d_1p)}}
   \Big)^{-(\frac{N-d_1p}{d_1p})}}.
 \end{equation*}
Moreover, $y_{\epsilon}$ satisfies
\begin{equation}\label{zzz1}
   {\int_{\mathbb{R}^N}}|x|^{-ap}|\nabla y_{\epsilon}|^{p}dx=
   {\int_{\mathbb{R}^N}}|x|^{-c_1p^*}|y_{\epsilon}|^{p^*}dx.
  \end{equation}
See also  Cl\'ement, Figueiredo and Mitidieri
\cite[Proposition 1.4]{Djairo}.

  The  next lemma can be  proved arguing as in
\cite{BrezisNirenberg} (see also \cite[Lemma 5.1]{Xuan1}). For the
sake of the completeness we will give the  proof in the appendix.

\begin{lemma}\label{Xuan-O(e)}
  In addition to \eqref{HOmega} and \eqref{Hexp}, assume
that $p_1=q_1\in(p,p^*)$, $\theta/p_1+\delta/q_1=1$,
$\beta_2=c_1p^*=c_2q^*$, and $\beta_1=(a+1)p_1-c$ with
 \begin{equation*}
   -N\left[1-({p_1}/{p})\right]<c.
 \end{equation*}
  Let $R_0\in (0,1)$ be such that $B(0,2R_0)\subset\Omega$
and $\psi\in C_0^{\infty}(B(0,2R_0))$ with $\psi\geq 0$ in
$B(0,2R_0)$ and $\psi\equiv 1$ in $B(0,R_0)$, then the
function
 \begin{equation*}
   u_{\epsilon}(x)=\frac{\psi(x)U_{a,p,\epsilon}(x)}
   {\|\psi U_{a,p,\epsilon}\|_{L^{p^*}(
   \Omega,|x|^{-c_1p^*})}}
 \end{equation*}
satisfies
 \begin{equation*}
   \|u_{\epsilon}\|^{p^*}_{L^{p^*}(\Omega,|x|^{-c_1p^*})} = 1,
   \quad
   \|\nabla u_{\epsilon}\|^p_{L^{p}(\Omega,|x|^{-ap})}
   \leq \tilde{S}_{a,p,R} + O(\epsilon^{(N-d_1p)/d_1p}),
 \end{equation*}
 and
 \begin{equation}\label{ff1}
   \|u_{\epsilon}\|^{p_1}_{L^{p_1}(\Omega,|x|^{-\beta_1})}
   \geq
   \begin{cases}
   O(\epsilon^{{(N-d_1p)p_1}/{d_1p^2}})
   \text{ if } c>\frac{(p_1-p+1)N-(a+1)p_1}{p-1},
\\[3pt]
   O(\epsilon^{{(N-d_1p)p_1}/{d_1p^2}}|ln (\epsilon)|)
   \text{ if } c=\frac{(p_1-p+1)N-(a+1)p_1}{p-1},
\\[3pt]
   O\Big(\epsilon^{\frac{(N-d_1p)(p-1)(N-p_1-ap_1+c)}{d_1p(N-p-ap)}
   -\frac{(N-d_1p)(p-1)p_1}{d_1p^2}}\Big)
\\
 \text{if } c<\frac{(p_1-p+1)N-(a+1)p_1}{p-1}\,.
   \end{cases}
 \end{equation}
 \end{lemma}


  The following result, which will be useful in the proof of
our results, was proved by Kavian in \cite[Lemma 4.8]{Kavian}.


 \begin{lemma}\label{Kavian}
  Let $\Omega$ be an open subset of $\mathbb{R}^N$,
$\{f_n\}\in L^{r}(\Omega)$,
for some $1<r<\infty$, a bounded sequence such that
$f_n(x)\to f(x)$, for a.e. $x\in \Omega$, $\text{as}\ n
\to \infty$. Then, $f\in L^{r}(\Omega)$ and
$f_n\rightharpoonup f$ weakly in $L^{r}(\Omega)$ as $n\to
\infty$.
 \end{lemma}

\subsection*{Definition}
  Let us consider $\{(u_n,v_n)\}$ in $W_0^{1,p}(
\Omega,|x|^{-ap})\times W_0^{1,q}(\Omega,|x|^{-bq})$. We say that
the  sequence $\{(u_n,v_n)\}$ is a Palais Smale sequence for
operator $I$ at the level $c$ (or simply, $(PS)_c$-sequence) if
  $$
I(u_n,v_n)\to  c   \quad\text{and}\quad
   I'(u_n,v_n) \to  0,
   \quad\text{as } n \to \infty.
$$


  Our approach will be to use variational techniques; that is, we have to
find the critical points of the Euler-Lagrange functional
 $$
I:W_0^{1,p}(\Omega,|x|^{-ap})\times W_0^{1,q}(\Omega,|x|^{-bq})
   \to  \mathbb{R}
$$
given by
 \begin{align*}
   I(u,v) &= \dfrac{1}{p}{\int_{\Omega}}|x|^{-ap}    |\nabla u|^p\,dx
   + \dfrac{1}{q}{\int_{\Omega}}    |x|^{-bq}|\nabla v|^q\,dx    \\
&\quad    -\lambda {\int_{\Omega}}|x|^{-\beta_1}
   u_{+}^{\theta}v_{+}^{\delta}\,dx
   -\mu {\int_{\Omega}}|x|^{-\beta_2}
   u_{+}^{\alpha}v_{+}^{\gamma} \,dx,
 \end{align*}
which is well defined and is of class $C^1$, with the G\^ateaux
derivative
 \begin{align*}
\langle I'(u,v),(w,z)\rangle
   &= {\int_{\Omega}}|x|^{-ap} |\nabla u|^{p-2}\nabla u \nabla w\,dx
   + {\int_{\Omega}}|x|^{-bq}
   |\nabla v|^{q-2}\nabla v \nabla z \,dx \\
&\quad  -\lambda\theta {\int_{\Omega}}|x|^{-\beta_1}
   u_{+}^{\theta-1}v_{+}^{\delta}w \,dx -\lambda \delta
   {\int_{\Omega}}|x|^{-\beta_1}
   u_{+}^{\theta}v_{+}^{\delta-1} z\,dx \\
&\quad  -\mu \alpha {\int_{\Omega}}|x|^{-\beta_2}
   u_{+}^{\alpha-1}v_{+}^{\gamma} w \,dx -\mu
   \gamma{\int_{\Omega}}|x|^{-\beta_2}
   u_{+}^{\alpha}v_{+}^{\gamma-1}z \,dx,
 \end{align*}
where $u_{\pm}=\max\{0,\pm u\}$ which is in $W_0^{1,p}(\Omega,|x|^{-ap})$
(Similarly $v_{\pm}=\max\{0,\pm v\}$ which is in
$W_0^{1,q}(\Omega,|x|^{-bq})$; see \cite{Akdim}).

 First of all, we are going to show the geometric
conditions of the mountain pass theorem.

\begin{lemma}\label{GC}
In addition to  \eqref{HOmega} and \eqref{Hexp},
assume that one of the following conditions hold:
\begin{itemize}
\item[(i)] the case \eqref{H1}, $p_i\in
(p,p^*)$, $q_i\in (q,q^*)$, with $\theta/p_1+\delta/q_1=\alpha/p_2+
\gamma/q_2=1$, and $\beta_i$ as in \eqref{hipotese555}, for
$i=1,2$.

\item[(ii)] the case \eqref{HH1}, $p_1\in (p,p^*)$, $q_1\in (q,q^*)$,
with $\theta/p_1+\delta/q_1=1$,
$\beta_1$ as in \eqref{hipotese555}, $p_2=p^*$, $q_2=q^*$, and
$\beta_2=c_1p^*=c_2q^*$.

\end{itemize}
Then the Euler-Lagrange functional $I$ satisfies:
\begin{itemize}
\item[(a)] There exist $\sigma, \rho> 0$ such that
 \begin{equation}\label{geom2}
   I(u,v) \geq \sigma \text{\, if \,} \|(u,v)\|=\rho.
 \end{equation}
\item[(b)] There exists $e \in W_0^{1,p}(\Omega,|x|^{-ap})\times
W_0^{1,q}(\Omega,|x|^{-bq}) $ such that
$$
I(e)\leq 0, \quad \|e\|\geq R \quad \text{for some } R > \rho.
$$
\end{itemize}
\end{lemma}

\begin{proof}
 Part (a). For $(u,v)\in W_0^{1,p}(\Omega,|x|^{-ap})\times W_0^{1,q}(\Omega,|x|^{-bq})$ with
$\|(u,v)\|\leq 1$, we have
 \begin{align*}
 I(u,v) &\geq
   \Big(\frac{1}{p}\|u\|^p
   - \lambda\frac{\theta C^{p_1/p}}{p_1}   \|u\|^{p_1}    -
   \mu \frac{\alpha C^{p_2/p}}{p_2}    \|u\|^{p_2} \Big) \\
&\quad +\Big(   \frac{1}{q}\|v\|^q
   -\lambda  \frac{\delta C^{q_1/q}}{q_1}   \|v\|^{q_1}
    -\mu \frac{\gamma C^{q_2/q}}{q_2}\|v\|^{q_2}   \Big) \\
&\geq    \frac{1}{p}\|u\|^{p}
   -\Big(\lambda \frac{\theta C^{p_1/p}}{p_1}
   + \mu \frac{\alpha C^{p_2/p}}{p_2}\Big)   \|u\|^{\min\{p_1,p_2\}} \\
&\quad   +\frac{1}{q}\|v\|^{q}
   - \Big(\lambda \frac{\delta C^{q_1/q}}{q_1}
   +\mu \frac{\gamma C^{q_2/q}}{q_2}\Big)
   \|v\|^{\min\{q_1,q_2\}}.
 \end{align*}
Hence, as $p<\min\{p_1,p_2\}$ and $q<\min\{q_1,q_2\}$, we can choose
$\rho\in (0,1)$ such that
 \begin{equation*}
   I(u,v) \geq\sigma  \quad\text{if } \|(u,v)\|=\rho.
 \end{equation*}

Part (b).  The proof follows by taking
$(u_0,v_0)\in W_0^{1,p}(\Omega,|x|^{-ap})\times
W_0^{1,q}(\Omega,|x|^{-bq})$ with $u_{0_+}.v_{0_+}\not\equiv 0$.
Then, defining $(u_t,v_t)=(t^{1/p}u_0,t^{\frac{1}{q}}v_0)$,
for $t>0$, we obtain
 \begin{eqnarray}\label{geom 1}
   I(u_t,v_t) \leq
   \Big(\frac{1}{p}\|u_0\|^{p}+
   \frac{1}{q}\|v_0\|^q\Big) t-
   \mu t^{\frac{\alpha}{p}+ \frac{\gamma}{q}}
   {\int_{\Omega}}|x|^{-\beta_2}
   u_{0_+}^{\alpha}v_{0_+}^{\gamma} dx
   \to -\infty,
 \end{eqnarray}
 \noindent as  $ t\to \infty$.
\end{proof}

 From the mountain pass theorem \cite{AR}
we get a $(PS)_c$-sequence $\{(u_n,v_n)\}$ in
$W_0^{1,p}(\Omega,|x|^{-ap})\times W_0^{1,q}(\Omega,|x|^{-bq})$,
where
 \begin{equation}\label{Carmo1}
   0<\sigma\leq c={\inf_{h\in \Gamma}} {\max_{t\in [0,1]}}I(h(t))
 \end{equation}
and
 \begin{equation}\label{Carmo}
   \Gamma =   \big\{h\in C([0,1],W_0^{1,p}(\Omega,|x|^{-ap})
    \times    W_0^{1,q}(\Omega,|x|^{-bq})) :
    h(0) = 0,\,    h(1) =  e\big\},
 \end{equation}
with $I(e)\equiv I(t_0u_0,t_0v_0)<0$.


  \begin{lemma}\label{u+}
In addition to  \eqref{HOmega} and \eqref{Hexp},
assume that one of the two following conditions hold:
\begin{itemize}
\item[(i)] the case \eqref{H1}, $p_i\in
(p,p^*)$, $q_i\in (q,q^*)$, with $\theta/p_1+\delta/q_1=\alpha/p_2+
\gamma/q_2=1$, and $\beta_i$ as in \eqref{hipotese555}, for
$i=1,2$.

\item[(ii)] the case \eqref{HH1}, $p_1\in
(p,p^*)$, $q_1\in (q,q^*)$, with $\theta/p_1+\delta/q_1=1$,
$\beta_1$ as in \eqref{hipotese555}, $p_2=p^*$, $q_2=q^*$, and
$\beta_2=c_1p^*=c_2q^*$.

\end{itemize}
Let $\{(u_n,v_n)\} \subset
W^{1,p}_{0}(\Omega,|x|^{-ap})\times W^{1,q}_{0}(\Omega,|x|^{-bq})$
be a $(PS)_c$-sequence.
Then $\{(u_{n_+},v_{n_+})\}$ is a
$(PS)_c$-sequence which is bounded uniformly in $\mu>0$.
\end{lemma}

\begin{proof}
 Let $\theta_1=\min\{p_1,p_2\}$ and
$\theta_2=\min\{q_1,q_2\}$, we have
 \begin{align*}
   c + \|(u_n,v_n)\|+O_n(1)
&\geq  I(u_n,v_n)-\langle I'(u_n,v_n),({u_n}/{\theta_1},
   {v_n}/{\theta_2})\rangle \\
 &\geq    ( \frac{1}{p}-\frac{1}{\theta_1})\|u_n\|^p
   +  ( \frac{1}{q}-\frac{1}{\theta_2})\|v_n\|^q \\
&\quad  +\lambda   (\frac{\theta}{\theta_1}+\frac{\delta}{\theta_2}-1)
   {\int_{\Omega}}|x|^{-\beta_1}
   u_{n_+}^{\theta}v_{n_+}^{\delta}dx \\
&\quad +   \mu(\frac{\alpha}{\theta_1}+\frac{\gamma}{\theta_2}-1)
   {\int_{\Omega}}|x|^{-\beta_2}
   u_{n_+}^{\alpha}v_{n_+}^{\gamma}dx \\
&\geq    (\frac{1}{p}-\frac{1}{\theta_1})\|u_n\|^p
   +(\frac{1}{q}-\frac{1}{\theta_2})\|v_n\|^q.
 \end{align*}
Therefore, independently of $\lambda\geq 0$ and $\mu>0$, we conclude
that $\{(u_n,v_n)\}$ is a bounded sequence in
$W^{1,p}_{0}(\Omega,|x|^{-ap})\times W^{1,q}_{0}(\Omega,|x|^{-bq})$.
In particular, we have that $\{(u_{n_-},v_{n_-})\}$ and
$\{(u_{n_+},v_{n_+})\}$ are bounded sequences in
$W^{1,p}_{0}(\Omega,|x|^{-ap})\times W^{1,q}_{0}(\Omega,|x|^{-bq})$,
then
 \begin{equation}\label{l11}
   -\|u_{n_-}\|^p
   = \left\langle I'(u_n,v_n),(u_{n_-},0)\right\rangle
   \to  0\ \quad\text{as }\ n\to \infty
 \end{equation}
and similarly
 \begin{equation}\label{l11.1}
   -\|v_{n_-}\|^q
   = \left\langle I'(u_n,v_n),(0,v_{n_-})\right\rangle
   \to  0\ \quad\text{as }\ n\to \infty.
 \end{equation}
Moreover, we get
 \begin{equation*}
   I(u_{n_+},v_{n_+}) =
   I(u_n,v_n)+\frac{1}{p}\|u_{n_-}\|^p
   + \frac{1}{q}\|v_{n_-}\|^q
   = I(u_n,v_n)+ O_n(1).
 \end{equation*}
Therefore, from $(\ref{l11})$ and $(\ref{l11.1})$, we obtain
 $I(u_{n_+},v_{n_+}) \to  c$ as $n \to \infty$.
 Similarly, if $(w,z)\in W_0^{1,p}(\Omega,|x|^{-ap})
\times W_0^{1,q}(\Omega,|x|^{-bq})$, we prove that
 $$
\langle I'({u_n}_{+},{v_n}_{+}),(w,z)\rangle
   = \langle I'(u_n,v_n),(w,z)\rangle +O_n(1),
$$
hence $I'(u_{n_+},u_{n_+})\to 0$ as $n\to \infty$.
\end{proof}

\section{Proof of theorem \ref{thm1}}

 \begin{lemma}\label{PS}
Suppose that \eqref{HOmega} and \eqref{Hexp} hold.
Assume that $p_i\in (p,p^*)$, $q_i\in (q,q^*)$, $i=1,2$, with
$\theta/p_1+\delta/q_1=\alpha/p_2+ \gamma/q_2=1$, and $\beta_i,
i=1,2$, as in \eqref{hipotese555}. Then, every $(PS)_c$-sequence
$\{(u_n,v_n)\}$ with $u_n,v_n\geq 0$, for a.e. in $\Omega$, is
precompact.
\end{lemma}

\begin{proof}
  From lemma \ref{u+}, the sequence
$\{(u_{n},v_{n})\}$ is  bounded  in
$W^{1,p}_{0}(\Omega,|x|^{-ap})\times W^{1,q}_{0}(\Omega,|x|^{-bq})$.
 We can assume, passing to a subsequence if necessary,  there exists
$(u,v)\in W_0^{1,p}(\Omega,|x|^{-ap})\times
W_0^{1,q}(\Omega,|x|^{-bq})$ satisfying $u_{n}\rightharpoonup u$ and
$v_{n}\rightharpoonup v$ weakly, as $n \to \infty$. From the
compact embedding theorem \cite[Theorem 2.1]{Xuan1}, we obtain
 \begin{gather*}
   u_{n} \to u \quad\text{in } L^{p_1}(\Omega, |x|^{-\beta_1})
   \cap L^{p_2}(\Omega,|x|^{-\beta_2})\,\quad\text{as }\, n \to \infty,\\
   v_{n} \to v \quad\text{in } L^{q_1}(\Omega,|x|^{-\beta_1})
   \cap L^{q_2}(\Omega,|x|^{-\beta_2})\,\quad\text{as }\, n\to
   \infty .
 \end{gather*}
Since there exist $f\in L^{p_1}(\Omega, |x|^{-\beta_1})$ and
$g\in L^{q_1}(\Omega, |x|^{-\beta_1})$ such that $|u_n|(x)\leq f(x)$ and
$|v_n|(x)\leq g(x)$, for a.e. $x\in \Omega$ and all $n\in
\mathbb{N}$, applying the Lebesgue's dominated convergence theorem
we infer that
 \begin{equation}\label{fjl1}
   \lim_{n\to \infty}{\int_{\Omega}}
   |x|^{-\beta_1}u_{n}^{\theta-1}v_{n}^{\delta}(u_n-u)dx
   =0\,,
 \end{equation}
and similarly
 \begin{equation}\label{fjl2}
   \lim_{n\to \infty}{\int_{\Omega}}
   |x|^{-\beta_2}u_{n}^{\alpha-1}v_{n}^{\gamma}(u_n-u)dx
   =0.
 \end{equation}
  Now, taking the upper limit in the equation
 \begin{align*}
& {\int_{\Omega}}|x|^{-ap}\left(|\nabla u_n|^{p-2}
   \nabla u_n-|\nabla u|^{p-2}\nabla u \right)\nabla(u_n-u)dx\\
&=\langle I'(u_n,v_n),(u_n-u,0)\rangle
   -{\int_{\Omega}}|x|^{-ap}|\nabla u|^{p-2}\nabla u \nabla(u_n-u)dx \\
&\quad +\lambda \theta {\int_{\Omega}}|x|^{-\beta_1}
   u_{n}^{\theta-1}v_{n}^{\delta}(u_n-u)dx
   +\mu \alpha{\int_{\Omega}}|x|^{-\beta_2}
   u_{n}^{\alpha-1}v_{n}^{\gamma}(u_n-u)dx\,.
\end{align*}
Using the definition of $(PS)_c$-sequence, the weak convergence,
$(\ref{fjl1})$, and $(\ref{fjl2})$, we obtain
 \begin{equation*}
   \limsup_{n\to \infty}
   {\int_{\Omega}}|x|^{-ap}\left(|\nabla u_n|^{p-2}
   \nabla u_n-|\nabla u|^{p-2}\nabla u \right)\nabla(u_n-u)dx = 0.
 \end{equation*}
Consequently, by a well known lemma (see e.g.
\cite[lemma 4.1]{Ghoussoub}) we achieve, up to a subsequence, that
$u_n\to u$ strongly in $W_0^{1,p}(\Omega,|x|^{-ap})$ as
$n\to \infty$.
  Analogously, we get  $v_n\to v$ strongly in
$W_0^{1,q}(\Omega,|x|^{-bq})$ as $n \to \infty$.
\end{proof}



\begin{proof}[Proof of theorem $\ref{thm1}$]
 By combining lemmata \ref{GC} and \ref{u+}, there
exists a $(PS)_c$-sequence $\{(u_n,v_n)\}$ in
$W_0^{1,p}(\Omega,|x|^{-ap}) \times W_0^{1,q}(\Omega,|x|^{-bq})$
with $u_n,v_n\geq 0$, for a.e. in $\Omega$. Moreover, from lemma
$\ref{PS}$ there exist $(u,v) \in W_0^{1,p}(\Omega,|x|^{-ap}) \times
W_0^{1,q}(\Omega,|x|^{-bq})$ and a subsequence of $\{(u_n,v_n)\}$,
that we will denote by $\{(u_n,v_n)\}$, such that $u_n\to u$
strongly in $W_0^{1,p}(\Omega,|x|^{-ap})$ and $v_n\to v$
strongly in $W_0^{1,q}(\Omega,|x|^{-bq})$, as $n \to
\infty$. Then, we conclude that
\begin{equation*}
   I(u,v)=c>0 \quad\text{and}\quad I'(u,v) = 0,
 \end{equation*}
that is, $(u,v)$ is a nonnegative weak solution of system
\eqref{sistemaperturbado1}. Moreover, it is easy to check that
$u,v\not \equiv 0$.
\end{proof}

\section{Proof of theorem \ref{edr}}

First of all, notice that by lemma \ref{GC} the geometric
conditions of the mountain pass theorem for the functional $I$ are
satisfied.

The next three lemmata are crucial in the proof of this theorem.

\begin{lemma}\label{existenciaCriticoProdFraco}
  Let $\{(u_n,v_n)\}\subset (W^{1,p}_{0}(\Omega,|x|^{-ap}))^2$ be
a bounded $(PS)_c$-sequence such that $u_n,v_n\geq 0$, for a.e. in $\Omega$,
and there exists $(u,v)\in (W^{1,p}_{0}( \Omega,|x|^{-ap}))^2$
satisfying $u_n\rightharpoonup u$ and $v_n\rightharpoonup v$ weakly
in $W^{1,p}_{0}(\Omega,|x|^{-ap})$, as $n \to \infty$. Then,
$(u,v)$ is a weak solution of system \eqref{sistemaperturbado1}
and $u,v \geq 0$ for a.e. in $\Omega$.
 \end{lemma}

\begin{proof}
 Arguing as in the proof of lemma $\ref{PS}$,
by combining the compact embedding theorem \cite[Theorem 2.1]{Xuan1}
with the Lebesgue's dominated convergence theorem, we obtain that
$u,v\geq 0$ for a.e. in $\Omega$,
 \begin{equation}\label{ed1}
   {\lim_{n\to \infty}}
   {{\int_{\Omega}}}
   |x|^{-\beta_1}u_{n}^{\theta-1}v_{n}^{\delta}wdx
   =    {{\int_{\Omega}}}
   |x|^{-\beta_1}u^{\theta-1}v^{\delta}wdx,
   \quad \forall w\in W^{1,p}_{0}(\Omega,|x|^{-ap}),
 \end{equation}
and
 \begin{equation}\label{ed2}
   {\lim_{n\to \infty}}
   {\int_{\Omega}}
   |x|^{-\beta_1} u_{n}^{\theta}v_{n}^{\delta-1}zdx
   =   {\int_{\Omega}}
   |x|^{-\beta_1} u^{\theta}v^{\delta-1}zdx,
   \quad \forall z\in W^{1,p}_{0}(\Omega,|x|^{-ap}).
 \end{equation}

 Notice  that $\nabla u_n(x) \to \nabla u(x)$ and
$\nabla v_n(x)\to \nabla v(x)$, for a.e. $x\in \Omega$,
as $n \to \infty$. These
facts can be proved arguing as in \cite{Boccardo} (see also
\cite{Assuncao(Critical), Ghoussoub, Rakotoson}).

  Since $\{(u_n,v_n)\}$ is bounded in $(W_0^{1,p}(\Omega,|x|^{-ap}))^2$,
we have $\{|\nabla u_n|^{p-2}\nabla u_n\}$ and $\{|\nabla
v_n|^{p-2}\nabla v_n\}$ are bounded in $(L^{\frac{p}{p-1}}
(\Omega,|x|^{-ap}))^{N}{ .}$ On the other hand,
since  $\alpha+\gamma=p^*$, by the H\"older's inequality, we infer
that $\{{u_n}^{\alpha-1}{v_n}^{\gamma}\}$ and
$\{{u_n}^{\alpha}{v_n}^{\gamma-1}\}$ are bounded in
$L^{\frac{p^*}{p^*-1}}(\Omega,|x|^{-e_1p^*})$.
 Therefore, by lemma $\ref{Kavian}$ we get
 \begin{equation}\label{ed3}
  \nabla u_n\rightharpoonup\nabla u
   \quad\text{and }
   \nabla v_n\rightharpoonup\nabla v
   \quad\text{weakly in }
   (L^{\frac{p}{p-1}}(\Omega,|x|^{-ap}))^{N}
 \end{equation}
and
 \begin{equation}\label{ed4}
   u_{n}^{\alpha} v_{n}^{\gamma-1}
   \rightharpoonup u^{\alpha}v^{\gamma-1},
   \quad
   u_{n}^{\alpha-1} v_{n}^{\gamma}
   \rightharpoonup u^{\alpha-1}v^{\gamma}
   \quad\text{weakly in }
   L^{\frac{p^*}{p^*-1}}(\Omega,|x|^{-c_1p^*}),
 \end{equation}
as $n\to \infty$.
Consequently, using $(\ref{ed1})-(\ref{ed4})$ we obtain
 \begin{equation*}
   \langle I'(u,v),(w,z)\rangle=    {\lim_{n\to\infty}}
   \langle I'(u_n,v_n),(w,z)\rangle   =0,
   \quad \forall (w,z)\in (W_0^{1,p}(\Omega,|x|^{-ap}))^2,
 \end{equation*}
that is, $(u,v)$ is a weak solution of system \eqref{sistemaperturbado1}.
\end{proof}


\begin{lemma}\label{PScriticoRN}
 In addition to \eqref{HOmega}, \eqref{Hexp}, and \eqref{HH1},
assume that $p=q$, $0\leq a=b$, $p_1=q_1\in(p,p^*)$, with
$\theta/p_1+\delta/q_1=1$, $p^*=q^*$, and $\beta_2=c_1p^*$.
Then, all the Palais Smale sequences
$\{(u_n,v_n)\}\subset(W^{1,p}_{0}(\Omega,|x|^{-ap}))^2$
for the operator $I$ at the  level $c$, with
$u_n,v_n\geq 0$ for a.e. in $\Omega$, are
precompact provided that
 \begin{equation}\label{mmC7R}
   c< (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}\tilde{S}
   ^{\frac{p^*}{p^*-p}}+ K(\lambda),
 \end{equation}
where
 \begin{equation*}
   K(\lambda)=\lambda p_1(\frac{1}{p}-\frac{1}{p_1})
    \lim_{n\to
   \infty}{\int_{\Omega}}|x|^{-\beta_1}
   {u}_{n}^{\theta}{v}_{n}^{\delta}dx.
 \end{equation*}
 \end{lemma}

\begin{proof}
 By Lemma \ref{u+} the sequence
$\{(u_n,v_n)\}$ is  bounded in
$(W^{1,p}_{0}(\Omega,|x|^{-ap}))^2;$ consequently,
there exists
$(u,v)\in(W^{1,p}_{0}(\Omega,|x|^{-ap}))^2$ such
that $u_n\rightharpoonup u$ and $v_n\rightharpoonup v$ weakly in
$W^{1,p}_{0}(\Omega,|x|^{-ap})$, as $n \to \infty$. Then, by
combining the compact embedding theorem \cite[Theorem 2.1]{Xuan1}
with the Lebesgue's dominated convergence theorem, we infer that
$u_n(x)\to u(x)$, $v_n(x)\to v(x)$, for a.e. in $\Omega$,
as $n \to \infty$, and
 \begin{equation}\label{RR2}
   {\lim_{n\to \infty}}
   {\int_{\Omega}}|x|^{-\beta_1}
   {u}_{n}^{\theta}{v}_{n}^{\delta}dx
   = {\int_{\Omega}}|x|^{-\beta_1}
   {u}^{\theta}{v}^{\delta}dx.
 \end{equation}
Moreover, as in Lemma
$\ref{existenciaCriticoProdFraco}$ we can suppose that $\nabla
u_n(x) \to \nabla u(x)$ and $\nabla v_n(x)\to \nabla
v(x)$, for a.e. $x\in \Omega$, as $n \to \infty$.

 Define $\tilde{u}_n=u_n-u$ and $\tilde{v}_n=v_n-v$. By
Brezis and Lieb  \cite[Theorem 1]{BrezisLieb} we have
\begin{itemize}
\item [(i)]\hspace{0.17cm} $\|u_n\|^p=\|\tilde{u}_n\|^p+\|u\|^p+O_n(1)$,
as $ n \to\infty$.
\item [(ii)]\hspace{0.05cm} $\|v_n\|^p=\|\tilde{v}_n\|^p+\|v\|^p+O_n(1)$,
as $ n\to \infty$.
\item[(iii)] {${}$}\vspace{-.57cm}
 \begin{align*}
   &{ \int_{\Omega}|x|^{-c_1p^*}|u_n|^{\alpha}|v_n|^{\gamma}dx
   -\int_{\Omega}|x|^{-c_1p^*}|\tilde{u}_n|^{\alpha}
   |\tilde{v}_n|^{\gamma}dx}\\
&= \int_{\Omega}
   |x|^{-c_1p^*}|u|^{\alpha}|v|^{\gamma}dx+
    O_{n}(1), \quad\text{as } n\to \infty.
 \end{align*}
\end{itemize}
We recall that the proof  of identity {\bf iii.} follows arguing as
in \cite[Lemma 8]{MoraisSouto}.

 By Lemma  \ref{existenciaCriticoProdFraco} we have that
$(u,v)$ is a weak solution of system \eqref{sistemaperturbado1},
that is, $\langle I'(u,v),(w,z)\rangle=0$ for all
$(w,z)\in (W_0^{1,p}(\mathbb{R}^N,|x|^{-ap}))^2$.
 By using $(\ref{RR2})$ and (i)--(iii),
we get
 \begin{align*}
&\|\tilde{u}_n\|^p-\mu\alpha
   {\int_{\Omega}}|x|^{-c_1p^*}
   |\tilde{u}_{n}|^{\alpha}|\tilde{v}_{n}|^{\gamma}dx \\
&=\|u_n\|^p-\|u\|^p
   -\mu\alpha\Big[
   {\int_{\Omega}}|x|^{-c_1p^*}
   {u}_{n}^{\alpha}{v}_{n}^{\gamma}dx
   -{\int_{\Omega}}|x|^{-c_1p^*}
   {u}^{\alpha}{v}^{\gamma}dx
   \Big]  + O_n(1)\\
& = \langle I'(u_n,v_n),(u_n,0)\rangle
   -\langle I'(u,v),(u,0)\rangle   + O_n(1) \\
& = O_n(1), \quad \text{as } n \to \infty.
 \end{align*}
Analogously, we obtain
 \begin{equation*}
   \|\tilde{v}_n\|^p-\mu\gamma
   {\int_{\Omega}}|x|^{-c_1p^*}
   |\tilde{u}_{n}|^{\alpha}|\tilde{v}_{n}|^{\gamma}dx
   = O_n(1).
 \end{equation*}
Thus, we can take $l\geq 0$ such that
 \begin{equation*}
   l={\lim_{n\to \infty}}
   \frac{\|\tilde{u}_n\|^p}{\alpha}
   = {\lim_{n\to \infty}}
   \frac{\|\tilde{v}_n\|^p}{\gamma}
   =\mu{\lim_{n\to \infty}}
   {\int_{\Omega}}|x|^{-c_1p^*}
   |\tilde{u}_{n}|^{\alpha}|\tilde{v}_{n}|^{\gamma} dx.
 \end{equation*}
 If $l=0$ the result is proved. Suppose by contradiction that
$l>0$. By the definition of $(PS)_c$-sequence we get
 \begin{equation}\label{RR3}
 \begin{aligned}
&c+O_n(1) \\
&= I(u_n,v_n)   -\frac{1}{p_1}\langle I'(u_n,v_n),   (u_n,v_n)\rangle\\
&= (\frac{1}{p}-\frac{1}{p_1})(\|u_n\|^p+\|v_n\|^p)
   +\mu(\frac{\alpha+\gamma}{p_1}-1) {\int_{\Omega}}|x|^{-c_1p^*}
   u_n^{\alpha}v_n^{\gamma}dx\\
&=(\frac{1}{p}-\frac{1}{p_1})
   (\|\tilde{u}_n\|^p+\|\tilde{v}_n\|^p)
   +(\frac{1}{p}-\frac{1}{p_1}) (\|u\|^p+\|v\|^p)\\
&\quad   +\mu(\frac{p^*}{p_1}-1)\Big[
   {\int_{\Omega}}|x|^{-c_1p^*}
   \tilde{u}_n^{\alpha}\tilde{v}_n^{\gamma}dx
   +{\int_{\Omega}}|x|^{-c_1p^*}
   u_n^{\alpha}v_n^{\gamma}dx\Big]   +O_n(1)  \\
&= (\frac{1}{p}-\frac{1}{p_1})p^* l   +(\frac{1}{p}-\frac{1}{p_1})
  (\|u\|^p+\|v\|^p) \\
&\quad  +(\frac{1}{p_1}-\frac{1}{p^*})p^*\Big[l
   +\mu{\int_{\Omega}}|x|^{-c_1p^*}
   u_n^{\alpha}v_n^{\gamma}dx\Big]   +O_n(1)   \\
&=   (\frac{1}{p}-\frac{1}{p^*})p^* l
   +(\frac{1}{p}-\frac{1}{p_1})
  \Big[\lambda p_1 {\int_{\Omega}}|x|^{-\beta_1}
   u^{\theta}v^{\delta}dx
   + \mu p^*{\int_{\Omega}}|x|^{-c_1p^*}
   u^{\alpha}v^{\gamma}dx\Big]\\
&\quad   +\mu(\frac{1}{p_1}-\frac{1}{p^*})p^*
  {\int_{\Omega}}|x|^{-c_1p^*}
   u_n^{\alpha}v_n^{\gamma}dx   +O_n(1) \\
&= (\frac{1}{p}-\frac{1}{p^*})p^* l
   +\lambda p_1(\frac{1}{p}-\frac{1}{p_1})
  {\int_{\Omega}}|x|^{-\beta_1} u^{\theta}v^{\delta}dx\\
&\quad   +\mu(\frac{1}{p}-\frac{1}{p^*})p^*
  {\int_{\Omega}}|x|^{-c_1p^*}
   u_n^{\alpha}v_n^{\gamma}dx  +O_n(1)\\
&\geq (\frac{1}{p}-\frac{1}{p^*})p^*l
   +\lambda p_1(\frac{1}{p}-\frac{1}{p_1})
  {\int_{\Omega}}|x|^{-\beta_1}
   u^{\theta}v^{\delta}dx+O_n(1).
 \end{aligned}
 \end{equation}
Using the definition of $\tilde{S}$ we have
 \begin{equation*}
\Big({\int_{\Omega}}|x|^{-c_1p^*}
   u_{n}^{\alpha}v_{n}^{\gamma} dx  \Big)^{p/p^*}\tilde{S}
   \leq \|u_n\|^{p}+\|v_n\|^p, \, \forall n.
\end{equation*}
 Hence, taking the limit in the above inequality we get
 \begin{equation*}
 \big(\frac{l}{\mu}\big)^{p/p^*}\tilde{S}
   \leq (\alpha + \gamma)l=p^*l
 \end{equation*}
 then
 \begin{equation}\label{ss13R}
  l\geq (\mu)^{\frac{-p}{p^*-p}}(p^*)^{\frac{-p^*}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}.
 \end{equation}
 Substituting \eqref{ss13R} in \eqref{RR3} and
taking the limit, we obtain
 \begin{equation*}
   c\geq   (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}
   +K(\lambda),
 \end{equation*}
which contradicts the inequality (\ref{mmC7R}).
\end{proof}

 \begin{lemma}\label{claim}
We can choose $e$ in $(\ref{Carmo})$ such
that $c$ given by $(\ref{Carmo1})$ satisfies
 \begin{equation}\label{c}
   c<(\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}.
 \end{equation}
 \end{lemma}

\begin{proof}
 Let us consider $s_0={s_1}{(s_1^{\alpha}
t_1^{\gamma})^{\frac{-1}{p^*}}}$ and
$t_0={t_1}{(s_1^{\alpha}t_1^{\gamma})^{\frac{-1}{p^*}}}$, where
$s_1,t_1>0$ and $s_1/t_1=(\alpha/\gamma)^{1/p}$, and $u_{\epsilon}$
the function defined in lemma
$\ref{Xuan-O(e)}$. Then, it is suffices to prove that there exists
$\epsilon>0$ such that
 \begin{equation*}
   {\sup_{t\geq 0}} I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))
   <(\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}.
 \end{equation*}
 Due to the geometric conditions of the mountain pass
theorem, for each $\epsilon>0$, there exists $t_{\epsilon}>0$ such
that
 \begin{equation*}
   0< \sigma\leq{\sup_{t\geq 0}}
   I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))=
   I(t_{\epsilon}(s_0u_{\epsilon}),t_{\epsilon}(t_0u_{\epsilon})).
 \end{equation*}
 Moreover, supposing by contradiction that there exists a
subsequence $\{t_{\epsilon_n}\}$ with $t_{\epsilon_n}\to 0$
as $n \to \infty$, we obtain
 \begin{align*}
   0&<\sigma \leq
   I(t_{\epsilon_n}(s_0u_{\epsilon_n}),
   t_{\epsilon_n}(t_0u_{\epsilon_n}))\\
&\leq    \frac{t_{\epsilon_n}^ps_0^p}{p}\|u_{\epsilon_n}\|^p
   +\frac{t_{\epsilon_n}^pt_0^p}{p}\|u_{\epsilon_n}\|^p \\
&\leq    \frac{t_{\epsilon_n}^p}{p}(s_0^p+t_0^p)
   (\tilde{S}_{a,p,R}+O(\epsilon_n^{{(N-d_1p)}/{d_1p}}))
    \to  0  \quad\text{as } n\to \infty,
  \end{align*}
which is an absurd. Then, there exists  $l>0$ with
$t_{\epsilon}\geq l$, for all $\epsilon>0$. Consequently, by using
lemma $\ref{Xuan-O(e)}$ and putting
$c_0=l^{p_1}s_0^{\theta}t_0^{\delta}$, we get
 \begin{equation}\label{gggg232}
   {\sup_{t\geq 0}}
   I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))
   \leq \frac{t_{\epsilon}^p}{p}\Big(
   \frac{s_1^p+t_1^p}{\left(s_1^{\alpha}t_1^{\gamma}
   \right)^{p/p^*}}\|u_{\epsilon}\|^p
    \Big)   -   \lambda c_0   {\int_{\Omega}}|x|^{-\beta_1}
   u_{\epsilon}^{p_1}dx    -   \mu t_{\epsilon}^{p^*}.
 \end{equation}
Note that
 \begin{equation}\label{oo333}
   t_{1_\epsilon} = (\mu p^*)^{\frac{-1}{p^*-p}}
   \Big(\frac{s_1^p+t_1^p}
   {(s_1^{\alpha}t_1^{\gamma})^{{p}/{p^*}}}
   \Big)^{\frac{1}{p^*-p}}\|u_{\epsilon}\|^{\frac{p}{p^*-p}}
 \end{equation}
is the unique maximum point of $f_{\epsilon}:(0,\infty)\to \mathbb{R}$,
given by
 \begin{equation*}
   f_{\epsilon}(t) = \frac{(s_1^p+t_1^p)\, t^p}
   {(s_1^{\alpha}t_1^{\gamma})^{{p}/{p^*}}p}
   \|u_{\epsilon}\|^p -\mu t^{p^*}.
 \end{equation*}
Also we know that
 \begin{equation}\label{oo444}
   (A+B)^{k}\leq A^k +k(A+B)^{k-1}B,
 \end{equation}
for all $A,B\geq 0$ and $k\geq 1$  \cite{Miyagaki}.
Observe that the following identity holds
 \begin{equation}\label{ksd}
   \big[\frac{s_1^p+t_1^p}{(s_1^{\alpha}
   t_1^{\gamma})^{{p}/{p^*}}}\big]
   = \left[({\alpha}/{\gamma})^
   {{\gamma}/{p^*}} + ({\alpha}/
   {\gamma})^{{-\alpha}/{p^*}}\right].
 \end{equation}
 By the Caffarelli-Kohn-Nirenberg's inequality,
$W_0^{1,p}(\Omega,|x|^{-ap})\subset W_{a,c_1}^{1,p}(\mathbb{R}^N)$.
Then
 \begin{equation}\label{SSS}
   \tilde{S}_{a,p}\leq C_{a,p}^*.
 \end{equation}
Substituting \eqref{oo333} in $(\ref{gggg232})$, from
\eqref{oo444}, \eqref{ksd}, \eqref{SSS}, and using lemma
\ref{Xuan-O(e)}, we obtain
 \begin{equation}\label{kkk101}
 \begin{aligned}
 { \sup_{t\geq 0}}   I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))
 &\leq (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \Big\{[(\frac{\alpha}{\gamma})^   {\gamma/p^*}
  +   (\frac{\alpha}{\gamma})^{-\alpha/p^*}]
   \tilde{S}_{a,p,R}\\
&\quad +O(\epsilon^{\frac{N-d_1p}{d_1p}})
   \Big\}^{\frac{p^*}{p^*-p}}
   -\lambda c_0
   {\int_{\Omega}}|x|^{-\beta_1}
   u_{\epsilon}^{p_1}dx
\\
&\leq (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \Big\{[(\frac{\alpha}{\gamma})^
   {\frac{\gamma}{p^*}} +
   (\frac{\alpha}{\gamma})^{\frac{-\alpha}{p^*}}]
   \tilde{S}_{a,p}\Big\}^{\frac{p^*}{p^*-p}}\\
&\quad +O(\epsilon^{\frac{N-d_1p}{d_1p}})
   -\lambda c_0    {\int_{\Omega}}|x|^{-\beta_1}
   u_{\epsilon}^{p_1}dx\\
& \leq(\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \Big\{[(\frac{\alpha}{\gamma})^
   {\frac{\gamma}{p^*}} +
   (\frac{\alpha}{\gamma})^{\frac{-\alpha}{p^*}}]
   C^*_{a,p}\Big\}^{\frac{p^*}{p^*-p}} \\
&\quad +O(\epsilon^{\frac{N-d_1p}{d_1p}})
   -\lambda c_0    {\int_{\Omega}}|x|^{-\beta_1}
   u_{\epsilon}^{p_1}dx
\end{aligned}
 \end{equation}
Now, from lemma \ref{1.121} and (\ref{kkk101}), we get
\begin{equation}\label{kkk1}
 \begin{aligned}
 { \sup_{t\geq 0}}
   I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))
   &\leq (\frac{1}{p}-
   \frac{1}{p^*})(\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}
   +O(\epsilon^{\frac{N-d_1p}{d_1p}}) \\
 &\quad   -\lambda c_0
   {\int_{\Omega}}|x|^{-\beta_1}
   u_{\epsilon}^{p_1}dx.
 \end{aligned}
 \end{equation}
Supposing that
$c<\frac{(p_1-p+1)N-(a+1)p_1}{p-1}-\frac{(N-p-ap)(p_1-p)}{p(p-1)}$ we have
 \begin{equation*}
   \frac{(N-p_1+ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}-
   \frac{(N-d_1p)(p-1)p_1}{d_1p^2} <\frac{N-d_1p}{d_1p},
 \end{equation*}
then, by lemma \ref{Xuan-O(e)} and by (\ref{kkk1}), we can take
a $\epsilon>0$ small enough  such that
 \begin{align*}
   {\sup_{t\geq 0}}
   I(t(s_0u_{\epsilon}),t(t_0u_{\epsilon}))
  &\leq
   (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}
   +O(\epsilon^{\frac{N-d_1p}{d_1p}})\\
&\quad  - O(\epsilon^{\frac{(N-d_1p)(p-1)(N-p_1-ap_1+c)}{d_1p(N-p-ap)}
   -\frac{(N-d_1p)(p-1)p_1}{d_1p^2}})\\
&<  (\frac{1}{p}-\frac{1}{p^*})
   (\mu p^*)^{\frac{-p}{p^*-p}}
   \tilde{S}^{\frac{p^*}{p^*-p}}.
 \end{align*}
 This completes the proof.
\end{proof}

\begin{proof}[Proof of theorem \ref{edr}]
From lemmata \ref{GC}, \ref{u+}, and $\ref{claim}$,
there exists a bounded $(PS)_c$-sequence $\{(u_n,v_n)\}$ in
$(W_0^{1,p}(\Omega,|x|^{-ap}))^2$  with $c>0$ satisfying $(\ref{c})$
and $u_n,v_n\geq 0$ for a.e. in $\Omega$. Since that $p_1\in (p,p^*)$,
it follows that $c$ verifies $(\ref{mmC7R})$. Thus, we have by lemma
\ref{PScriticoRN} that there exists $(u,v)\in (W_0^{1,p}(\Omega,|x|^{-ap}))^2$
with $u_n\to u$ and $v_n\to v$ in $W_0^{1,p}(\Omega,|x|^{-ap})$,
as $n\to \infty$. Hence, we conclude
\begin{equation*}
   I(u,v)=c>0 \quad\text{and}\quad I'(u,v) = 0,
 \end{equation*}
that is, $(u,v)$ is a nontrivial and nonnegative weak solution of system
\eqref{sistemaperturbado1}.
\end{proof}

\section{Proof of theorem $\ref{dif}$}

 The proof follows the steps the proof of theorem \ref{edr}.
By lemmata \ref{GC} and \ref{u+}, there exists a
$(PS)_c$-sequence $\{(u_n,v_n)\}$ in
$W_0^{1,p}(\Omega,|x|^{-ap})\times W_0^{1,q}(\Omega,|x|^{-bq})$ with
$c>0$ given as in $(\ref{Carmo1})$ and $u_n,v_n\geq 0$ for a.e. in
$\Omega$. Moreover, $\{(u_n,v_n)\}$ is bounded uniformly in $\mu>0$,
that is, there exist $M>0$ such that $\|(u_n,v_n)\|\leq M$ for all
$n \in \mathbb{N}$, uniformly in $\mu>0$. Consequently, we get that
$c\leq \overline{M}$ uniformly in $\mu>0$.

  Due to the boundedness of $\{(u_n,v_n)\}$, there exists a
subsequence, that we will denote by $\{(u_n,v_n)\}$, and $(u,v)
\in W_0^{1,p}(\Omega,|x|^{-ap}) \times
W_0^{1,q}(\Omega,|x|^{-bq})$ with $u_n\rightharpoonup u$ weakly in
$W^{1,p}_{0}(\Omega,|x|^{-ap})$ and $v_n\rightharpoonup v$ weakly
in $W^{1,q}_{0}(\Omega,|x|^{-bq})$, as $n \to \infty$.
Then, arguing as in lemma $\ref{existenciaCriticoProdFraco}$ we
obtain that $(u,v)$ is a weak solution of system
 \eqref{sistemaperturbado1} with $u,v\geq 0$ for a.e. in $\Omega$.

  Now, we will prove that there exists $\mu_0>0$ such that
$u,v$ is nontrivial, provided that $0<\mu<\mu_0$.

 Supposing by contradiction that $u(x)\equiv 0$ for a.e.
$x\in \Omega$ and proceeding as in the proof of theorem
\ref{edr}, we obtain $l>0$ such that
 \begin{equation*}
   l={\lim_{n\to \infty}}
   \frac{\|u_n\|^p}{\alpha}
   ={\lim_{n\to \infty}}
   \frac{\|v_n\|^q}{\gamma}
   =\mu{\lim_{n\to \infty}}
   {\int_{\Omega}}|x|^{-c_1p^*}
   u_{n}^{\alpha}v_{n}^{\gamma} dx
 \end{equation*}
and
 \begin{equation}\label{dr 223}
   c= {{\lim_{n\to \infty}}}  I(u_n,v_n)
   = (\frac{\alpha}{p}+\frac{\gamma}{q}-1)l  >0.
 \end{equation}
On the other hand, by Young's inequality and definitions
of $C^*_{a,p}$ and $C^*_{b,q}$, we obtain
 \begin{align*}
   \frac{l}{\mu} &\leq \frac{\alpha^{({p^*+p})/{p}}}{p^*}
   (C^*_{a,p})^{{-p^*}/{p}}l^{{p^*}/{p}}
   +\frac{\gamma^{({q^*+q})/{q}}}{q^*}
   (C^*_{b,q})^{{-q^*}/{q}}l^{{q^*}/{q}} \\
&\leq    \big[\frac{\alpha^{(p^*+p)/p}}{p^*}
   (C^*_{a,p})^{{-p^*}/{p}}+\frac{\gamma^{(q^*+q)/q}}{q^*}
   (C^*_{b,q})^{{-q^*}/{q}}\big]l^{\tau}  ,
 \end{align*}
where $\tau=\max\{p^*/p, q^*/q\}$ if $l>1$, and $\tau=\min\{p^*/p,
q^*/q\}$ if $l\leq 1$. Therefore,
 \begin{equation*} %\label{dr 224}
   l\geq \Big[\mu
   \Big(\frac{\alpha^{(p^*+p)/p}}{p^*}
   (C^*_{a,p})^{{-p^*}/{p}}+\frac{\gamma^{(q^*+q)/q}}{q^*}
   (C^*_{b,q})^{{-q^*}/{q}}\Big)\Big]^{\frac{-1}{\tau-1}} .
 \end{equation*}
Thus substituting the above inequality in \eqref{dr 223} and
taking $\mu_0>0$ small enough we conclude
 \begin{equation*}
   c\geq(\frac{\alpha}{p}+\frac{\gamma}{q}-1)\Big[\mu
   \Big(\frac{\alpha^{(p^*+p)/p}}{p^*}
   (C^*_{a,p})^{{-p^*}/{p}}+\frac{\gamma^{(q^*+q)/q}}{q^*}
   (C^*_{b,q})^{{-q^*}/{q}}\Big)\Big]^{\frac{-1}{\tau-1}}
   \geq \overline{M},
 \end{equation*}
for all $0<\mu<\mu_0$, which is an absurd.
% \end{proof}



 \section{Appendix}

 \subsection*{Proof of lemma \ref{Xuan-O(e)}}
 From equation (\ref{zzz1}) we obtain
 \begin{gather*}
  \|\nabla y_{\epsilon}\|^p_{L^p(\mathbb{R}^N,|x|^{-ap})}
=   (\tilde{S}_{a,p,R})^{p^*/(p^*-p)} = (k_{a,p}(\epsilon))^{p}
   \|\nabla U_{a,p,\epsilon}\|^p_{L^p(\mathbb{R}^N,|x|^{-ap})},
\\
  \|y_{\epsilon}\|^{p^*}_{L^{p^*}(\mathbb{R}^N,|x|^{-c_1p^*})}
=  (\tilde{S}_{a,p,R})^{p^*/(p^*-p)} = (k_{a,p}(\epsilon))^{p^*}
   \|U_{a,p,\epsilon}\|^{p^*}_{L^{p^*}(\mathbb{R}^N,|x|^{-c_1p^*})}.
 \end{gather*}
We observe that
 \begin{equation*}
\nabla (\psi(x)U_{a,p,\epsilon}(x))
   =   \begin{cases}
   \nabla U_{a,p,\epsilon}(x)& \text{if } |x|<R_0 \\
   U_{a,p,\epsilon}(x)\nabla \psi(x)
   +\psi(x)\nabla U_{a,p,\epsilon}(x)
   &\text{if }  R_0\leq |x|<2R_0 \\
   0 &\text{if } |x|\geq 2R_0
   \end{cases}
 \end{equation*}
and
 \begin{equation*}
   \nabla U_{a,p,\epsilon}(x)= -\frac{N-p-ap}{p-1}\cdot
   \frac{|x|^{[d_1p(N-p-ap)/(p-1)(N-d_1p)] -2}x}
   {(\epsilon+|x|^{d_1p(N-p-ap)/(p-1)(N-d_1p)})^{N/d_1p}}\cdot
 \end{equation*}
Therefore,
 \begin{align*}
   {\int_{\Omega}}|x|^{-ap}|\nabla
   (\psi U_{a,p,\epsilon})(x)|^pdx
 &= O(1) + {\int_{\mathbb{R}^N}}|x|^{-ap}
   |\nabla U_{a,p,\epsilon}(x)|^pdx    \\
 &= O(1) + (\tilde{S}_{a,p,R})^{\frac{p^*}{p^*-p}}
   (k_{a,p}(\epsilon))^{-p}
\end{align*}
and
 \begin{align*}
  {\int_{\Omega}}|x|^{-c_1p^*}|
   \psi(x)U_{a,p,\epsilon}(x)|^{p^*}dx
 &=    O(1) + {\int_{\mathbb{R}^N}}
   |x|^{-c_1p^*}   |U_{a,p,\epsilon}(x)|^{p^*}dx \\
 &= O(1) + (\tilde{S}_{a,p,R})^{\frac{p^*}{p^*-p}}
   (k_{a,p}(\epsilon))^{-p^*}.
 \end{align*}
 Consequently,
 \begin{align*}
 {\int_{\Omega}}|x|^{-ap}   |\nabla u_{\epsilon}(x)|^pdx
 &= \frac{O(1)+(\tilde{S}_{a,p,R})^{p^*/(p^*-p)}
   (k_{a,p}(\epsilon))^{-p}}
   {[O(1)+(\tilde{S}_{a,p,R})^{p^*/(p^*-p)}
   (k_{a,p}(\epsilon))^{-p^*}]^{p/p^*}} \\
&= \frac{(k_{a,p}(\epsilon))^{p}[O(1)
   +(\tilde{S}_{a,p,R})^{p^*/(p^*-p)}(k_{a,p}(\epsilon))^{-p}]}
   {[O(k_{a,p}(\epsilon)^{p^*})
   +(\tilde{S}_{a,p,R})^{p^*/(p^*-p)}]^{p/p^*}} \\
 &\leq  \tilde{S}_{a,p,R}+O(k_{a,p}(\epsilon)^p)\\
 &= \tilde{S}_{a,p,R}+O(\epsilon^{(N-d_1p)/d_1p}).
 \end{align*}

  Now, we  prove that  $\|u_{\epsilon}\|^{p_1}_{
L^{p_1}(\Omega,|x|^{-\beta_1})}$ is as in (\ref{ff1}). Considering
the changes of variables by the polar coordinates and
$s=R_0^{-1}\epsilon^{-1/\alpha}r$ with $\alpha = \frac{d_1p(N-p-ap)}
{(p-1)(N-d_1p)}$, we obtain
 \begin{equation}\label{k1}
 \begin{aligned}
 &{ \int_{\Omega}}|x|^{-(a+1)p_1+c}
   |\psi U_{a,p,\epsilon}|^{p_1}dx \\
& \geq O(1)+ { \int_{|x|<R_0}}
   \hspace{-.5cm}|x|^{-(a+1)p_1+c}
   |U_{a,p,\epsilon}|^{p_1}\, dx \\
&  =O(1) +{  \omega_N \int_{0}^{R_0}}
   \hspace{-.3cm} \frac{r^{-(a+1)p_1+c+N-1}}
   {(\epsilon + r^{\alpha})^{{(N-d_1p)p_1}/{d_1p}}}\,dr \\
&  = O(1)  +  \omega_N
   (R_0^{\alpha}\epsilon)^{\frac{-(N-d_1p)p_1}{d_1p}+ \frac
   {(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}} \\
&\quad    \times{ \int_{0}^{\epsilon^{-1/\alpha}}}
   \hspace{-.6cm}\frac{s^{-(a+1)p_1+c+N-1}}
   {( R_0^{-\alpha} +s^{\alpha}
   )^{{(N-d_1p)p_1}/{d_1p}}}\, ds.
  \end{aligned}
 \end{equation}
Assuming that $c=[(p_1-p+1)N-(a+1)p_1]/({p-1})$, we see that
 \begin{gather*}
   \frac{-(N-d_1p)p_1}{d_1p}
   +\frac{(p-1)(N-p_1-ap_1+c)(N-d_1p)}{d_1p(N-p-ap)}
   =0,\\
 -(a+1)p_1+c+N-\frac{(N-p-ap)p_1}{(p-1)}
   =0\cdot
\end{gather*}
 Therefore, by (\ref{k1}),
 \begin{align*}
\int_{\Omega}|x|^{-(a+1)p_1+c}
   |\psi U_{a,p,\epsilon}|^{p_1}dx
&\geq \omega_N   { \int_{1}^{\epsilon^{-1/\alpha}}}
   \frac{s^{-(a+1)p_1+c+N-1-\alpha\frac{(N-d_1p)p_1}{d_1p}}}
   {[(R_0s)^{-\alpha}+1]^{{(N-d_1p)p_1}/{d_1p}}}\,ds \\
&\geq\frac{\omega_N}{\left(R_0^{-\alpha}
   +1\right)^{{(N-d_1p)p_1}/{d_1p}}} |\ln(\epsilon)|
  = O(|\ln(\epsilon)|).
 \end{align*}
 Hence, we obtain
 \begin{align*}
   { \int_{\Omega}}
   |x|^{-(a+1)p_1+c}|u_{\epsilon}|^{p_1}dx
&\geq   \frac{ O(|\ln(\epsilon)|)}
   {\left[O(1)+(S_{a,p,R})^{p^*/(p^*-p)}
   (k_{a,p}(\epsilon))^{-p^*}\right]^{p_1/p^*}} \\
&=   \frac{ O(|\ln(\epsilon)|)}
   {(k_{a,p}(\epsilon))^{-p_1}
   \left[O(k_{a,p}(\epsilon)^{p^*})
   +(S_{a,p,R})^{p^*/(p^*-p)}\right]^{p_1/p^*}} \\
&\geq O(\epsilon^{(N-d_1p)p_1/d_1p^2}
   |\ln(\epsilon)|).
 \end{align*}
   Assuming that $c >[(p_1-p+1)N-(a+1)p_1]/({p-1})$, we have
 \begin{gather*}
  \frac{-(N-d_1p)p_1}{d_1p}+ \frac
     {(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}>0, \\
  -(a+1)p_1+c+N-\frac{(N-p-ap)p_1}{(p-1)}>0.
 \end{gather*}
 Consequently, by $(\ref{k1})$,
 \begin{align*}
& \int_{\Omega}|x|^{-(a+1)p_1+c}
   |\psi U_{a,p,\epsilon}|^{p_1}dx \\
& \geq O(1) +  \omega_N
   (R_0^{\alpha}\epsilon)^{\frac{-(N-d_1p)p_1}{d_1p}+
   \frac{(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}} \\
&\quad  \times \frac{1}{( R_0^{-\alpha} +1
   )^{{(N-d_1p)p_1}/{d_1p}}}
    \int_{1/2}^{1}    s^{-(a+1)p_1+c+N-1}\,ds
 \geq  O(1).
\end{align*}
 Hence, we get
 \begin{align*}
   { \int_{\Omega}}
   |x|^{-(a+1)p_1+c}|u_{\epsilon}|^{p_1}dx
&\geq    \frac{O(1)}{(k_{a,p}(\epsilon))^{-p_1}[
   O(k_{a,p}(\epsilon)^{p^*})+(S_{a,p,R})^{p^*/(p^*-p)}
   ]^{p_1/p^*}} \\
& \geq   \frac{O(k_{a,p}(\epsilon)^{p_1})}
   {[O(1)+(S_{a,p,R})^{p^*/(p^*-p)}]^{p_1/p^*}} \\
& \geq   O(\epsilon^{(N-d_1p)p_1/d_1p^2}).
 \end{align*}

 If $c < [(p_1-p+1)N-(a+1)p_1]/({p-1})$, we see that
 \begin{gather*}
  \frac{-(N-d_1p)p_1}{d_1p}+ \frac
   {(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}<0,\\
  -(a+1)p_1+c+N-\frac{(N-p-ap)p_1}{(p-1)}<0.
 \end{gather*}
 Using (\ref{k1}) we obtain
 \begin{align*}
&   { \int_{\Omega}|x|^{-(a+1)p_1+c}}  |\psi U_{\epsilon}|^{p_1}dx \\
& \geq O(1)\epsilon^{\frac{-(N-d_1p)p_1}{d_1p}
   +\frac{(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}}
   { \int_{1/2}^{1}}
   s^{-(a+1)p_1+c+N-1}ds \\
& \geq O\Big(\epsilon^{\frac{-(N-d_1p)p_1}{d_1p}
   +\frac{(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}}
   \Big)\,.
 \end{align*}
  Hence, we conclude
 \begin{align*}
 { \int_{\Omega}|x|^{-(a+1)p_1+c}}    |u_{\epsilon}|^{p_1}dx
 &\geq    \frac{O\big(\epsilon^{\frac{-(N-d_1p)p_1}{d_1p}
   + \frac{(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}}\big)}
   {\big[O(1)+(S_{a,p,R})^{p^*/(p^*-p)}(k_{a,p}
   (\epsilon))^{-p^*}\big]^{p_1/p^*}} \\
&\geq O\Big(\epsilon^{\frac{(N-p_1-ap_1+c)(p-1)(N-d_1p)}{d_1p(N-p-ap)}
   -\frac{(N-d_1p)(p-1)p_1}{d_1p^2}}\Big).
 \end{align*}
%\end{proof}



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\end{document}