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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 69, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/69\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions to
 strongly indefinite Hamiltonian system involving critical
 Hardy-Sobolev exponents}

\author[F. O. de Paiva, R. S. Rodrigues \hfil EJDE-2013/69\hfilneg]
{Francisco Odair de Paiva, Rodrigo S. Rodrigues}  % in alphabetical order

\address{Francisco Odair de Paiva \newline
 Departamento de Matem\'atica, Universidade Federal de S\~ao Carlos
 13565-905, S\~ao Carlos, SP, Brazil}
\email{odair@dm.ufscar.br}

\address{ Rodrigo da Silva 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 September 10, 2012. Published March 13, 2013.}
\thanks{Supported by Fapesp-Brazil.}
\subjclass[2000]{35B25, 35B33, 35J55, 35J70}
\keywords{Hamiltonian systems; strongly indefinite variational structure;
\hfill\break\indent  critical Hard-Sobolev exponents}

\begin{abstract}
 In this article, we  study the existence and multiplicity of
  nontrivial solutions for a class of Hamiltoniam systems with
  weights and nonlinearity involving the Hardy-Sobolev exponents.
  Results are proved using variational methods for strongly
  indefinite functionals.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{introduction}

 Elliptic problems involving general operators,
such as the degenerate quasilinear elliptic equation
$-\operatorname{div}(|x|^{-2a}\nabla u)=|x|^{\zeta}f(u)$,
were motivated by the Caffarelli, Kohn, and Nirenberg's inequality
\cite{Caff}
\begin{equation}\label{Caffarelli-Kohn-Nirenberg}
   \Big(\int_{\mathbb{R}^N}
   |x|^{-2^*_ae_1}|u|^{2^*_a} dx\Big)^{2/2^*_a}
   \leq C_{a,e_1}\Big(
   {\int_{\mathbb{R}^N}|x|^{-2a}|\nabla u|^2 dx}
   \Big), \quad
   \forall  u \in C_0^{\infty}(\mathbb{R}^N),
 \end{equation}
where $N\geq3$, $-\infty <a<(N-2)/2$, $a\leq e_1\leq a+1$,
$2^*_a:=2N/(N-2d_a)$, $d_a=1+a-e_1$, and $C_{a,e_1}>0$.
Note that several papers have  appeared on this subject. Mainly, the works
about the existence of solution for quasilinear equations and
systems of the gradient type with nonlinearity involving critical
growth. See, for instance, \cite{Alves, Catrina, MR-1, MR-2, Xuan1}
and references therein.
 In particular, for $a=e_1=0$, Smets, Willem, and Su \cite{SWS} studied
the existence of non-radial ground states for the H\'enon equation
  \begin{gather*}
   -\Delta u  =  |x|^{\zeta} u^{l-1} \quad \text{in } B,   \\
   u  = 0 \quad \text{on } \partial B,
\end{gather*}
where $B$ denotes the unit ball in $\mathbb{R}^N$ with $\zeta\geq 0$
and $l\in (2, 2^*)$. More general H\'enon-Type problems has been
studied by Carri\~ao, de Figueiredo, and Miyagaki
\cite{CDM}, for example. Also, we would like to refer to \cite{Ni} for
H\'enon equation, and to \cite{FJ} for Hardy-H\'enon system.

 In this article, we study the following class of quasilinear elliptic
 systems with weights and nonlinearity involving critical
Hardy-Sobolev exponent
 \begin{equation}\label{Sist-Hamil}
 \begin{gathered}
   -\operatorname{div}(|x|^{-2a}\nabla u)
 =  \mu_1 \frac{|u|^{\tau-2}u}{|x|^{\beta_0}}
   +\frac{H_u(x,u,v)}{|x|^{\beta_2}}
   + \alpha\frac{|u|^{\alpha-2}|v|^{\gamma}u}{|x|^{2^*_a e_1}}
   \quad \text{in } \Omega,
   \\
   -\operatorname{div}(|x|^{-2b}\nabla v)
 =  -\mu_2\frac{|v|^{\xi-2}v}{|x|^{\beta_1}}
   - \frac{H_v(x,u,v)}{|x|^{\beta_2}}
   - \gamma\frac{|u|^{\alpha}|v|^{\gamma-2}v}{|x|^{2^*_b e_2}}
   \quad \text{in } \Omega,
   \\
   u=v  = 0 \quad \text{on }\partial \Omega,
 \end{gathered}
  \end{equation}
where
\begin{itemize}
\item[(H1)] $\Omega$   is a bounded smooth
domain in $\mathbb{R}^N$  ($N\geq 3$) with $0\in \Omega$
and $H:\Omega\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is of
the class $C^1$.

\item[(H2)] The exponents satisfy
\[
  0\leq a,\quad b<\frac{N-2}{2},\quad \xi \in (1,\frac{2N}{N-2}),\quad
   \tau \in (2,\frac{2N}{N-2}), \quad
 \alpha, \gamma>1,
\]
$2^*_a=\frac{2N}{N-2d_a}$  and $2^*_b=\frac{2N}{N-2d_b}$
are the Hardy-Sobolev exponents,
\begin{gather*}
    d_a=1+a-e_1, \quad
   0\leq  a\leq e_1<a+1, \\
  d_b=1+b-e_2,\quad
   0\leq b\leq e_2<b+1,
 \end{gather*}
with $2^*_ae_1 = 2^*_be_2$ and
\[
   \frac{\alpha}{2^*_a}+\frac{\gamma}{2^*_b}=1.
\]
\end{itemize}


Note that problem \eqref{Sist-Hamil} belongs to the class of
Hamiltonian elliptic systems
 \begin{gather*}
    -L_1(x,u)  =  \frac{\partial F}{\partial u}(x,u,v) \quad \text{in }  \Omega,
    \\
    -L_2(x,v)  =  -\frac{\partial F}{\partial v}(x,u,v) \quad \text{in }  \Omega,
    \\
    u=v  =  0 \quad \text{on }  \partial \Omega,
  \end{gather*}
where  $L_1$ and $L_2$ are self-adjoint elliptic operators of second
order. It is well known that the Euler-Lagrange functional
associated is strongly indefinite. For this type of system with
$L_1=L_2=\Delta$, the Laplacian operator, and subcritical growth, we
would like to refer Benci and Rabinowitz \cite{BR}, Costa and
Magalh\~aes \cite{CM-1}, and de Figueiredo and Ding
\cite{Djairo-Ding}. For critical and supercritical growth, we cite
the de Figueiredo and Ding \cite{Djairo-Ding} and Hulshof,
Mitidieri, and van der Vorst \cite{HMV}. We also cite the papers
\cite{Clapp} and \cite{Yan}. Our results will be obtained as an
application of some critical point results to strongly indefinite
functionals proved in \cite{Djairo-Ding} and certain Galerkin
approximations.


For the rest of this article we will assume that
$H:\Omega\times\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is
of the class $C^1$ and satisfies:
\begin{itemize}
\item[(H4)] There exist $K_0, K_1>0$,
$p_i,q_i,r_i\in (1,{2N}/({N-2}))$, for $i=1,2$, $p_0,q_0\in
(1,{2N}/({N-2}))$, such that
 \begin{gather*}
  |H(x,s,t)|  \leq  K_0(|s|^{p_0}+|t|^{q_0}),
  \\
  |H_s(x,s,t)l|  \leq  K_1(|s|^{p_1}+|t|^{q_1}+|l|^{r_1}),
  \\
  |H_t(x,s,t)l|  \leq  K_1(|s|^{p_2}+|t|^{q_2}+|l|^{r_2}),
 \end{gather*}
for all $s,t,l\in \mathbb{R}$, $x\in \Omega$;

\item[(H5)] $\beta_0$, $\beta_1$, $\beta_2$ satisfy
 \begin{gather*}
   \beta_0 <(a+1)\tau+N(1-\frac{\tau}{2}),
   \\
   \beta_1 \leq (b+1)\xi+N(1-\frac{\xi}{2}),
   \\
\begin{aligned}
 \beta_2 <\min\big\{&(a+1)p_i+N(1-\frac{p_i}{2}),
   (a+1)r_1+N(1-\frac{r_1}{2}),\\
 &   (b+1)q_i+N(1-\frac{q_i}{2} ),  (b+1)r_2+N(1-\frac{r_2}{2}) :
   i=1,2\big\};
 \end{aligned}
 \end{gather*}

\item[(H6)] for all $s,t\in \mathbb{R}$, almost
everywhere $x\in \Omega$,
\[
   H(x,s,t)
 \geq    -\Big(\frac{\mu_2}{\xi}|x|^{\beta_2-\beta_1}|t|^{\xi}
   +|x|^{\beta_2-2^*_a e_1}|s|^{\alpha}|t|^{\gamma}\Big);
\]

\item[(H7)] there exist $\theta_1\in (2,\tau]$ and
$\theta_2\in (1,2)$ such that, for all $s,t\in\mathbb{R}$,
almost everywhere $x\in\Omega$,
\[
   \frac{1}{\theta_1}H_u(x,s,t)s+\frac{1}{\theta_2}H_v(x,s,t)t
   \geq  H(x,s,t), \quad
   \frac{\alpha}{\theta_1}+\frac{\gamma}{\theta_2} \geq 1.
\]
\end{itemize}

Under the assumptions (H1), (H2)--(H5),
$\mu_1\leq 0$, $\mu_2\geq 0$, and
$\gamma H_u(x,s,t)s=\alpha H_v(x,s,t)t$ for all $s,t\in\mathbb{R}$,
almost everywhere $x\in\Omega$, we note that
system \eqref{Sist-Hamil} does not possess any
nontrivial weak solution.
Indeed, supposing by contradiction that
$(u,v)$ is a nontrivial weak solution, we obtain
\[
   \int_{\Omega}\frac{|\nabla u|^2}{|x|^{2a}}\,dx
   -\mu_1{\int_{\Omega}}\frac{|u|^{\tau}}{|x|^{\beta_0}}\,dx
   =
   \alpha \Big[
   \frac{1}{\alpha}{\int_{\Omega}} \frac{H_u(x,u,v)u}{|x|^{\beta_2}} \,dx
   +{\int_{\Omega}} \frac{|u|^{\alpha}|v|^{\gamma}}{|x|^{2^*_a e_1}}\, dx
   \Big]
\]
and
\[
   \int_{\Omega}\frac{|\nabla v|^2}{|x|^{2b}}\,dx
   +\mu_2{\int_{\Omega}}\frac{|v|^{\xi}}{|x|^{\beta_1}}\,dx
   =
   -\gamma \Big[
   \frac{1}{\gamma}{\int_{\Omega}} \frac{H_v(x,u,v)v}{|x|^{\beta_2}} \,dx
   +{\int_{\Omega}} \frac{|u|^{\alpha}|v|^{\gamma}}{|x|^{2^*_a e_1}}\, dx
   \Big].
\]
So, we conclude that
\[
   \int_{\Omega}\frac{|\nabla u|^2}{|x|^{2a}}\,dx
   -\mu_1{\int_{\Omega}}\frac{|u|^{\tau}}{|x|^{\beta_0}}\,dx
   =-\frac{\alpha}{\gamma}\Big({\int_{\Omega}}\frac{|\nabla v|^2}{|x|^{2b}}\,dx
   +\mu_2{\int_{\Omega}}\frac{|v|^{\xi}}{|x|^{\beta_1}}\,dx
   \Big),
 \]
hence,  $u= v= 0$ almost everywhere in $\Omega$, which is a
contradiction.

  Before enunciating our results, we recall that Xuan
\cite{Xuan-eigenvalue}, under the assumption (H1), proved
that if $0\leq a <({N-2})/{2}$, and $\beta_2<2(a+1)$, then there
exists the first eigenvalue $\lambda_{1_{\beta_2}}>0$ of problem
\begin{equation}
 \begin{gathered}
   -\operatorname{div}(|x|^{-2a}\nabla u)= \lambda |x|^{-\beta_2} u
\quad\text{in } \Omega,   \\
   u  =  0 \quad \text{on } \partial \Omega,
 \end{gathered}
 \end{equation}
which is associated to an eigenfunction
$\varphi_{1_{\beta_2}}\in C^{1,\alpha_1}(\Omega\setminus\{0\})$ with
$\varphi_{1_{\beta_2}}>0$ in $\Omega \setminus\{0\}$ for some $\alpha_1>0$.

\begin{theorem}\label{teorema-1}
  Assume {\rm (H1), (H2)--(H7)}, and $\theta_2\in (1,2)\cap (1,\xi]$.
Then system \eqref{Sist-Hamil}
possesses a nontrivial weak solution for each $\mu_1>0$ and $\mu_2\geq 0$,
provided that one of the following conditions is satisfied
\begin{itemize}
\item[(i)] $p_0\in (2,\frac{2N}{N-2})$;

\item[(ii)] $p_0=2$ and
$K_0\in (0,\frac{\lambda_{1_{\beta_2}}}{2})$.

\end{itemize}
Moreover, if $p_0\in(1,2)$ there exists $\bar{\mu}_0>0$ such that
system \eqref{Sist-Hamil} possesses a nontrivial weak solution for
each $\mu_1\in (0,\bar{\mu}_0)$ and $\mu_2\geq 0$.
 \end{theorem}

\begin{theorem}\label{teorema-2.2}
 Assume {\rm (H1), (H2)--(H7)},
$\xi<2$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$, and $\theta_2\in [\xi,2)$.
Then system \eqref{Sist-Hamil} possesses a nontrivial weak solution for
each $\mu_1>0$ and $\mu_2<0$, provided that one of the following conditions
is satisfied
\begin{itemize}
 \item[(i)] $p_0\in (2,\frac{2N}{N-2});$

 \item[(ii)] $p_0=2$ and
$K_0\in (0,\frac{\lambda_{1_{\beta_2}}}{2})$.

\end{itemize}
 Moreover, if $p_0\in(1,2)$ there exists $\bar{\mu}_0>0$ such that
system \eqref{Sist-Hamil} possesses a nontrivial weak solution
for each $\mu_1\in(0,\bar{\mu}_0)$ and $\mu_2<0$.
 \end{theorem}

\begin{theorem}\label{teorema-3}
   In addition to {\rm (H1), (H2)--(H7)},
$\theta_2\in (1,2)\cap (1,\xi]$, and $H$ even in the variables $s,t$,
suppose either $H(x,s,0)\leq0$ for all $s\in \mathbb{R}$, $x\in \Omega$ or
$p_0=\tau$. Then system \eqref{Sist-Hamil} possesses a sequence
$\{(u_n,v_n)\}$ of nontrivial weak solutions with energies
$I(u_n,v_n)\to\infty$ as $n\to\infty$ for each $\mu_1>0$
and $\mu_2\geq 0$. Moreover, this result still held if $\xi<2$,
$\beta_1<(b+1)\xi+N[1-(\xi/2)]$, $\theta_2\in [\xi,2)$, $\mu_1>0$,
and $\mu_2<0$. See the definition of $I$ in \eqref{Euler-I}.
 \end{theorem}

Now we present some complementary results, for which we use
following condition:
\begin{itemize}
\item[(H9)] Assume that
\[
   H(x,s,t)    \geq    -(|x|^{\beta_2-2^*_a e_1}|s|^{\alpha}|t|^{\gamma}),
\]
\end{itemize}
instead of the condition (H6). Notice that if $\mu_2<0$ then
(H6) is more restrictive than (H9). To obtain similar
results we will impose that $-\mu_2$ is small or that $\xi<2$.


\begin{theorem}\label{teorema-4}
 Assume {\rm (H1), (H2)--(H5), (H7), (H9)},
 $\xi<2$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$,
and $\theta_2\in [\xi,2)$. Then, there exists $\tilde{\mu}_0>0$ such
that system \eqref{Sist-Hamil} possesses a nontrivial weak solution
for each $\mu_1>0$ and $\mu_2\in(-\tilde{\mu}_0,0)$, provided that one
of the following conditions is satisfied
\begin{itemize}
\item[(i)] $p_0\in (2,\frac{2N}{N-2})$;

\item[(ii)] $p_0=2$ and
$K_0\in (0,\frac{\lambda_{1_{\beta_2}}}{2})$.

\end{itemize}
 Moreover, if $p_0\in(1,2)$ there exist $\tilde{\mu}_0,\bar{\mu}_0>0$
such that system \eqref{Sist-Hamil} possesses a nontrivial weak solution
for each $\mu_1\in (0,\bar{\mu}_0)$ and $\mu_2\in(-\tilde{\mu}_0,0)$.
 \end{theorem}

\begin{theorem}\label{teorema-5}
   In addition to {\rm (H1), (H2)--(H5), (H7), (H9)},
 $\xi<2$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$,
$\theta_2\in [\xi,2)$, and $H$ even in the variables $s,t$,
suppose either $H(x,s,0)\leq0$ for all $s\in \mathbb{R}$, $x\in \Omega$ or
$p_0=\tau$. Then, system \eqref{Sist-Hamil} possesses a sequence
$\{(u_n,v_n)\}$ of nontrivial weak solution with energies $I(u_n,v_n)\to\infty$
as $n\to\infty$ for each $\mu_1>0$ and $\mu_2<0$. See the
definition of $I$ in \eqref{Euler-I}.
 \end{theorem}

 \begin{remark} \label{rmk1.1} \rm
The Theorems \ref{teorema-1}--\ref{teorema-5} still  hold
for system \eqref{Sist-Hamil} with subcritical growth; that is,
when $\xi\in (1,2N/(N-2))$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$, and
we consider $\beta$ instead $2^*_ae_1=2^*_be_2$, where
$\beta<\min\{(a+1)p_3+N[1-(p_3/2)],\, (b+1)p_4+N[1-(p_4/2)]\}$
for some $p_3,p_4\in (1,\frac{2N}{N-2})$ with
 \begin{equation*}
   \frac{\alpha}{p_3}+\frac{\gamma}{p_4}=1.
 \end{equation*}
\end{remark}


         \section{Preliminaries}\label{Preliminaries}

  Consider $\Omega$ a bounded smooth domain in
 $\mathbb{R}^N$ ($N\geq 3$) with $0\in\Omega$. If $\alpha \in
\mathbb{R}$ and $l\in (0,+\infty)$, let $L^l(\Omega, |x|^{\alpha})$
be 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 $-\infty<a<(N-2)/2$, we define
$W_0^{1,2}(\Omega,|x|^{-2a})$ as being the completion of
$C_0^{\infty}(\Omega)$ with respect to the norm $\| \cdot \|_a$ defined by
 \begin{equation*}
   \|u\|_a=\|u\|_{W_0^{1,2}(\Omega,|x|^{-2a})}
   :=\Big(\int_{\Omega}|x|^{-2a}|\nabla u|^2 dx\Big)^{1/2},
 \end{equation*}
which is induced by inner product
 \begin{equation*}
 \begin{array}{c}
  \langle u,w\rangle_{W_0^{1,2}(\Omega,|x|^{-2a})}
  :={\int_{\Omega}|x|^{-2a}\nabla u \nabla w}\, dx.
 \end{array}
 \end{equation*}


First of all, by using  inequality \eqref{Caffarelli-Kohn-Nirenberg}
and the boundedness of $\Omega$, in \cite{Xuan1} was proved that there
exists $C>0$ such that
 \begin{equation}\label{CKN}
 \Big( \int_{\Omega}|x|^{-\delta}
   |u|^r dx\Big)^{2/r}\leq C \Big(
   {\int_{\Omega}|x|^{-2a} |\nabla u|^{2}dx}
   \Big), \quad \forall  u \in W_0^{1,2}(\Omega,|x|^{-2a}),
 \end{equation}
where $1\leq r \leq {2N}/({N-2})$ and
$\delta \leq (a+1)r +N\left[1-({r}/{2})\right]$, which is the
Caffarelli, Kohn, Nirenberg's inequality.
In other words, the embedding
$W_0^{1,2}(\Omega,|x|^{-2a}) \hookrightarrow
L^{r}(\Omega,|x|^{-\delta})$
is continuous if $1\leq r \leq 2N/(N-2)$ and
$\delta \leq (a+1)r+N [1-({r}/{2})]$.
Moreover, this embedding is compact if $1\leq r < 2N/(N-2)$ and
$\delta < (a+1)r+N [1-({r}/{2})]$, see \cite[Theorem 2.1]{Xuan1}.

 Due to Theorem \ref{base}, see Appendix, we can consider
$$
\{\frac{\varphi_{a,n}}{\sqrt{\lambda_{a,n}}}\} \subset
C^1(\overline{\Omega}\setminus\{0\})\cap C^0(\overline{\Omega})
$$
and
$$
\{\frac{\varphi_{b,n}}{\sqrt{\lambda_{b,n}}}\} \subset
C^1(\overline{\Omega}\setminus\{0\})\cap C^0(\overline{\Omega})
$$
the Hilbertian bases of spaces $W_0^{1,2}(\Omega,|x|^{-2a})$ and
$W_0^{1,2}(\Omega,|x|^{-2b})$, respectively.  We define
\[
    E:=W_0^{1,2}(\Omega,|x|^{-2a})\times W_0^{1,2}(\Omega,|x|^{-2b}),
\]
endowed with the norm $\|(u,v)\|:=\|u\|_a+\|v\|_b$.
We will denote $\varphi^a_{n}=(\varphi_{a,n},0)$, and
$\varphi^b_{n}=(0,\varphi_{b,n})$. Evidently, $\{\varphi^a_{n}\}$
(resp. $\{\varphi^b_{n}\}$) is a basis for space
$E^+:=W_0^{1,2}(\Omega,|x|^{-2a})\times \{0\}$
(resp. $E^-:=\{0\}\times W_0^{1,2}(\Omega,|x|^{-2b})$) and
$E=E^-\oplus E^+$.
We define the spaces
 \begin{equation*}
   X^m:=\operatorname{span}\{\varphi^a_{1},\dots ,\varphi^a_{m}\}\oplus E^-,
  \quad
   X_n:=E^+\oplus \operatorname{span}\{\varphi^b_{1},\dots ,\varphi^b_{n}\},
 \end{equation*}
and we denote by $(X^m)^{\bot}$ (resp. $(X_n)^{\bot}$) the complement
of $X^m$ (resp. $X_n$) in $E$.

 Our approach is variational, so we will study the critical
 points of the Euler-Lagrange functional
$I:E\to \mathbb{R}$ given by
 \begin{equation}\label{Euler-I}
 \begin{aligned}
   I(u,v) &= \frac{1}{2}(\|u\|^2_a-\|v\|^2_b)
   -\frac{\mu_1}{\tau} {\int_{\Omega}} |x|^{-\beta_0}|u|^{\tau} \,dx
   -\frac{\mu_2}{\xi} {\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi} \,dx
   \\
  &\quad -{\int_{\Omega}} |x|^{-\beta_2}H(x,u,v)\,dx
   -{\int_{\Omega}} |x|^{-2^*_a e_1}|u|^{\alpha}|v|^{\gamma}\, dx,
 \end{aligned}
 \end{equation}
which belongs to the class $C^1$.

  Now, we will proof that $I'$ is weakly sequentially continuous.

 \begin{theorem}\label{sol-fraca}
 Let $\{(u_j,v_j)\}\subset E$ be a sequence and $(u,v)\in E$
 such that $(u_j,v_j)\rightharpoonup (u,v)$ weakly in
$E$ as $j\to\infty$. Assume {\rm (H1), (H2)--(H5)}.
Then, $I'(u_j,v_j) \rightharpoonup I'(u,v)$ weakly in $E^*$
as $j\to \infty$.
 \end{theorem}

\begin{proof}
 By definition of weak convergence in $E$, we
have for $(w, z)\in E$ that
 \begin{gather}\label{lim1}
   {\lim_{j\to\infty}\int_{\Omega}}
   |x|^{-2a}\nabla u_{j}\nabla w \, dx =
   {\int_{\Omega}}  |x|^{-2a}\nabla u\nabla w \, dx,
\\
  {\lim_{j\to\infty}\int_{\Omega}}
   |x|^{-2b}\nabla v_{j}\nabla z \, dx =
   {\int_{\Omega}}   |x|^{-2b}\nabla v\nabla z\, dx.
 \end{gather}
 By compact embedding, we have
 \begin{gather*}
\text{$u_j\to u$  strongly in $L^{\tau}(\Omega,|x|^{-\beta_0})$
 and  $L^{p_1}(\Omega,|x|^{-\beta_2})$  as $j\to\infty$},
   \\
\text{$v_j\to v$ strongly in $L^{q_1}(\Omega,|x|^{-\beta_2})$
  as $j\to\infty$.}
 \end{gather*}
In particular, there exist functions  $h\in L^{\tau}(\Omega, |x|^{-\beta_0})$,
$f\in L^{p_1}(\Omega, |x|^{-\beta_2})$, and
$g\in L^{q_1}(\Omega, |x|^{-\beta_2})$ such that
$|u_j|(x)\leq \min\{f(x), h(x)\}$ and
$|v_j|(x)\leq g(x)$  almost everywhere $x\in \Omega$. Passing to
a subsequence, if necessary, we obtain $u_j(x) \to u(x)$ and
$v_j(x) \to v(x)$, as $j\to \infty$, for almost
everywhere $x\in \Omega$. Therefore, we obtain
 \begin{gather*}
 [H_u(x,u_j,v_j)w](x)\to [H_u(x,u,v)w](x)
   \quad \text{as } j\to \infty \text{ almost everywhere }
    x\in \Omega,
\\
   (|u_{j}|^{\tau-2}u_jw)(x)
   \to    (|u|^{\tau-2}uw)(x) \quad   \text{ as }
   j\to \infty    \text{ almost everywhere } x\in \Omega,
\\
\begin{aligned}
  |H_u(x,u_j,v_j)w| & \leq
   K_1(|u_j|^{p_1}+|v_j|^{q_1}+|w|^{r_1})   \\
   & \leq    K_1(f^{p_1}+g^{q_1}+|w|^{r_1})  \in L^{1}(\Omega,|x|^{-\beta_2}),
 \end{aligned}\\
\begin{aligned}
   \|u_{j}|^{\tau-2}u_jw|
   & \leq    h^{\tau-1}|w|   \\
   & \leq    \frac{\tau-1}{\tau} h^\tau
   +\frac{1}{\tau} |w|^{\tau}
   \in L^1(\Omega,|x|^{-\beta_0}).
 \end{aligned}
 \end{gather*}
Consequently, the Lebesgue Theorem implies that
  \begin{gather*}
    {\lim_{j\to \infty}}   {\int_{\Omega}}
    |x|^{-\beta_2}H_u(x,u_j,v_j)w\,dx
    =   {\int_{\Omega}}    |x|^{-\beta_2}H_u(x,u,v)w\,dx,\\
    {\lim_{j\to \infty}}
    {\int_{\Omega}}
    |x|^{-\beta_0}|u_{j}|^{\tau-2}u_jw\,dx
    =    {\int_{\Omega}}
    |x|^{-\beta_0}|u|^{\tau-2}uw    \,dx.
  \end{gather*}
 Analogously, we obtain
 \begin{equation}\label{lim2}
   {\lim_{j\to \infty}}
   {\int_{\Omega}}
   |x|^{-\beta_2}H_v(x,u_j,v_j)z\,dx
   =   {\int_{\Omega}}
   |x|^{-\beta_2}H_v(x,u,v)z\,dx.
 \end{equation}
 Due to weak convergence, $\{(u_j,v_j)\}$ is bounded
in $E$. Also, since that $(\alpha/2^*_a)+(\gamma/2^*_b)=1$, we obtain
\[
\frac{\alpha-1}{2^*_a-1}+\frac{2^*_a\, \gamma}{2^*_b(2^*_a-1)}
=   \frac{\gamma-1}{2^*_b-1}+\frac{2^*_b\, \alpha}{2^*_a(2^*_b-1)}=1,
 \quad
   \frac{2^*_a-1}{\alpha-1},\frac{2^*_b-1}{\gamma-1}>1\,.
\]
Then, by H\"older's inequality,
 \begin{align*}
&\big|{\int_{\Omega}}|x|^{-2^*_a e_1}
   (|u_{j}|^{\alpha-2} |v_{j}|^{\gamma}u_j
   )^{\frac{2^*_a}{2^*_a-1}}\,dx\big|\\
& \leq \Big(\|u_{j}\|_{L^{2^*_a}(\Omega,|x|^{-2^*_ae_1})}
   \Big)^{\frac{2^*_a(\alpha-1)}{2^*_a-1}}
  \Big(\|v_{j}\|_{L^{2^*_b}(\Omega,|x|^{-2^*_be_2})}
   \Big)^{{\frac{2^*_a\, \gamma }{2^*_a-1}}}.
 \end{align*}
Then $\{|u_{j}|^{\alpha-2} |v_{j}|^{\gamma}u_{j}\}$ is a bounded
sequence in $L^{\frac{2^*_a}{2^*_a-1}}(\Omega,|x|^{-2^*_a e_1})$. Also,
the sequence 
$\{|v_{j}|^{\xi-2}v_j\}$ is bounded in
$L^{\frac{\xi}{\xi-1}}(\Omega,|x|^{-\beta_1})$.
Moreover,   
\[
(|u_{j}|^{\alpha-2}|v_{j}|^{\gamma}u_j)(x)\to
(|u|^{\alpha-2} |v|^{\gamma}u)(x)\quad\text{and}\quad
(|v_{j}|^{\xi-2}v_j)(x)\to (|v|^{\xi-2}v)(x),
\]
 as $j\to\infty$,
for almost everywhere $x\in \Omega$. Then, by
\cite[Lemma 4.8]{Kavian}, we obtain
 \begin{gather*}
   |u_{j}|^{\alpha-2}|v_{j}|^{\gamma} u_j
   \rightharpoonup |u|^{\alpha-2}|v|^{\gamma}u
   \quad \text{weakly in }L^{\frac{2^*_a}{2^*_a-1}}(\Omega,|x|^{-2^*_a e_1})
   \text{ as } j\to \infty,
\\
|v_{j}|^{\xi-2} v_j \rightharpoonup |v|^{\xi-2}v
\quad \text{weakly in }
 L^{\frac{\xi}{\xi-1}}(\Omega,|x|^{-\beta_1})
 \text{ as } j\to \infty.
\end{gather*}
 In particular, we have
 \begin{gather}
   \lim_{n\to \infty}
   {\int_{\Omega}}|x|^{-2^*_a e_1}|u_{j}|^{\alpha-2}
   |v_{j}|^{\gamma}u_jw\,dx
   ={\int_{\Omega}}|x|^{-2^*_a e_1}|u|^{\alpha-2}
   |v|^{\gamma}uw\,dx,
\\
   {\lim_{j\to \infty}}    {\int_{\Omega}}
   |x|^{-\beta_1}|v_{j}|^{\xi-2}v_jz\,dx
   =    {\int_{\Omega}}
   |x|^{-\beta_1}|v|^{\xi-2}vz
   \,dx.
 \end{gather}
  Similarly, we obtain
 \begin{equation}\label{lim6}
   \lim_{j\to \infty}
   {\int_{\Omega}}
   |x|^{-2^*_a e_1}|u_{j}|^{\alpha}
   |v_{j}|^{\gamma-2}v_jz\,dx
   ={\int_{\Omega}}
   |x|^{-2^*_a e_1}|u|^{\alpha}
   |v|^{\gamma-2}vz\,dx.
  \end{equation}
By combining the limits \eqref{lim1}-\eqref{lim6}, we conclude that
\[
   {\lim_{j\to\infty}}
   \langle I'(u_j,v_j),(w,z)\rangle
   = \langle I'(u,v),(w,z)\rangle, \;\; \forall \, (w,z)\in E.
\]
\end{proof}

\begin{definition} \label{(PS)} \rm
 We say that $\{(u_j,v_j)\}\subset E$ is a
$(PS)_c^*$-sequence with relation to the functional $I$
if $(u_j,v_j)\in X_{n_j}$, $n_j\to \infty$ as $j\to \infty$,
$I(u_j,v_j)\to c$, and  $\|I'|_{X_{n_j}}(u_j,v_j)\|_{(X_{n_j})^*}\leq
\epsilon_{n_j}$, $\epsilon_{n_j}\to 0$ as $j\to \infty$. Moreover,
if all $(PS)_c^*$-sequence be precompact, we say that functional $I$
satisfies the
$(PS)_c^*$-condition.
 \end{definition}


 \begin{lemma}\label{limitacao}
 Assume {\rm (H1), (H2)--(H5)}.
  Then, all $(PS)_c^*$-sequence is bounded in $E$, if one of the
following conditions occurs:
\begin{itemize}
\item[(i)] $\mu_1> 0$, $\mu_2\geq 0$, and $(H7)$ are satisfied with
$\theta_2\in (1,2)\cap(1,\xi]$;


\item[(ii)] $\xi<2$, $\beta_1<(b+1)\xi[1-(\xi/2)]$,
$\mu_1>0$, $\mu_2<0$, and {\rm (H7)}
are satisfied with $\theta_2\in [\xi,2)$.
\end{itemize}
 \end{lemma}

\begin{proof} Let $\{(u_j,v_j)\}$ be a $(PS)_c^*$-sequence
with relation to the functional $I$.
  We consider $\theta_1\in(2,\tau]$ and
$\theta_2\in(1,2)\cap(1,\xi]$ if $(i)$ is satisfied and, for (ii),
we consider $\theta_1\in (2,\tau]$ and $\theta_2\in [\xi,2)$.
In both cases, we obtain
 \begin{align*}
  c+o(1)\|(u_{j},v_{j})\|+o(1)
   & \geq    I(u_{j},v_{j})
   -   \langle I'|_{ X_{n_j}}(u_{j},v_{j}),
   (\frac{1}{\theta_1}u_{j},\frac{1}{\theta_2}v_{j})
   \rangle
   \\
   & \geq    (\frac{1}{2}-\frac{1}{\theta_1})\|u_{j}\|_a^2
   +(\frac{1}{\theta_2}-\frac{1}{2})\|v_{j}\|_b^2,
 \end{align*}
so, $\{(u_{j},v_{j})\}$ is bounded in $E$.
\end{proof}

\begin{theorem}\label{sol-nao-trivial}
 Assume {\rm (H1), (H2)--(H5)}.
Let $\{(u_j,v_j)\}\subset E$ be a $(PS)_c^*$-sequence with relation
to the functional $I$ such that $(u_j,v_j)\rightharpoonup (u,v)$
weakly in $E$ as $j\to\infty$. Then, $(u,v)$ is a weak solution
of system \eqref{Sist-Hamil} and $(u_j,v_j)\to (u,v)$ strongly
in $E$ as $j\to\infty$, provided that one of the following conditions
is satisfied
\begin{itemize}
\item[(i)] $\mu_1> 0$
and $\mu_2\geq 0$;

\item[(ii)] $\xi<2$, $\beta_1<(b+1)\xi+[1-(\xi/2)]$,
$\mu_1>0$, and $\mu_2<0$.
\end{itemize}
 \end{theorem}

\begin{proof}
Due to weak convergence, $\{(u_j,v_j)\}$ is bounded in $E$.

\noindent{\bf Step I.} We will prove that $(u_j,v_j)\to (u,v)$
strongly in $E$ as $j\to\infty$.
 For each $z\in W_0^{1,2}(\Omega,|x|^{-2b})$, we can write
$z=\sum_{k=1}^{\infty}a_k\varphi_k^b$. Thus, we have the projection
$P^0_{n_j}:W_0^{1,2}(\Omega,|x|^{-2b}) \to
\operatorname{span}\{\varphi^b_1,\dots ,\varphi^b_{n_j}\}$ given by
$P^0_{n_j}(z) = \sum_{k=1}^{n_j}a_k\varphi_k^b$. Moreover,
it is easy to see that $P^0_{n_j}(z)\to z$ strongly in
$W_0^{1,2}(\Omega,|x|^{-2b})$ as $j\to\infty$.

 By definition of $(PS)_c^*$-sequence, we obtain
 \begin{equation}\label{pre-compacta-1}
 \begin{aligned}
&{\int_{\Omega}}|x|^{-2b}  \nabla v_{j}\nabla(v-v_{j})\, dx   \\
& =   \langle I'|_{X_{n_j}}(u_{j},v_{j}),
   (0,v_{j}-P^0_{n_j}(v))\rangle
   - \langle I'(u_{j},v_{j}),   (0,v -P^0_{n_j}(v))\rangle   \\
&\quad    +\mu_2   {\int_{\Omega}} |x|^{-\beta_1}|v_{j}|^{\xi-2}v_j(v_j-v) \,dx
   +{\int_{\Omega}}|x|^{-\beta_2}
   H_v(x,u_j,v_j)(v_j-v) \,dx
   \\
&\quad    +\gamma{\int_{\Omega}} |x|^{-2^*_a e_1}
   |u_{j}|^{\alpha}|v_{j}|^{\gamma-2}v_j(v_j-v) \,dx.
 \end{aligned}
 \end{equation}
 Since that $(0,v_{j}-P^0_{n_j}(v))\in X_{n_j}$ and
 $\{(0,v_{j}-P^0_{n_j}(v))\}$ is
bounded in $E$, we have
 \begin{equation}\label{pre-compacta-3.1}
  \langle I'|_{X_{n_j}}(u_{j},v_{j}),
   (0,v_{j}-P^0_{n_j}(v))\rangle
   \to  0 \quad \text{as }   j\to  \infty.
 \end{equation}
From $P^0_{n_j}(v)\to v$ strongly in $W_0^{1,2}(\Omega,|x|^{-2b})$
as $j\to \infty$ and boundedness of $\{(u_j,v_j)\}$ in $E$
follow that
 \begin{equation}\label{pre-compacta-3.2}
   \langle I'(u_{j},v_{j}),
   (0,v -P^0_{n_j}(v))\rangle\to  0
   \quad \text{as }   j\to  \infty.
 \end{equation}
 Similarly to proof of Theorem \ref{sol-fraca}, we obtain
  \begin{gather}\label{pre-compacta-3.178}
  {\lim_{j\to \infty}}
  {\int_{\Omega}} |x|^{-\beta_1}|v_{j}|^{\xi-2}v_jv \,dx
  ={\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi} \,dx,
\\
\label{pre-compacta-3.18}
   {\lim_{j\to \infty}}
   {\int_{\Omega}}|x|^{-\beta_2}
   H_v(x,u_j,v_j)(v_j-v)\,dx
   = 0,
\\
\label{pre-compacta-3.28}
{\lim_{j\to \infty}}
   {\int_{\Omega}}
   |x|^{-2^*_a e_1}|u_{j}|^{\alpha}|v_{j}|^{\gamma-2}v_jv\,dx
   =    {\int_{\Omega}}
   |x|^{-2^*_a e_1}|u|^{\alpha}|v|^{\gamma}\,dx.
 \end{gather}

  By compact embedding, $u_j(x)\to u(x)$ and $v_j(x)\to v(x)$,
as $j\to\infty$, for almost everywhere $x\in \Omega$. Then
$|x|^{-2^*_a e_1}|u_{j}|^{\alpha}(x)|v_{j}|^{\gamma}(x)\to
|x|^{-2^*_a e_1}|u|^{\alpha}(x)|v|^{\gamma}(x)$,
as $j\to\infty$, for almost everywhere $x\in \Omega$. Hence, we obtain
by Fatou's Lemma that
 \begin{equation}\label{pre-compacta-4.02}
   {\int_{\Omega}} |x|^{-2^*_a e_1}
   |u|^{\alpha}|v|^{\gamma}\, dx
   \leq    {\liminf_{j\to\infty}\int_{\Omega}}
   |x|^{-2^*_a e_1}|u_{j}|^{\alpha}|v_{j}|^{\gamma} \, dx.
 \end{equation}

 Hence, taking the lower limit in equation \eqref{pre-compacta-1} and
by using \eqref{pre-compacta-3.1}-\eqref{pre-compacta-4.02}, we obtain
 \begin{equation}\label{bip-1}
 \begin{aligned}
   \|v\|_b^2-{\limsup_{j\to\infty}}\,
   \|v_{j}\|_b^2
   & =   {\liminf_{j\to\infty}}
   {\int_{\Omega}}|x|^{-2a}
   \nabla v_{j}\nabla(v-v_{j})\, dx
   \\
   & \geq   {\liminf_{j\to\infty}} \Big(\mu_2
   {\int_{\Omega}} |x|^{-\beta_1}|v_{j}|^{\xi-2}v_j(v_j-v)
   \,dx\Big)
   \\
   & \geq     {\liminf_{j\to\infty}} \Big(\mu_2
   {\int_{\Omega}} |x|^{-\beta_1}|v_{j}|^{\xi}
   \,dx\Big)
   -\mu_2 {\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi} \,dx.
 \end{aligned}
 \end{equation}
  Consider $\mu_2\geq 0$. Then, since
$|x|^{-\beta_1}|v_{j}|^{\xi}(x)\to |x|^{-\beta_1}|v|^{\xi}(x)$
as $j\to\infty$ for almost everywhere $x\in \Omega$, we obtain by Fatou's
Lemma that
 \begin{equation*}
   {\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi}\, dx
   \leq
   {\liminf_{j\to\infty}\int_{\Omega}}
    |x|^{-\beta_1}|v_j|^{\xi}\, dx;
 \end{equation*}
therefore,  from \eqref{bip-1} we obtain
\[
   \|v\|_b^2-{\limsup_{j\to\infty}}\,
   \|v_{j}\|_b^2  \geq  0.
\]
But, if $\mu_2<0$, $\xi<2$, and $\beta_1<(b+1)\xi+[1-(\xi/2)]$,
then the embedding
$W_0^{1,2}(\Omega,|x|^{-2b})\hookrightarrow L^{\xi}(\Omega,|x|^{-\beta_1})$
is compact. Therefore,
\begin{equation*}
   {\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi}\, dx
   =
   {\lim_{j\to\infty}\int_{\Omega}}
    |x|^{-\beta_1}|v_j|^{\xi}\, dx,
 \end{equation*}
and  from \eqref{bip-1} it follows that
\[
   \|v\|_b^2-{\limsup_{j\to\infty}}\,
   \|v_{j}\|_b^2 \geq 0.
\]
Then, in both cases, we have
\[
   {\limsup_{j\to\infty}}\,
   \|v_{j}\|_b^2
   \leq    \|v\|_b^2
   \leq   {\liminf_{j\to\infty}}\,
   \|v_{j}\|_b^2,
\]
so, $v_j\to v$ strongly in $W_0^{1,2}(\Omega,|x|^{-2b})$
as $j\to\infty$.

  Define $\tilde{u}_j:=u_j-u$ and $\tilde{v}_j:=v_j-v$. From definition
of $(PS)_c^*$-sequence and by Brezis-Lieb's Lemma follow
 \begin{equation}\label{Tufu}
 \begin{aligned}
  & \|\tilde{u}_j\|_a^2-\alpha
   {\int_{\Omega}}|x|^{-2^*_a e_1}
   |\tilde{u}_{j}|^{\alpha}|\tilde{v}_{j}|^{\gamma}dx
   \\
&   =\langle I'|_{X_{n_j}}(u_j,v_j),(u_j,0)\rangle
   -\langle I'(u,v),(u,0)\rangle    + o(1),
 \end{aligned}
 \end{equation}
where $o(1)\to 0$ as $j\to\infty$.

 As $\{(u_j,0)\}$ is bounded in $E$,
$(u_j,0),(w,0)\in X_{n_j}:
=E^+\oplus \operatorname{span}\{\varphi^b_{1},\dots ,\varphi^b_{n_j}\}$ where
$E^+:=W_0^{1,2}(\Omega,|x|^{-2a})\times \{0\}$, we have by definition
of $(PS)_c^*$-sequence that
 $\langle I'|_{X_{n_j}}(u_j,v_j),(u_j,0)\rangle\to 0$
and $\langle I'|_{X_{n_j}}(u_j,v_j),(w,0)\rangle\to 0$
as $j\to \infty$ for all $w\in W_0^{1,2}(\Omega,|x|^{-2a})$.
On the other hand, by Theorem \ref{sol-fraca}, 
$\langle I'|_{X_{n_j}}(u_n,v_n),(w,0)\rangle
\to \langle I'(u,v),(w,0)\rangle$ as $j\to \infty$
for all $w\in W_0^{1,2}(\Omega,|x|^{-2a})$. Then,
 \begin{equation}\label{ml-1}
   \langle I'(u,v),(w,0)\rangle=0,\quad \forall
w\in W_0^{1,2}(\Omega,|x|^{-2a}).
 \end{equation}
Thus, we obtain by H\"older's inequality, Caffarelli, Kohn,
and Nirenberg's inequality, boundedness of $\{\tilde{u}_n\}$
in $W_0^{1,2}(\Omega,|x|^{-2a})$, and \eqref{Tufu} that
 \begin{equation}\label{Tufu-1}
   \|\tilde{u}_j\|_a^2
    =   \alpha {\int_{\Omega}}|x|^{-2^*_a e_1}
   |\tilde{u}_{j}|^{\alpha}|\tilde{v}_{j}|^{\gamma}dx    +o(1) \\
  \leq    M \|v\|^{\gamma}_b    +o(1),
 \end{equation}
so, as $\tilde{v}_j\to 0$ strongly in $E$ as $j\to\infty$, it
follows that $\tilde{u}_j\to 0$ strongly in $W_0^{1,2}(\Omega,|x|^{-2a})$
as $j\to \infty$. Hence, we conclude that $(u_j,v_j)\to (u,v)$
strongly in $E$ as $n\to \infty$.


\noindent{\bf Step II.} We will prove that $(u,v)$ is a weak solution of system
\eqref{Sist-Hamil}.
 Consider $z\in W_0^{1,2}(\Omega,|x|^{-2b})$. Then, we have
 \begin{equation}\label{pre-compacta-980}
   \langle I'(u_{j},v_{j}),
   (0,z)\rangle
   =   \langle I'|_{X_{n_j}}(u_{j},v_{j}),
   (0,P^0_{n_j}(z))\rangle
   + \langle I'(u_{j},v_{j}),
   (0,z-P^0_{n_j}(z))\rangle.
 \end{equation}
However, as $(0,P^0_{n_j}(z))\in X_{n_j}$ and $\{(0,P^0_{n_j}(z))\}$ is
bounded in $E$, we have
 \begin{equation}\label{pre-compacta-90}
   \langle I'|_{X_{n_j}}(u_{j},v_{j}),
   (0,P^0_{n_j}(z))\rangle
   \to  0 \text{\; as \;}
   j\to  \infty.
 \end{equation}
Also, follows similar to \eqref{pre-compacta-3.2} that
 \begin{equation}\label{pre-compacta-91}
   \langle I'(u_{j},v_{j}),(0,z -P^0_{n_j}(z))\rangle\to  0
   \quad\text{as }  j\to  \infty.
 \end{equation}
 Hence, by \eqref{pre-compacta-980}, \eqref{pre-compacta-90}, and \eqref{pre-compacta-91},
we obtain
\[
   \langle I'(u_{j},v_{j}),   (0,z)\rangle   \to  0
  \quad \text{as }   j\to  \infty.
\]
But, by Theorem \ref{sol-fraca}, we have
$\langle I'(u_j,v_j),(0,z)\rangle \to \langle I'(u,v),(0,z)\rangle$ as
$j\to \infty$ for all $z\in W_0^{1,2}(\Omega,|x|^{-2b})$. Then,
 \begin{equation}\label{ml-2}
   \langle I'(u,v),(0,z)\rangle=0,\quad \forall z\in W_0^{1,2}(\Omega,|x|^{-2b}).
 \end{equation}
Hence, by using \eqref{ml-1} and \eqref{ml-2}, we conclude that
$(u,v)$ is a weak solution of system \eqref{Sist-Hamil}.
\end{proof}

 \section{Proof of main results}


\begin{lemma}\label{link-1}
  Assume {\rm (H1), (H2)--(H5), (H7)},
$\mu_1>0$, and $\mu_2\in \mathbb{R}$.
 Then, there exist $r, \sigma>0$ such that
 \begin{equation}\label{bin-1}
   \inf I(\partial B_r(E^+))\geq \sigma,
 \end{equation}
provided that one of the following conditions is
satisfied
\begin{itemize}
 \item[(i)] $p_0\in (2,\frac{2N}{N-2})$;

 \item[(ii)] $p_0=2$ and
$K_0\in (0,\frac{\lambda_{1_{\beta_2}}}{2})$.
\end{itemize}
Moreover, if $p_0\in (1,2)$, there exist $\bar{\mu}_0, r, \sigma>0$
such that \eqref{bin-1} is held for each $\mu_1\in (0,\bar{\mu}_0)$
and $\mu_2\in \mathbb{R}$.
\end{lemma}
\begin{proof} 
If (i) is satisfied, we obtain
 \begin{align*}
I(u,0)   & \geq    \frac{1}{2}\|u\|_a^2
   -\frac{\mu_1}{\tau} C^{\frac{\tau}{2}}\|u\|_a^{\tau}
   -K_0{\int_{\Omega}} |x|^{-\beta_2} |u|^{p_0} \,dx\\
 & \geq    \frac{1}{2}\|u\|_a^2
   -\frac{\mu_1}{\tau} C^{\frac{\tau}{2}}\|u\|_a^{\tau}
   -K_0C^{\frac{p_0}{2}}\|u\|_a^{p_0},
 \end{align*}
so, as $\mu_1>0$ and $\tau,p_0>2$, there exist $r,\sigma\in(0,1)$ such that
$I(u,0)  \geq  \sigma$ for all $(u,0)\in E^+$ with $\|(u,0)\|=r$.

Assuming (ii), we obtain
 \begin{align*}
 I(u,0)
   & \geq    \frac{1}{2}\|u\|_a^2
   -\frac{\mu_1}{\tau} C^{\frac{\tau}{2}}\|u\|_a^{\tau}
   -K_0{\int_{\Omega}} |x|^{-\beta_2} |u|^{2} \,dx   \\
   & =    (\frac{1}{2}-\frac{K_0}{\lambda_{1_{\beta_2}}})\|u\|_a^{2}
   -\frac{\mu_1}{\tau} C^{\frac{\tau}{2}}\|u\|_a^{\tau},
 \end{align*}
so, as $\mu_1>0$, $K_0\in (0,\frac{\lambda_{1_{\beta_2}}}{2})$, and $\tau>2$,
there exist $r,\sigma\in(0,1)$ such that
$I(u,0) \geq \sigma$ for all $(u,0)\in E^+$ with $\|(u,0)\|=r$.

 Now, for $p_0\in (1,2)$, we have
  \begin{align*}
  I(u,0)
   & \geq    \frac{1}{2}\|u\|_a^2
   -\frac{\mu_1}{\tau} C^{\frac{\tau}{2}}\|u\|_a^{\tau}
   -K_0C^{\frac{p_0}{2}}\|u\|_a^{p_0}
   \\
   & =    (\frac{1}{4}\|u\|_a^2-K_0C^{\frac{p_0}{2}}\|u\|_a^{p_0})
   +(\frac{1}{4}\|u\|_a^2-\frac{\mu_1}{\tau}
   C^{\frac{\tau}{2}}\|u\|_a^{\tau}).
  \end{align*}
 Since $p_0\in(1,2)$, there exist $r,\sigma>0$ such that
 \begin{equation*}
 \begin{array}{c}
   (\frac{1}{4}r^2-K_0C^{\frac{p_0}{2}}r^{p_0})\geq \sigma.
 \end{array}
 \end{equation*}
 We choose $\bar{\mu}_0>0$ such that
\[
  (\frac{1}{4}r^2-\frac{\mu_1}{\tau}
   C^{\frac{\tau}{2}}r^{\tau}) \geq 0
\]
for all $\mu_1\in (0,\bar{\mu}_0)$.

 Hence, we conclude that $I(u,0) \geq  \sigma$ for all $(u,0)\in E^+$
with $\|(u,0)\|=r$, provided that $\mu_1\in (0,\bar{\mu}_0)$ and 
$\mu_2\in \mathbb{R}$.
\end{proof}

Consider $(e,0)\in E^+$ with $\|(e,0)\|=r$. We define the sets
 \begin{gather*}
   M = M(\rho):=
   \{(se,v): v\in W_0^{1,2}(\Omega,|x|^{-2b}), \|(se,v)\|\leq \rho\},\\
\begin{aligned}
   M_0 = M_0(\rho)  := 
   \{&(se,v): v\in W_0^{1,2}(\Omega,|x|^{-2b}), \|(se,v)\|= \rho
   \\
 &   \text{and } s>0 \text{ or } \|v\|_b \leq \rho \text{ and } s=0 \}.
 \end{aligned}
 \end{gather*}

\begin{lemma}\label{link-2}
 Assume {\rm (H1) and (H2)-(H7)}.
 Then, there exists $\rho>r>0$ such that
$I(u,v)\leq 0$ for all $(se,v)\in M_0$, for each $\mu_1>0$
and $\mu_2\in \mathbb{R}$.
 \end{lemma}

\begin{proof} 
If $(se,v)\in M_0$, then, by using (H6), we obtain
 \begin{equation}\label{upa-1}
   I(se,v) \leq \frac{r^2}{2}s^2
   -s^\tau \frac{\mu_1}{\tau} {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx -\frac{1}{2}\|v\|^2_b.
 \end{equation}
Fix $\rho_0>r>0$ such that
 \begin{equation}\label{upa-2}
   \frac{r^2}{2}s^2
   -s^\tau\; \frac{\mu_1}{\tau} {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx \leq 0, \forall  s\geq \rho_0,
 \end{equation}
and, define
 \begin{equation*}
   0\, <\, b_0:={\max_{s\geq 0}} (\frac{r^2}{2}s^2
   -s^\tau \frac{\mu_1}{\tau} {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx)
    < \infty.
  \end{equation*}
Then, we choose $\rho>\max\{\rho_0, \, r \rho_0\}>r$ such that
 \begin{equation}\label{upa-3}
   \frac{1}{2}\|v\|_b^2\geq b_0,\quad \text{for all $v$ with } 
\|v\|_b\geq \rho-\rho_0r.
 \end{equation}
Thus, if $s=0$ and $\|v\|_b\leq \rho$, follows by \eqref{upa-1}
that $I(0,v)\leq 0$.

  If $s>0$ and $\|(se,v)\|=\rho$, we have $\|v\|_b=\rho -s\|e\|_a=\rho -sr$.
Then, for $s\geq \rho_0$, we obtain by \eqref{upa-1} and \eqref{upa-2} that
$I(0,v)\leq 0$. However, if $s < \rho_0$, we have
 $\|v\|_b=\rho -sr\geq\rho -\rho_0r$,
so, by \eqref{upa-1} and \eqref{upa-3}, we obtain
$I(se,v)\leq 0$. Note that $\frac{1}{2}\|v\|_b^2\leq b_0$ and $s>0$ imply
$s \geq (\rho-\sqrt{2b_0})/r> \rho_0$.
\end{proof}

\begin{proof}[Proof of Theorems \ref{teorema-1} and \ref{teorema-2.2}]
 We have 
$$
X_n=E^+\oplus \operatorname{span}\{\varphi_1^b,\cdots,\varphi_n^b\}.
$$
We define
 \begin{gather*}
   M_n:=M\cap X_n,\quad M_{0,n}:=M_0\cap X_n,\quad
   N_n:= \partial B_r(E^+), \\
   c_n:=\inf_{h\in \Gamma_n}\max I(h(M_n)),
 \end{gather*}
where
 \begin{equation*}
   \Gamma_n:=\{h\in C(M_n,X_n): h|_{M_{0,n}}\equiv id_{M_{0,n}}\}.
 \end{equation*}
Similar to the proof of \cite[Theorem 2.12]{Willem}, we obtain
 \begin{equation*}
   h(M_n)\cap \partial B_r(E^+) \neq \emptyset, \quad \forall  h\in \Gamma_n.
 \end{equation*}
Then, by using Lemmata \ref{link-1}
and \ref{link-2}, we obtain
 \begin{equation*}
     \sup I(M_{0,n}) \leq 0<\sigma
     \leq \inf I(\partial B_rE^+)
     \leq c_n \leq k_0:=\sup I(M_n)<\infty.
 \end{equation*}
 In particular, we obtain a subsequence $\{c_{n_j}\}$ of $\{c_{n}\}$
and $c\in [\sigma,k_0]$ such that $c_{n_j}\to c$ as
$j\to \infty$.

 Then, by applying \cite[Theorem 2.8]{Willem}, we obtain
$(u_n,v_n)\in X_n$ with $|I(u_n,v_n)-c_n|\leq 1/n$ and
$\|I'|_{X_n}(u_n,v_n)\|_{({X_n})^*}\leq 1/n$ for each $n\in\mathbb{N}$.
Thus, $\{(u_{n_j},v_{n_j})\}$ is a $(PS)_c^*$-sequence with relation to
the functional $I$. Due to Lemma \ref{limitacao},
$\{(u_{n_j},v_{n_j})\}$ is bounded in $E$.
Therefore, there exists $(u,v)\in E$ such that $(u_{n_j},v_{n_j})
\rightharpoonup (u,v)$ weakly in $E$ as $j\to \infty$.
Hence, by Theorem \ref{sol-nao-trivial}, we conclude
that $(u,v)$ is a weak solution of system \eqref{Sist-Hamil} and
$(u_{n_j},v_{n_j}) \to (u,v)$ strongly in $E$ as $j\to \infty$.
In particular, $I(u,v)=c>0$, then $(u,v)$ is nontrivial.
\end{proof}



%\section{Proof of Theorem \ref{teorema-3}}

\begin{lemma}\label{lema-1-teo-2}
 Assume {\rm (H1), (H2)--(H6)}, $\mu_1>0$, and $\mu_2\in\mathbb{R}$.
 Then, there exists $R_m>0$ such that $I(u,v)\leq 0$
for all $(u,v)\in X^m$ with $\|(u,v)\|\geq R_m$.
 \end{lemma}

\begin{proof} 
We recall that $X^m \approx \operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}\times
W_0^{1,2}(\Omega,|x|^{-2b})$. Thus, as
$\operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}$ has 
finite dimension, all
norms in this space are equivalent. From Caffarelli, Kohn, and
Nirenberg's inequality 
$\|w\|_{L^{\tau}(\Omega,|x|^{-\beta_0})}\leq
C^{1/2}\|w\|_a$ for all $w\in W_0^{1,2}(\Omega,|x|^{-2a})$
and
$\|z\|_{L^{\xi}(\Omega,|x|^{-\beta_1})}\leq C^{1/2}\|z\|_b$ for all
$z\in W_0^{1,2}(\Omega,|x|^{-2b})$. In particular,
$\|\cdot\|_{L^{\tau}(\Omega,|x|^{-\beta_0})}$ define a norm on the
space $\operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}$. 
Then, there exists $K_m>0$ such that
 \begin{equation*}
   \|w\|_{L^{\tau}(\Omega,|x|^{-\beta_0})}\geq K_m \|w\|_a,
   \quad \forall  w\in \operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}.
 \end{equation*}
 Hence, we obtain
\[
   I(u,v)
    \leq 
   (\frac{1}{2}\|u\|^2_a-\frac{\mu_1}{\tau} K_m^\tau\|u\|_a^{\tau}
   ) -\frac{1}{2}\|v\|^2_b
 \leq    0, 
\]
for all $ (u,v)\in X^m$, $\|(u,v)\|\geq R_m$,
for some $R_m>0$ large enough, because $\tau>2$.
\end{proof}


\begin{lemma}\label{lema-2-teo-2}
 In addition to {\rm (H1), (H2)--(H5)},
$\mu_1>0$, and $\mu_2\in \mathbb{R}$, suppose either 
$H(x,s,0)\leq0$ for all $s\in \mathbb{R}, \, x\in \Omega$ or 
$p_0=\tau$. Then, there exist $r_m, a_m>0$
such that $a_m\to\infty$ as $m\to\infty$ and $I(u,v)\geq a_m$
for all $(u,v)\in (X^{m-1})^\bot$ with  $\|(u,v)\| = r_m$.
 \end{lemma}

\begin{proof} 
We have $(X^{m-1})^\bot\approx
\operatorname{span}\{\varphi_{a,j}:j\geq m\}\times \{0\} \approx
\operatorname{span}\{\varphi_{a,j}:j\geq m\}$. Thus, we can consider
$(X^{m-1})^\bot\subset W_0^{1,2}(\Omega,|x|^{-2a})$. Let
 \begin{gather*}
   \sigma_m:=\sup_{u\in (X^{m-1})^\bot , \, \|u\|_a=1}
   \|u\|_{L^{\tau}(\Omega,|x|^{-\beta_0})} , \\
 \rho_m:=\sup_{ u\in (X^{m-1})^\bot,\, \|u\|_a=1}
   \|u\|_{L^{p_0}(\Omega,|x|^{-\beta_2})}.
 \end{gather*}
 Since that $(X^{m})^\bot\subset (X^{m-1})^\bot$, it follows that
$\sigma_{m}\geq\sigma_{m+1}$ for all $m\in\mathbb{N}$. Thus,
$\sigma_m\searrow\sigma\geq 0$ as $m\to\infty$. We will prove that
$\sigma=0$. By definition of $\sigma_m$, for each $m\in \mathbb{N}$,
there exists $u_m\in (X^{m-1})^\bot$ with $\|u_m\|_a=1$ and
 \begin{equation*}
   \|u_m\|_{L^{\tau}(\Omega,|x|^{-\beta_0})}
   \geq \frac{\sigma_m}{2}\cdot
 \end{equation*}
 Moreover, as $(X^{m-1})^\bot\approx \operatorname{span}\{\varphi_{a,j}:j\geq m\}$,
we obtain $u_m\rightharpoonup 0$ weakly in $W_0^{1,2}(\Omega,|x|^{-2a})$
as $m\to\infty$. We have, from compact embedding $W_0^{1,2}(\Omega,|x|^{-2a})
\hookrightarrow L^{\tau}(\Omega,|x|^{-\beta_0})$, that $u_m\to 0$
strongly in $L^{\tau}(\Omega,|x|^{-\beta_0})$ as $m\to\infty$. Then,
$\sigma_m\searrow\sigma=0$ as $m\to\infty$. Similarly, we have
that $\rho_m \searrow 0$ as $m\to\infty$.

 If $H(x,s,0)\leq0$ for all $s\in \mathbb{R}, \, x\in \Omega$, then, for each
$(u,0)\in (X^{m-1})^\bot$, we obtain
\[
   I(u,0)    \geq    \frac{1}{2}\|u\|^2_a
   -\mu_1 (\sigma_m)^{\tau}\|u\|_a^{\tau},
\]
therefore, taking $r_m:=[\mu_1 (\sigma_m)^{\tau}]^{\frac{1}{2-\tau}}$
and $a_m:=(\frac{1}{2}-\frac{1}{\tau})r_m^2$, we conclude that
\[
   I(u,0) \geq  a_m,
\]
where  $a_m\to \infty$ as $m\to\infty$,
for all $(u,0)\in (X^{m-1})^\bot$ with $\|(u,0)\|=r_m$.

 However, if $p_0=\tau$, we define $l:=\max\{\mu_1/\tau,K_0\}$
and $\eta_m:=\max\{\sigma_m,\, \rho_m\}$. Then, for each $(u,0)\in (X^{m-1})^\bot$,
we obtain
 \begin{align*}
  I(u,0)
   & \geq    \frac{1}{2}\|u\|^2_a
   -\frac{\mu_1}{\tau} (\sigma_m)^{\tau}\|u\|_a^{\tau}
   - K_0 {\int_{\Omega}} |x|^{-\beta_2}|u|^{\tau} \,dx
   \\
   & \geq    \frac{1}{2}\|u\|^2_a
   -\frac{\mu_1}{\tau} (\sigma_m)^{\tau}\|u\|_a^{\tau}
   - K_0 (\rho_m)^{\tau}\|u\|_a^{\tau}
   \\
   & \geq    \frac{1}{2}\|u\|^2_a
   -2l(\eta_m)^{\tau}\|u\|_a^{\tau},
 \end{align*}
so, taking $r_m:=[2\, \tau\, l\, (\eta_m)^{\tau}]^{\frac{1}{2-\tau}}$
and $a_m:=(\frac{1}{2}-\frac{1}{\tau})r_m^2$, we conclude that
\[
   I(u,0) \geq  a_m,
\]
where  $a_m\to \infty$ as $m\to\infty$,
for all $(u,0)\in (X^{m-1})^\bot$ with $\|(u,0)\|=r_m$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{teorema-3}]
We remark that $I$ is an even functional in the variables $u$ and $v$.
By using Lemma \ref{lema-1-teo-2}, we obtain
 \begin{equation}\label{sp1}
   \sup_{X^m} I<\infty.
 \end{equation}
Then, by Lemmata \ref{limitacao}, \ref{lema-1-teo-2}, and
\ref{lema-2-teo-2}, Theorem \ref{sol-nao-trivial}, and by \eqref{sp1},
we have the hypotheses of \cite[Proposition 2.1]{Djairo-Ding},
which concludes Theorem \ref{teorema-3}.
\end{proof}


%\section{Proof of Theorems \ref{teorema-4} and \ref{teorema-5}}



 \begin{proof}[Proof of Theorem \ref{teorema-4}]
 The proof is similar to prove of Theorem \ref{teorema-2.2},
the difference is that we apply the next lemma instead of Lemma
\ref{link-2}.
\end{proof}

\begin{lemma}\label{LIMKIMG}
 Assume {\rm (H1), (H2)--(H5), (H7), (H9)},
 $\xi<2$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$,
and $\theta_2\in [\xi,2)$.
 Then, there exist $\tilde{\mu}_0>0$ and $\rho>r>0$ such that 
$\sup I(M_0)<\sigma$ for all $\mu_1>0$ and $\mu_2\in (-\tilde{\mu}_0,0)$, 
where $\sigma,r>0$ are coming from Lemma \ref{link-1}.
 \end{lemma}


 \begin{proof}
If $(se,v)\in M_0$, then
 \begin{equation}\label{upa-13}
\begin{aligned}
   I(se,v)
  &\leq    \frac{r^2}{2}s^2
   -s^\tau\; \frac{\mu_1}{\tau} {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx
   -\frac{1}{2}\|v\|_b^2 + \frac{|\mu_2|}{\xi}
   {\int_{\Omega}} |x|^{-\beta_1}|v|^{\xi} \,dx
   \\
 &   \leq   \Big(\frac{r^2}{2}s^2-s^\tau\; \frac{\mu_1}{\tau}
   {\int_{\Omega}} |x|^{-\beta_0}|e|^{\tau} \,dx\Big)
   +\Big(\frac{|\mu_2|}{\xi}C^{\frac{\xi}{2}}\|v\|_b^{\xi}
   -\frac{1}{2}\|v\|_b^2 \Big).
 \end{aligned}
 \end{equation}
It is easy verify that
\[
   t_{\mu_1}:=\Big(\frac{r^2}{\mu_1{\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx}\Big)^{\frac{1}{\tau-2}},\quad
   t_{\mu_2}:=\big(|\mu_2|C^{\frac{\xi}{2}}\big)^{\frac{1}{2-\xi}}
\]
are the respective maximum points of the functions $f:(0,\infty)\to \mathbb{R}$
and $g:(0,\infty)\to \mathbb{R}$ given by
\[
  f(s):=\frac{r^2}{2}s^2-s^\tau \frac{\mu_1}{\tau}
  {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx,\quad 
    g(t):= \frac{|\mu_2|}{\xi}C^{\frac{\xi}{2}} t^\xi - \frac{t^2}{2}\cdot
\]
Moreover, we have
 \begin{gather*}
 f(t_{\mu_1})=(\frac{1}{2}-\frac{1}{\tau})
   ( \mu_1 {\int_{\Omega}}
   |x|^{-\beta_0}|e|^{\tau} \,dx)^{\frac{-2}{\tau-2}}
   (r^2)^{\frac{\tau}{\tau-2}}, \\
 g(t_{\mu_2})=(\frac{1}{\xi}-\frac{1}{2})
   (|\mu_2| C^{\frac{\xi}{2}})^{\frac{2}{2-\xi}}.
 \end{gather*}
Let us fix $\tilde{\mu}_0>0$ and $\rho_0>r>0$ such that
  \begin{gather}\label{upa-0}
    g(t_{\mu_2})<\sigma \quad\text{if  }    0<|\mu_2|<\tilde{\mu}_0, \\
 \label{upa-23}
   f(s) \leq 0 \quad \text{for all } s\geq \rho_0.
 \end{gather}
Also, we choose $\rho>\max\{\rho_0, \, r \rho_0\}>r$ such that
 \begin{equation}\label{upa-33}
   g(\|v\|_b)+f(t_{\mu_1})\leq 0,\quad \forall  \|v\|_b\geq \rho-\rho_0r.
 \end{equation}

Thus, if $s=0$ and $\|v\|_b\leq \rho$, it follows by \eqref{upa-13} and 
\eqref{upa-0} that $I(0,v)\leq g(t_{\mu_2})<\sigma$ if $0<|\mu_2|<\tilde{\mu}_0$.

 If $s>0$ and $\|(se,v)\|=\rho$, we have $\|v\|_b=\rho -s\|e\|_a=\rho -sr$.
Then, for $s\geq \rho_0$, we obtain by \eqref{upa-13}, \eqref{upa-0}, and
\eqref{upa-23} that $I(se,v)\leq g(t_{\mu_2})<\sigma$  if $0<|\mu_2|<\tilde{\mu}_0$.
However, if $s < \rho_0$, we have $\|v\|_b=\rho -sr\geq\rho -\rho_0r$,
so, by \eqref{upa-13} and \eqref{upa-33}, we obtain $I(se,v)\leq f(t_{\mu_1})
+g(\|v\|_b)\leq 0<\sigma$. Note that $\|v\|_b\leq \rho-r\rho_0$ and $s>0$ imply
$s \geq \rho_0$.
\end{proof}


To prove Theorem \ref{teorema-5}, we will need of the following lemma.

 \begin{lemma}\label{lema-1-teo-44}
 Assume {\rm (H1), (H2)--(H5), (H7), (H9)},
 $\xi<2$, $\beta_1<(b+1)\xi+N[1-(\xi/2)]$,
$\theta_2\in [\xi,2)$, $\mu_1>0$, and $\mu_2<0$. Then, there exists
$R_m>0$ such that $I(u,v)\leq 0$ for all $(u,v)\in X^m$ with
$\|(u,v)\|\geq R_m$.
 \end{lemma}

 We remark that $I$ is an even functional in the variables $u$ and $v$.
By using Lemma \ref{lema-1-teo-44}, we obtain
 \begin{equation}\label{sp144}
   \sup_{X^m} I<\infty.
 \end{equation}
  Then, by Lemmata \ref{lema-1-teo-44} and
\ref{lema-2-teo-2}, Theorem \ref{sol-nao-trivial}, and \eqref{sp144},
we have the hypotheses of \cite[Proposition 2.1]{Djairo-Ding},
which concludes Theorem \ref{teorema-5}.


\begin{proof}[Proof of Lemma \ref{lema-1-teo-44}]
 We have
$X^m \approx \operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}\times
W_0^{1,2}(\Omega,|x|^{-2b})$. Following as in Lemma \ref{lema-1-teo-2},
we obtain that there exists
$K_m>0$ such that
 \begin{equation*}
   \|w\|_{L^{\tau}(\Omega,|x|^{-\beta_0})}\geq K_m \|w\|_a,
   \quad \forall  w\in \operatorname{span}\{\varphi_{a,1},\dots ,\varphi_{a,m}\}.
 \end{equation*}
 Hence, we obtain
 \begin{align*}
   I(u,v)
   & \leq  (\frac{1}{2}\|u\|^2_a-\frac{\mu_1}{\tau} K^\tau_m\|u\|_a^{\tau}
   )
   -\frac{1}{2}\|v\|^2_b + \frac{|\mu_2|}{\xi} C^{\frac{\xi}{2}}\|v\|_b^\xi
   \\
   & \leq    -(\mu_1 K^\tau\|u\|_a^{\tau-2}-\frac{1}{2})\|u\|^2_a
   -(\frac{1}{2}\|v\|_b^{2-\xi}-|\mu_2| C^{\frac{\xi}{2}})\|v\|_b^\xi
   \\
   & \leq    0, \quad \forall (u,v)\in X^m,\; \|(u,v)\|\geq R_m,
 \end{align*}
for some $R_m>0$ large enough, because $\tau>2>\xi$.
\end{proof}


\section{Appendix: Basis for
$L^2(\Omega,|x|^{-2a})$ and $W_0^{1,2}(\Omega,|x|^{-2a})$}

The next result was proved in  \cite{Iturriaga}.

 \begin{theorem}\label{Iturriaga} 
For each $f\in L^2(\Omega,|x|^{-2a})$, the problem
 \begin{gather*}
  -\operatorname{div}(|x|^{-2a}\nabla u) = |x|^{-2(a+1)+c} f \quad
\text{in } \Omega,   \\
   u=0 \quad\text{on }\partial \Omega,
 \end{gather*}
has an unique weak solution $u \in W_0^{1,2}(\Omega,|x|^{-2a})$ for
each $c>0$. Moreover, the operator
$T_c:L^2(\Omega,|x|^{-2a}) \to  L^2(\Omega,|x|^{-2a})$,
$T_cf=u$ is continuous and nondecreasing.
 \end{theorem}


\begin{lemma}\label{operador T} 
If $c=2$, then the operator $T:=T_2$
is a compact self-adjoint operator and $N(T)=\{0\}$.
 \end{lemma}

\begin{proof} 
Let $\{f_n\}\subset L^2(\Omega,|x|^{-2a})$ a
bounded sequence and $f \in L^2(\Omega,|x|^{-2a})$ such that
$f_n\rightharpoonup f$ weakly in $L^2(\Omega,|x|^{-2a})$ as
$n\to\infty$.
  By using definition of  $T$, we obtain
 \begin{align*}
  {\int_{\Omega}|x|^{-2a}|\nabla (Tf_n)|^2 \, dx}
   & =    {\int_{\Omega}|x|^{-2a} f_n (Tf_n) \, dx}   \\
   &\leq    M ({\int_{\Omega}|x|^{-2a}(Tf_n)^2\, dx})^{1/2}   \\
   &\leq    M  ({\int_{\Omega}|x|^{-2a}   |\nabla(Tf_n)|^2\, dx})^{1/2},
 \end{align*}
where $M$ is a positive constant; therefore
 \begin{equation*}
   \|Tf_n\|_{W_0^{1,2}(\Omega,|x|^{-2a})}
   \leq M,\quad \forall n\in \mathbb{N}.
 \end{equation*}
 Consequently, there exists $g\in W_0^{1,2}(\Omega,|x|^{-2a})$ such that
$Tf_n\rightharpoonup g$ weakly in $W_0^{1,2}(\Omega,|x|^{-2a})$ as
$n\to\infty$. Hence form the compact embedding
$W_0^{1,2}(\Omega,|x|^{-2a}) \hookrightarrow
L^{2}(\Omega,|x|^{-2a})$ we conclude that $Tf_n\to g$ strongly in
$L^2(\Omega,|x|^{-2a})$ as $n\to\infty$, in other words, $T$
is compact.

  Now, we prove that $T$ is self-adjoint.
  Let $f,g\in L^2(\Omega,|x|^{-2a})$. Then, we have
\[
   {\int_{\Omega}|x|^{-2a} g (Tf)  \, dx}
   = {\int_{\Omega}|x|^{-2a}\nabla (Tf)\nabla (Tg) \, dx}
  =  {\int_{\Omega}|x|^{-2a} f\, (Tg) \, dx};
\]
that is,
 \begin{equation*}
  \langle Tf, g\rangle_{L^2(\Omega,|x|^{-2a})}
  = \langle f,Tg\rangle_{L^2(\Omega,|x|^{-2a})},
 \end{equation*}
so, $T$ is self-adjoint.

Let $f\in N(T)$. We have $Tf = 0$ almost everywhere $x\in\Omega$, then
 \begin{equation*}
   0={\int_{\Omega}|x|^{-2a}\nabla (Tf)\nabla w \, dx}
   = {\int_{\Omega}|x|^{-2a} f w \, dx},
   \; \forall  w\in W_0^{1,2}(\Omega,|x|^{-2a}).
 \end{equation*}
 Then, we obtain $f\equiv 0$.
\end{proof}


 \begin{theorem}\label{base}
 The normalized eigenfunctions $\{\varphi_{a,n}\}\subset
C^1(\overline{\Omega}\setminus\{0\})\cap C^0(\overline{\Omega})$ of
eigenvalue problem
 \begin{equation}\label{autovalor-problema}
 \begin{gathered}
   -\operatorname{div}(|x|^{-2a}\nabla u)
 = \lambda |x|^{-2a} u \quad\text{in } \Omega,
   \\
   u  =  0 \quad \text{on } \partial \Omega,
 \end{gathered}
 \end{equation}
is a Hilbertian basis of space $L^2(\Omega,|x|^{-2a})$. Moreover, if
$\{\lambda_{a,n}\}$ are the respective eigenvalues of
$\{\varphi_{a,n}\}$, the sequence
$\{\frac{\varphi_{a,n}}{\sqrt{\lambda_{a,n}}}\} \subset
C^1(\overline{\Omega}\setminus\{0\})\cap C^0(\overline{\Omega})$ is
a Hilbertian basis for space $W_0^{1,2}(\Omega,|x|^{-2a})$.
 \end{theorem}

\begin{proof}
 First of all, we recall that Xuan 
\cite{Xuan-eigenvalue} proved that the eigenvalue problem
\eqref{autovalor-problema} has a sequence of eigenfunctions
$\{\varphi_{a,n}\}$ associated to eigenvalues $\{\lambda_{a,n}\}$
with $0<\lambda_{a,1}<\lambda_{a,2}\leq \lambda_{a,3}\leq
\dots \nearrow +\infty$. Changing $\{\varphi_{a,n}\}$ by
$\varphi_{a,n}/\|\varphi_{a,n}\|_{L^2(\Omega,|x|^{-2a})}$, if
necessary, we can consider
$\|\varphi_{a,n}\|_{L^2(\Omega,|x|^{-2a})}=1$. Moreover,
\[
\{\varphi_{a,n}\}\subset L^{\infty}(\Omega,|x|^{-2a}) \cap
C^1(\Omega\setminus\{0\})
\]
 and $\varphi_1>0$ in $\Omega\setminus\{0\}$.
 Then, by \cite[Theorem 2.1]{Iturriaga}, we obtain
$\{\varphi_{a,n}\}\subset C^0(\overline{\Omega})$. Thus, by applying
the \cite[Theorem 1]{Lieberman} follows that
$\{\varphi_{a,n}\}\subset C^1(\overline{\Omega}\setminus\{0\})$.
Also, by strong maximum principle, see 
\cite[Theorem 2.1]{Rodrigues2}, we obtain $\varphi_{a,1}>0$ in $\Omega$.

 By the definition of $T$, we have
 \begin{align*}
  { \int_{\Omega}}|x|^{-2a}\nabla (T\varphi_{a,n})\nabla w \, dx
  & =   { \int_{\Omega}}|x|^{-2a}\varphi_{a,n} w \, dx  \\
  & =   \lambda_{a,n}^{-1}{ \int_{\Omega}}|x|^{-2a}
  \nabla \varphi_{a,n} \nabla w \, dx  \\
  & =   { \int_{\Omega}}|x|^{-2a}\nabla (\lambda_{a,n}^{-1}
  \varphi_{a,n}) \nabla w \, dx
 \end{align*}
for all $w\in W_0^{1,2}(\Omega,|x|^{-2a})$. Hence, we conclude
 \begin{equation*}
   T \varphi_n =\lambda_{a,n}^{-1}\varphi_{a,n}, \quad
 \forall  n\in \mathbb{N};
 \end{equation*}
that is, $\{\lambda_{a,n}^{-1}\}$ and $\{\varphi_{a,n}\}$ are the
eigenvalues and eigenfunctions of $T$, respectively. But, by Lemma
\ref{operador T} and \cite[Theorem V.I.11]{Brezis-Analise}, 
we obtain that the eigenfunctions of $T$ is a Hilbertian basis for space
$L^2(\Omega,|x|^{-2a})$.

To prove the second claim, we remark that
$\{\frac{\varphi_{a,n}}{\sqrt{\lambda_{a,n}}}\}\subset
W_0^{1,2}(\Omega,|x|^{-2a})$. Moreover, from
\eqref{autovalor-problema}, we obtain that
$\{\frac{\varphi_{a,n}}{\sqrt{\lambda_{a,n}}}\}$ is an orthonormal
set with respect to inner product of $W_0^{1,2}(\Omega,|x|^{-2a})$.

 Now, we  prove that the space spanned by $\{\varphi_{a,n}\}$
is dense in $W_0^{1,2}(\Omega,|x|^{-2a})$. Indeed, if
 $u\in W_0^{1,2}(\Omega,|x|^{-2a})$ is such that $\langle
u,\varphi_{a,n}\rangle_{W_0^{1,2}(\Omega,|x|^{-2a})}=0$ for all
$n\in \mathbb{N}$. Then, by \eqref{autovalor-problema}, we obtain
 \begin{equation*}
   {\int_{\Omega}}|x|^{-2a} u \varphi_{a,n} \, dx=0,
   \quad \forall  n\in \mathbb{N}.
 \end{equation*}
Then, since that $\{\varphi_{a,n}\}$ is a Hilbertian basis of
$L^2(\Omega,|x|^{-2a})$, we conclude that $u=0$ for almost everywhere in
$\Omega$. Hence, by \cite[Corollary I.8]{Brezis-Analise} follows
that $\{\varphi_{a,n}\}$ is dense in $W_0^{1,2}(\Omega,|x|^{-2a})$.
\end{proof}

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\end{document}