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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 113, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/113\hfil Existence of a nontrivial solution]
{Existence of  solutions for Kirchhoff equations
involving $p$-linear and $p$-superlinear therms and with critical growth}

\author[M. B. Guimar\~aes, R. D. S. Rodrigues \hfil EJDE-2016/113\hfilneg]
{Mateus Balbino Guimar\~aes, Rodrigo da Silva Rodrigues}

\address{Mateus Balbino Guimar\~aes \newline
Departamento de Matem\'atica,
Universidade Federal de S\~ao Carlos
13565-905, S\~ao Carlos, SP, Brazil}
\email{mateusbalbino@yahoo.com.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{rodrigo@dm.ufscar.br}

\thanks{Submitted October 7, 2015. Published May 3, 2016.}
\subjclass[2010]{35A15, 35B33, 35B25, 35J60}
\keywords{Variational methods; critical exponents; singular perturbations;
\hfill\break\indent Kirchhoff equation; nonlinear elliptic equations}

\begin{abstract}
 In this article we establish the existence of a nontrivial weak solution
 to a class of nonlinear boundary-value problems of Kirchhoff type involving
 $p$-linear and $p$-superlinear terms and with critical
 Caffaearelli-Kohn-Nirenberg exponent.
\end{abstract}

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

\section{Introduction}

In this article we study the existence of nontrivial solutions for the
nonlocal boundary-value problem of Kirchhoff type
\begin{equation}\label{problema1}
\begin{gathered}
 L(u) = \lambda |x|^{-\delta}f(x,u)+|x|^{-bp^{\ast}}|u|^{p^{\ast}-2}u \quad
 \text{in }  \Omega,\\
 u = 0 \quad \text{on } \partial\Omega,
 \end{gathered}
\end{equation}
where
$$
L(u) := -\Big[M\Big(\int_{\Omega}|x|^{-ap} |\nabla u|^p \,dx\
\Big)\Big]\operatorname{div}\big(|x|^{-ap}|\nabla u|^{p-2}\nabla u\big),
$$
and $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain with $N\geq 3$,
$1<p<N$, $a\leq \frac{N-p}{p}$, $p^*=\frac{Np}{N-dp}$ is the critical
 Caffarelli-Kohn-Nirenberg exponent,
where $d=1+a-b$ with $a\leq b\leq a+1$,
 $M:\mathbb{R}^{+}\cup \{0\}\to\mathbb{R}^+$ is a continuous function,
  and $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function.

Because of the integral over $\Omega$ in $L(u)$,
\eqref{problema1} is no longer a pointwise equation, so
it is called nonlocal problem.
The mathematical difficulties that comes with this phenomenon is what makes
the study of such  problems particularly interesting.
Also the physical motivation makes this problem interesting. 
Indeed,  \eqref{problema1} is related to the stationary version
 of the Kirchhoff equation
\begin{equation}\label{problema ref3}
\begin{gathered}
 u_{tt}-M\Big(\int_{\Omega}|\nabla u|^{2} dx \Big)\Delta u
 = g(x,u) \quad \text{in } \Omega \times (0,T)\\
u=0 \quad \text{on } \partial\Omega \times (0,T)\\
u(x,0)=u_0(x), \quad  u_{t}(x,0)=u_1(x),
\end{gathered}
\end{equation}
where $M(s) = a + bs$, $a,b>0$. It was proposed by Kirchhoff \cite{kirchhoff}
as an extension of the classical D'Alembert's wave equation for free
vibrations of elastic strings to describe the transversal oscillations
 of a stretched string, particularly, taking into account the subsequent
change in string length caused by oscillations.

Some early classical studies of Kirchhoff equations were done by 
Bernstein \cite{bernstein} and Pohozaev \cite{pohozaev}. 
However,  \eqref{problema ref3} received
great attention only after Lions \cite{lions} proposed an abstract framework
for the problem. After that, the study on nonlocal problems of the
type \eqref{problema ref3} grew exponentially. Some interesting results can be found,
for example, in
\cite{alvescorreama,anelo, correagiovany,correagiovany2,dai,Servadei, HeZou,liu,
MaRivera,Binlin,Radulescu,Pizzimenti,pererazhang}, and the references therein.

Problems involving a Kirchhoff equation with critical growth can be seen,
for example, in \cite{ACG,giovany,GJ}. In \cite{chungquoc}, the authors
studied a problem involving the $p$-Laplacian operator with weights,
but with subcritical growth. A version of a Kirchhoff type problem involving
the $p$-Laplacian operator with weights and critical growth was
studied in \cite{GMR}.

In our work we intent to complement the results obtained in \cite{GMR}.
There the authors studied problem \eqref{problema1} involving $p$-sublinear
 and $p$-superlinear therms. We treat the case in which 
\eqref{problema1} has a $p$-linear therm. Also, we extend the results for
the $p$-superlinar case by finding a weak solution for each $\lambda >0$.
We  use  the mountain pass theorem to find weak solutions for the problem.
Different from the techniques in \cite{GMR} and the other articles listed above,
we work with extremal functions to control the level of the Palais-Smale sequence
 obtained with the mountain pass theorem.
The lack of compactness due to the critical therm in the first equation
of \eqref{problema1} was bypassed using a technique in common with some
of the above papers: a version of the concentration-compactness principle
due to Lions \cite{Lio2}.

Because of the nonlocal terms in the equation \eqref{problema1},
it was necessary to make a truncation on the Kirchhoff type function that
appear on the operator, creating an auxiliary problem. By finding solutions
of the auxiliary problem we can find solutions for  \eqref{problema1}.
This truncation argument is similar to the one used in \cite{giovany}.

For enunciating the main result, we need to give some hypotheses on the
continuous function $M:\mathbb{R}^{+}\cup \{0\}\to \mathbb{R}^{+}$, and on the
 Caratheodory function $f:\Omega\times\mathbb{R}\to\mathbb{R}$:
\begin{itemize}

\item[(H1)] There exists $m_0>0$ such that
$ M(t)\geq  m_0$   for all $t\geq 0$.

\item[(H2)] The function $M$ is increasing.

\item[(H3)] $ f(x,-t)=-f(x,t)$ for all $ (x,t)\in \Omega\times\mathbb{R}$.

\item[(H4)] There exist $r \in [p,p^{\ast})$ and $C_1,C_2$ positive
constants with $C_1 < C_2$, such that
$$
C_1 |t|^{r-1}\leq f(x,t)\leq C_2|t|^{r-1}, \quad \forall
    (x,t)\in \Omega\times(\mathbb{R}^+\cup \{0\}).
$$
Moreover,  $\delta\leq (a+1)r+N(1-\frac{r}{p})$.

\item[(H5)] The well known Ambrosetti-Rabinowitz superlinear condition holds,
$$
0 < \xi\int_0^{t}f(x,s)ds \leq tf(x,t), \quad \forall (x,t) \in
\Omega\times \mathbb{R}^+, \text{ and some } \xi \in (p,p^{\ast}).
$$
\end{itemize}

We denote by $\lambda_1$ the first eigenvalue of the  problem
\begin{equation} \label{problema autovalor}
\begin{aligned}
-\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)
&= \lambda \int_{\Omega}|x|^{-\delta}|u|^{p-2}u dx \quad &\text{in }\Omega,\\
u&=0 \quad \text{on } \partial \Omega,
\end{aligned}
\end{equation}
Note that the first eigenvalue of \eqref{problema autovalor} is given by
\begin{equation}\label{primeiro autovalor}
\lambda_1 =  \inf\Big\{\int_{\Omega}|x|^{-ap}|\nabla u|^p dx
; u \in \mathcal{D}_{a}^{1,p}, \int_{\Omega}|x|^{-\delta}|u|^pdx = 1 \Big\},
\end{equation}
and it is positive (see for instance \cite{Xuan1}).
The main results of our paper are read as follows.

\begin{theorem}\label{thm1}
  Assume {\rm (H1)--(H5)} hold, and $r = p$. Then \eqref{problema1}
 has a nontrivial solution for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$.
\end{theorem}

\begin{theorem}\label{thm2}
Assume {\rm (H1)--(H5)} hold, and  and $p < r < p^{\ast}$. Then
 \eqref{problema1} has a nontrivial solution for each $\lambda >0$.
\end{theorem}

This article is organized as follows. In section \ref{preliminares}
we provide some preliminary results and the variational framework.
In section \ref{problemaauxiliar} we constructed the auxiliary problem.
Section \ref{condicaops} is devoted to the Palais-Smale condition for the
Euler-Lagrange functional associated to problem \eqref{problema1}.
In sections \ref{provateo1} and \ref{provateo2} we prove
Theorems \ref{thm1} and \ref{thm2}, respectively.

\section{Preliminary results and variational framework}\label{preliminares}

Consider $\Omega\subset\mathbb{R}^N$ a
smooth domain with $0\in \Omega$, $N\geq 3$, $1<p<N$, $a<(N-p)/p$, $a\leq b< a+1$,
and $p^* =Np/(N-dp)$, where $d=1+a-b$. From \cite{Caffarelli, Xuan} we have
 \begin{equation}\label{inequality}
   \Big({\int_{\Omega}|x|^{-\alpha} |u|^{r}dx}\Big)^{p/r}
   \leq C    {\int_{\Omega}|x|^{-ap} |\nabla u|^pdx},\quad
   \forall  u\in \mathcal{D}^{1,p}_a,
 \end{equation}
where $1\leq r\leq Np/(N-p)$, $\alpha\leq (a+1)r+N(1-\frac{r}{p})$,
 $\mathcal{D}^{1,p}_a$ is the completion of $C_0^\infty({\Omega})$ with
respect to the norm
 $$
\|u\|=\Big({\int_{\Omega}|x|^{-ap} |\nabla u|^pdx}\Big)^{1/p};
$$
thus we have the continuous embedding of $\mathcal{D}^{1,p}_a$ in
the weighted space $L^r(\Omega,|x|^{-\alpha})$.
This space is $L^r(\Omega)$ with the norm
 $$
\|u\|_{r,\alpha}=\Big({\int_{\Omega}|x|^{-\alpha} |u|^{r}dx}\Big)^{1/r}.
$$
 Moreover, this embedding is compact if $1\leq r< Np/(N-p)$ and
$\alpha < (a+1)r+N(1-\frac{r}{p})$.
 The best constant of the weighted Caffarelli-Kohn-Nirenberg type
(see \cite{Caffarelli}) inequality will be
denoted by $C_{a,p}^*$, which is characterized by
\begin{equation*}
 C_{a,p}^*
 =\inf_{u\in \mathcal{D}_a^{1,p}\setminus\{0\}}
\frac{\int_\Omega |x|^{-ap}|\nabla u|^pdx}
    {\big(\int_\Omega |x|^{-bp^{\ast}}|u|^{p^*}dx\big)^{p/p^*}}\,.
 \end{equation*}

We will look for solutions of  \eqref{problema1}  by finding critical
points of the Euler-Lagrange functional
$I:\mathcal{D}_a^{1,p}\to \mathbb{R}$ given by
$$
I(u) = \frac{1}{p}\widehat{M}(\|u\|^p)
- \lambda {\int_\Omega}|x|^{-\delta}F(x,u)\,dx -
\frac{1}{p^{\ast}}{\int_{\Omega}}|x|^{-bp^{\ast}}|u|^{p^{\ast}}\,dx,
$$
where $\widehat{M}(t):=\int_0^t M(s)ds$ and $F(x,t) = \int_0^{t}f(x,s)ds$.
Note that $I \in C^1$ and
\begin{align*}
I'(u)(\phi)
& =  M(\|u\|^p){\int_{\Omega}}|x|^{-ap}|\nabla u|^{p-2}\nabla u \nabla \phi \,dx\\
& \quad -  \lambda{\int_{\Omega}}|x|^{-\delta}f(x,u)\phi \,dx
               - {\int_{\Omega}}|x|^{-bp^{\ast}}|u|^{p^{\ast}-2}u\phi \,dx,
\end{align*}
for all $\phi \in \mathcal{D}_a^{1,p}$.

The next Lemma will be useful, and can be easily proved by
using \cite[Lemma 4.1]{GY}.

\begin{lemma}[$S_{+}$ condition] \label{S_{+}}
Suppose that $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain,
$0 \in \Omega$, $1<p<N$, $-\infty < a < \frac{N-p}{p}$, and
$(u_n) \subset \mathcal{D}_{a}^{1,p}$ such that
\begin{gather*}
u_n \rightharpoonup u, \quad \text{as } n \to \infty,\\
\limsup_{n \to \infty} \int_{\Omega}|x|^{-ap}|\nabla u_{n}|^{p-2}
\nabla u_{n}\nabla(u_{n}-u)dx \leq 0,
\end{gather*}
then there exists a subsequence strongly convergent in
$\mathcal{D}_{a}^{1,p}$.
\end{lemma}

\section{Auxiliary problem}\label{problemaauxiliar}

To proof Theorems \ref{thm1} and \ref{thm2}, we will  use a version
 of the mountain pass theorem due to Ambrosetti and Rabinowitz \cite{Ambrosetti},
but since we are working with critical growth and a nonlocal operator without
more information about the behavior of the function $M$ at infinity,
we need to make a truncation on function $M$. So we will prove that
the Euler-Lagrange functional associated to  \eqref{problema1} has the Mountain
Pass Geometry.

From (H2), there exists $t_0 > 0$ such that $m_0 = M(0) < M(t_0) < \frac{\xi}{p}m_0$,
where $\xi$ is given by (H5). We set
\begin{equation*}
M_0(t):= \begin{cases}
M(t), &\text{if } 0 \leq t \leq t_0,\\
M(t_0), &\text{if } t \geq t_0.
\end{cases}
\end{equation*}
From (H2) we obtain
\begin{equation}\label{trunc 2}
m_0 \leq M_0(t) \leq \frac{\xi}{p}m_0, \quad \forall t \geq 0.
\end{equation}


The proofs of the Theorems \ref{thm1} and \ref{thm2} are based
on a careful study of solutions of the  auxiliary problem
\begin{equation}\label{problema auxiliar}
\begin{gathered}
 L_0(u) = \lambda |x|^{-\delta}f(x,u)+|x|^{-bp^{\ast}}|u|^{p^{\ast}-2}u \quad
\text{in }  \Omega, \\
 u = 0 \quad \text{on } \partial\Omega,
 \end{gathered}
\end{equation}
where
$$
L_0(u) := -\Big[M_0\Big(\int_{\Omega}|x|^{-ap} |\nabla u|^p \,dx\
\Big)\Big]\operatorname{div}\big(|x|^{-ap}|\nabla u|^{p-2}\nabla u\big).
$$

We will look for solutions of  \eqref{problema auxiliar} by finding critical
points of the Euler-Lagrange functional $J: \mathcal{D}_a^{1,p}\to \mathbb{R}$
given by
$$
J(u) = \frac{1}{p}\widehat{M_0}(\|u\|^p)
- \lambda {\int_\Omega}|x|^{-\delta}F(x,u)\,dx -
\frac{1}{p^{\ast}}{\int_{\Omega}}|x|^{-bp^{\ast}}|u|^{p^{\ast}} dx,
$$
where $\widehat{M_0}(t):=\int_0^t M_0(s)ds$. Note that $J$ is $C^1$ and
\begin{align*}
J'(u)(\phi)
&=  M_0(\|u\|^p){\int_{\Omega}}|x|^{-ap}|\nabla u|^{p-2}\nabla u \nabla \phi \,dx\\
&\quad  -  \lambda{\int_{\Omega}}|x|^{-\delta}f(x,u)\phi \,dx
        - {\int_{\Omega}}|x|^{-bp^{\ast}}|u|^{p^{\ast}-2}u\phi \,dx,
\end{align*}
for all $\phi \in \mathcal{D}_a^{1,p}$.

\section{Palais-Smale Condition}\label{condicaops}

In this section we verify that, under the hypotheses
(H1)--(H4), the functional $J$ satisfies the Palais-Smale condition below
a given level.

\begin{lemma}\label{nivelabaixo}
Let $(u_{n})$ be a bounded sequence in $\mathcal{D}_{a}^{1,p}$ such that
$$
J(u_n) \to c \quad\text{and}\quad
J'(u_n) \to 0 \text{ in  } (\mathcal{D}_{a}^{1,p})^{-1}, \quad
\text{as } n\to \infty.
$$
Suppose {\rm (H1)--(H5)} hold, and
\begin{equation*}
c < \Big( \frac{1}{\xi}-\frac{1}{p^{\ast}} \Big)
\big( m_0C_{a,p}^* \big)^{\frac{p^{\ast}}{p^{\ast}-p}}.
\end{equation*}
Then there exists a subsequence strongly convergent in $\mathcal{D}_{a}^{1,p}$.
\end{lemma}

\begin{proof}
Since $(u_n)$ is bounded in $\mathcal{D}_{a}^{1,p}$,
passing to a subsequence, if necessary, we have
\begin{gather*}
 u_{n}\rightharpoonup u \quad \text{in }  \mathcal{D}_a^{1,p}, \\
u_{n}\to u \quad \text{in } \ L^{s}(\Omega,|x|^{-\sigma}), \\
u_{n}(x)\to u(x) \quad \text{ a.e. in } \Omega, \\
\|u_{n}\|\to t_0 \geq 0,
\end{gather*}
as $n\to +\infty$, where $1\leq s < p^{\ast}$ and $\sigma < (a+1)s + N(1-s/p)$.
Moreover, using the concentration-compactness principle due to Lions (cf.
\cite{Lio2, Xuan}), we obtain at most countable index set
$\Lambda$, sequences $(x_i) \subset \mathbb{R}^N$, $(\mu_i),
(\nu_i) \subset (0,\infty)$, such that
\begin{equation}\label{convergencia fraca medida}
|x|^{-ap}|\nabla u_n|^p \rightharpoonup |x|^{-ap}|\nabla u|^p + \mu
\quad\text{and}\quad
 |x|^{-bp^{\ast}}|u_n|^{p^*} \rightharpoonup |x|^{-bp^{\ast}}|u|^{p^*}+ \nu,
\end{equation}
as $n\to +\infty$, in weak$^*$-sense of measures where
\begin{equation}
\nu  =  \sum_{i \in \Lambda}\nu_{i}\delta_{x_{i}},\quad
\mu\geq \sum_{i \in \Lambda}\mu_{i}\delta_{x_{i}},\quad
C_{a,p}^* \nu_{i}^{p/p^{*}}\leq \mu_{i},
 \label{lema_infinito_eq11}
\end{equation}
for all $i \in\Lambda$, where $\delta_{x_i}$ is the Dirac mass at
$x_i \in \Omega$.

Now let $k\in \mathbb{N}$. Without loss of generality we can suppose
$B_2(0) \subset \Omega$, then for every $\varrho>0$, we set
$\psi_{\varrho}(x) := \psi((x-x_k)/\varrho)$ where
$\psi \in C_0^{\infty}(\Omega,[0,1])$
is such that $\psi \equiv 1$ on
$B_1(0)$, $\psi \equiv 0$ on $\Omega \setminus B_2(0)$, and $|\nabla \psi| \leq 1$.
Observe that $(\psi_{\varrho}u_n)$ is bounded in $\mathcal{D}_a^{1,p}$.
So we have $J'(u_n)(\psi_{\varrho}u_n)
\to 0$; that is,
\begin{align*}
&M_0(\|u_{n}\|^p){\int_{\Omega}} \frac{u_n|\nabla u_{n}|^{p-2}\nabla
u_n \nabla \psi_{\varrho}}{|x|^{ap}} dx  + o_n(1)\\
\\
&= -M_0(\|u_{n}\|^p){\int_{\Omega}}
\frac{|\nabla u_{n}|^p \psi_{\varrho}}{|x|^{ap}} dx
+ \lambda{\int_{\Omega}}\frac{f(x,u_n)\psi_{\varrho}u_n}{|x|^{\delta}} dx
+ {\int_{\Omega}}\frac{\psi_{\varrho}|u_n|^{p^*}}{|x|^{bp^{\ast}}} dx.
\end{align*}

Since $u_n \to u$ in $L^{r}\left(\Omega,|x|^{-\delta}\right)$,
it follows from \eqref{convergencia fraca medida}, (H1), (H4) and the Dominated
Convergence Theorem, that
\begin{align*}
&  { \limsup_{n \to \infty}}\Big[
  M_0(\|u_{n}\|^p){\int_{\Omega}} \frac{u_n|\nabla u_{n}|^{p-2}\nabla
  u_n \nabla \psi_{\varrho}}{|x|^{ap}} dx \Big] \\
&\leq  -m_0{\int_{\Omega}}
  \frac{|\nabla u|^p \psi_{\varrho}}{|x|^{ap}} dx
  -m_0{\int_{\Omega}}\psi_{\varrho}d\mu
  + \lambda{\int_{\Omega}}\frac{f(x,u)\psi_{\varrho}u}{|x|^{\delta}} dx
  + {\int_{\Omega}}\frac{\psi_{\varrho}|u|^{p^*}}{|x|^{bp^{\ast}}} dx
  + {\int_{\Omega}}\psi_{\varrho}d\nu.
\end{align*}
Using the Dominated Convergence Theorem again, we obtain
\begin{equation*}
{\int_{\Omega}}\frac{|\nabla u|^p \psi_{\varrho}}{|x|^{ap}} dx
= o_{\varrho}(1),\quad
{\int_{\Omega}}\frac{f(x,u)\psi_{\varrho}u}{|x|^{\delta}} dx = o_{\varrho}(1), \quad
 {\int_{\Omega}}\frac{\psi_{\varrho}|u|^{p^*}}{|x|^{bp^{\ast}}} dx
=o_{\varrho}(1),
\end{equation*}
where ${\lim_{\varrho\to 0^{+}}o_{\varrho}(1) =0}$.
So, we obtain
\begin{equation}\label{des.1}
\begin{aligned}
&{\lim_{\varrho\to 0^{+}}\Big\{ \limsup_{n \to \infty}
 \Big[{M_0(\|u_{n}\|^p){\int_{\Omega}}
 \frac{u_n|\nabla u_{n}|^{p-2}\nabla
 u_n \nabla \psi_{\varrho}}{|x|^{ap}} dx} \Big]\Big\}}\\
&\leq { \lim_{\varrho\to 0^+}
 \Big[ \int_{\Omega}\psi_{\varrho}d\nu -m_0\int_{\Omega}\psi_{\varrho}d\mu \Big]}.
\end{aligned}
\end{equation}
Now, we  show that
\begin{equation}\label{eq limsup}
\lim_{\varrho \to 0^{+}}
\Big[ \limsup_{n\to \infty} M_0(\|u_{n}\|^p)\int_{\Omega} |x|^{-ap}u_n|\nabla
u_{n}|^{p-2}\nabla u_n \nabla \psi_{\varrho} dx \Big] = 0.
\end{equation}
Indeed, by H\"{o}lder's Inequality,
\begin{equation*}
\Big|\int_{\Omega} \frac{u_n|\nabla u_{n}|^{p-2}
\nabla u_n \nabla \psi_{\varrho}}{|x|^{ap}} dx\Big|
\leq \|u_n\|^{p-1}\Big(\int_{\Omega}
\frac{|u_{n}\nabla\psi_{\varrho}|^p}{|x|^{ap}}dx \Big)^{1/p}.
\end{equation*}
Since $u_n$ is bounded in $\mathcal{D}_{a}^{1,p}$, $M_0$ is continuous, and
$\operatorname{supp}(\psi_{\varrho}) \subset B(x_{k};2\varrho)$,
 there exists $L>0$ such that
$$
M_0(\|u_{n}\|^p)\big|\int_{\Omega} \frac{u_n|\nabla u_{n}|^{p-2}\nabla u_n
\nabla \psi_{\varrho}}{|x|^{ap}} dx\big|
\leq L\Big(\int_{B(x_{k};2\varrho)} \frac{|u_{n}\nabla\psi_{\varrho}|^p}{|x|^{ap}}
dx \Big)^{1/p}.
$$
Using the dominated convergence theorem and H\"{o}lder's inequality, we obtain
\begin{align*}
&{\limsup_{n\to \infty}}\Big[ M_0(\|u_{n}\|^p)\big|{\int_{\Omega}}
\frac{u_n|\nabla u_{n}|^{p-2}\nabla u_n \nabla \psi_{\varrho}}{|x|^{ap}} dx\big|
\Big]\\
&\leq L\Big({\int_{B(x_{k};2\varrho)}} \frac{|u|^p|
\nabla\psi_{\varrho}|^p}{|x|^{ap}}dx \Big)^{1/p}
\\
&\leq L \Big( {\int_{B(x_{k};2\varrho)}}
|\nabla\psi_{\varrho}|^{N}dx \Big)^{1/N}
\Big( {\int_{B(x_{k};2\varrho)}}
\big(|x|^{-ap}|u|^p\big)^{\frac{N}{N-p}}dx \Big)^{\frac{N-p}{Np}}
\\
&\leq L|B(x_{k};2\varrho)|^{1/N} \Big(
{\int_{\Omega}}\chi_{B(x_{k};2\varrho)}\big(|x|^{-ap}|u|^p\big)^{\frac{N}{N-p}}dx
\Big)^{\frac{N-p}{Np}}.
\end{align*}
Letting $\varrho \to 0^{+}$ on the above expression, we obtain \eqref{eq limsup}.
Thus, we conclude from \eqref{des.1} that
$$
0 \leq \lim_{\rho\to 0^{+}}\Big[ \int_{\Omega}
\psi_{\varrho}d\nu -m_0\int_{\Omega}\psi_{\varrho}d\mu \Big].
$$
That is,
\begin{align*}
0
&\leq \lim_{\rho\to 0^{+}}
\Big[ \int_{B(x_{k};2\varrho)}\psi_{\varrho}d\nu -m_0
\int_{B(x_{k};2\varrho)}\psi_{\varrho}d\mu \Big]\\
&= \nu(\{x_{k}\}) -m_0\mu(\{x_{k}\})\\
&\leq \sum_{i \in \Lambda}\nu_{i}\delta_{x_{i}}(\{x_{k}\}) -m_0\sum_{i
\in \Lambda}\mu_{i}\delta_{x_{i}}(\{x_{k}\})\\
&= \nu_{k} - m_0\mu_{k}.
\end{align*}
So, we have
$m_0\mu_{k} \leq \nu_{k}$.
It follows from \eqref{lema_infinito_eq11} that
\begin{equation}\label{nik}
\nu_{k} \geq (m_0C_{a,p}^*)^{\frac{p^{\ast}}{p^{\ast}-p}}
\geq \big(\frac{1}{\theta}-\frac{1}{p^{\ast}}\big)
(m_0C_{a,p}^*)^{p^{\ast}/(p^{\ast}-p)}.
\end{equation}

Now we shall prove that the above expression can not occur,
and therefore the set $\Lambda$ is empty. Indeed, arguing by contradiction,
let us suppose that \eqref{nik} hold for some $k \in \Lambda$. Thus, once that
$m_0 \leq M_0(t) \leq \frac{\xi}{p}m_0$, for all $t \in \mathbb{R}$,
and by using $(f_3)$ we have
\begin{align*}
c &= J(u_n)- \frac{1}{\xi}J'(u_n)(u_n) + o_{n}(1)\\
&\geq \big(\frac{m_0}{p} - \frac{\xi m_0}{\xi p}\big)\|u_{n}\|^p
-\lambda {\int_{\Omega}}\frac{F(x,u_{n})
-\frac{1}{\xi}f(x,u_n) u_n}{|x|^{\delta}} dx
\\
&\quad +\big(\frac{1}{\xi}-\frac{1}{p^{\ast}}\big)
{\int_{\Omega}}\frac{|u_{n}|^{p^{\ast}}}{|x|^{bp^{\ast}}}dx +o_{n}(1)\\
&\geq \big(\frac{1}{\xi}-\frac{1}{p^{\ast}}\big){\int_{\Omega}}
\frac{|u_{n}|^{p^{\ast}}\psi_{\varrho}}{|x|^{bp^{\ast}}}dx +o_{n}(1).
\end{align*}
Letting $n\to +\infty$, we obtain
$$
c \geq \big(\frac{1}{\xi}-\frac{1}{p^{\ast}}\big)
(m_0C_{a,p}^*)^{\frac{p^{\ast}}{p^{\ast}-p}}.
$$
But this is a contradiction. Thus $\Lambda$ is empty and it follows that
$u_n \to u$ in $L^{p^{\ast}}\left(\Omega,|x|^{-bp^{\ast}}\right)$.

Now we will prove that $u_n \to u$ in $\mathcal{D}_{a}^{1,p}$.
Indeed, since $u_n \to u$ in $L^{r}(\Omega,|x|^{-\delta})$ and in
$L^{p^{\ast}}(\Omega,|x|^{-bp^{\ast}})$, it follows from the dominated convergence
theorem that
$$
\lim_{n\to+\infty}\int_{\Omega}\frac{f(x,u_{n})(u_{n}-u)}{|x|^{\delta}}dx
=\lim_{n\to+\infty}
\int_{\Omega}\frac{|u_{n}|^{p^{\ast}-2}u_{n}(u_{n}-u)}{|x|^{bp^{\ast}}}dx =0.
$$
Therefore, as $(u_n)$ is bounded in $\mathcal{D}_{a}^{1,p}$,
$J'(u_n)(u_n-u) \to 0 \text{ in  } (\mathcal{D}_{a}^{1,p})^{-1}$,
$\|u_n\| \to t_0$, as $n\to \infty$, and as $M$ is continuous and positive,
we conclude that
$$
\lim_{n\to\infty}\int_{\Omega}|x|^{-ap}|\nabla u_{n}|^{p-2}
\nabla u_{n}\nabla(u_{n}-u)dx = 0.
$$
It follows from Lemma \ref{S_{+}} that $u_n \to u$ in $\mathcal{D}_{a}^{1,p}$.
\end{proof}

\section{Proof of Theorem \ref{thm1}}\label{provateo1}

In this section we prove Theorem \ref{thm1}, which concerns to
problem \eqref{problema1} when $r=p$.
The next two lemmas show that the functional $J$ has the Mountain Pass geometry.

\begin{lemma}\label{mpg1 r=p}
Suppose that $r=p$ and let $\lambda_1$ be as in \eqref{primeiro autovalor}.
 Assume that the conditions {\rm (H1)--(H4)} hold. Then, there exist positive
numbers $\rho$ and $\alpha$ such that
$$
J(u) \geq \alpha >0, \forall u \in \mathcal{D}_{a}^{1,p}, \quad
\text{with } \|u\| = \rho,
$$
for all $\lambda \in (0,\frac{m_0}{C_2}\lambda_1)$.
\end{lemma}

\begin{proof}
Let $\lambda \in (0,\frac{m_0}{C_2}\lambda_1)$. From (H1), (H3), (H4),
\eqref{primeiro autovalor}, and Caffarelli-Khon-Nirenberg inequality, we obtain
$$
J(u) \geq \big(m_0 - \frac{\lambda C_2}{\lambda_1}\big)
\frac{1}{p}\|u\|^p - \frac{1}{p^{\ast}}\tilde{C}\|u\|^{p^{\ast}}.
$$
Since $p<p^{\ast}$ and $\lambda < \frac{m_0}{C_2}\lambda_1$.
The result follows by choosing $\rho >0$ small enough.
\end{proof}

\begin{lemma}\label{mpg2 r=p}
Suppose that $r=p$. Assume that the conditions {\rm (H1), (H3), (H4)}  hold.
For each $\lambda >0$, there exists $e \in \mathcal{D}_{a}^{1,p}$ with
$J(e)<0$ and $\|e\|>\rho$.
\end{lemma}

\begin{proof}
Fix $v_0 \in \mathcal{D}_{a}^{1,p}\backslash \{0\}$,
with $v_0 > 0$ in $\Omega$. Using \eqref{trunc 2} and (H4) we obtain
\begin{equation*}
J(tv_0) \leq \frac{\xi}{p^{2}}m_0t^p\|v_0\|^p
-\frac{\lambda C_1}{p}t^p\int_{\Omega}\frac{|v_0|^p}{|x|^{\delta}}dx
- \frac{t^{p^{\ast}}}{p^{\ast}}\int_{\Omega}\frac{|v_0|^{p^{\ast}}}{|x|^{bp^{\ast}}}dx.
\end{equation*}
Since $p<p^{\ast}$, we have ${\lim_{t\to +\infty}J(tv_0)} = -\infty$.
Thus, there exists $\bar{t}>0$ large enough, such that $\bar{t}\|v_0\|>\rho$
and $J(\bar{t}v_0) <0$.
The result follows by considering $e = \bar{t}v_0$.
\end{proof}

Using a version of the mountain pass theorem without the (PS) condition
(see \cite{willem}), for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$,
there exists a sequence $(u_{n}) \in \mathcal{D}_{a}^{1,p}$, satisfying
$$
J(u_{n}) \to c_{\lambda} \text{ and } J'(u_{n}) \to 0 \text{ in }
(\mathcal{D}_{a}^{1,p})^{-1},
$$
where
\begin{gather*}
0< c_{\lambda} = \inf_{\gamma \in \Gamma}\max_{t\in [0,1]}J(\gamma(t)), \\
\Gamma := \big\{\gamma \in C([0,1],\mathcal{D}_{a}^{1,p}): \gamma(0) = 0,
\gamma(1)=\bar{t}v_0\big\},
\end{gather*}
and $v_0 \in \mathcal{D}_{a}^{1,p}$ is such that $v_0 >0$.

To obtain the level $c_\lambda$ below the level given by Lemma \ref{nivelabaixo},
we will give some estimates.
We define the Sobolev space
$$
W_{a,b}^{1,p}(\Omega) = \big\{ u \in L^{p^{\ast}}(\Omega,|x|^{-bp^{\ast}})
: |\nabla u| \in L^p(\Omega,|x|^{-ap}) \big\},
$$
with respect to the norm
$$
\|u\|_{W_{a,b}^{1,p}(\Omega)} = \|u\|_{p^{\ast},bp^{\ast}} + \|\nabla u\|_{p,ap}.
$$
We consider the best constant of the weighted Caffarelli-Kohn-Nirenberg type
given by
$$
\tilde{S}_{a,p} = \inf_{u\in W_{a,b}^{1,p}(\mathbb{R}^{N})\backslash \{0\}}
\Big\{\frac{\int_{\mathbb{R}^{N}} |x|^{-ap}|\nabla u|^p dx}
{\big(\int_{\mathbb{R}^{N}} |x|^{-bp^{\ast}}|u|^{p^{\ast}}dx\big)^{p/p^*}}\Big\}
\,.
$$
We also set $R_{a,b}^{1,p}(\Omega)$ as the subspace of $W_{a,b}^{1,p}(\Omega)$
of the radial functions, more precisely
$$
R_{a,b}^{1,p}(\Omega) = \left\{u \in W_{a,b}^{1,p}(\Omega) : u(x) = u(|x|) \right\},
$$
with respect to the induced norm
$$
\|u\|_{R_{a,b}^{1,p}(\Omega)} = \|u\|_{W_{a,b}^{1,p}(\Omega)}.
$$
Horiuchi \cite{horiuchi}  proved that
$$
\tilde{S}_{a,p,R} = \inf_{u\in R_{a,b}^{1,p}(\mathbb{R}^{N})\backslash \{0\}}
\Big\{\frac{\int_{\mathbb{R}^{N}} |x|^{-ap}|\nabla u|^p dx}
{\big(\int_{\mathbb{R}^{N}} |x|^{-bp^{\ast}}|u|^{p^{\ast}}dx\big)^{p/p^*}}\Big\}
$$
is achieved by  functions of the form
$$
u_{\varepsilon}(x) = k_{a,p}(\varepsilon)v_{\varepsilon}(x), \quad\forall \varepsilon >0,
$$
where
$$
k_{a,p}(\varepsilon)=c\varepsilon^{(N-dp)/dp^{2}} \quad
 v_{\varepsilon}(x) = \Big( \varepsilon + |x|^{\frac{dp(N -p -ap)}{(p-1)(N-dp)}}
\Big)^{-(\frac{N-dp}{dp})}\cdot
$$
Moreover, $u_{\varepsilon}$ satisfies
\begin{equation}\label{S}
\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u_{\varepsilon}|^pdx
= \int_{\mathbb{R}^{N}}|x|^{-bp^{\ast}}|u_{\varepsilon}|^{p^{\ast}}dx
= (\tilde{S}_{a,p,R})^{\frac{p^{\ast}}{p^{\ast}-p}}.
\end{equation}
From \eqref{S} we obtain
\begin{gather}\label{eq.vepsilon1}
\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v_{\varepsilon}|^pdx
= [k_{a,p}(\varepsilon)]^{-p}(\tilde{S}_{a,p,R})^{\frac{p^{\ast}}{p^{\ast}-p}},\\
\label{eq.vepsilon2}
\int_{\mathbb{R}^{N}}|x|^{-bp^{\ast}}|v_{\varepsilon}|^{p^{\ast}}dx
= [k_{a,p}(\varepsilon)]^{-p^{\ast}}(\tilde{S}_{a,p,R})
^{\frac{p^{\ast}}{p^{\ast}-p}}.
\end{gather}

Let $R_0$ be a positive constant and set $\Psi(x) \in C_0^{\infty}(\mathbb{R}^{N})$
such that $0\leq \Psi(x) \leq 1$, $\Psi(x) = 1$, for all $|x| \leq R_0$, and
$\Psi(x) = 0$, for all $|x| \geq 2R_0$. Set
\begin{equation}\label{vtil}
\tilde{v}_{\varepsilon}(x) = \Psi(x)v_{\varepsilon}(x),
\end{equation}
for all $x \in \mathbb{R}^{N}$ and for all $\varepsilon >0$.
Without loss of generality we can consider $B(0;2R_0) \subset \Omega$.

\begin{lemma}\label{limitesistema}
With the above notation we have
$$
\lim_{\varepsilon \to 0^{+}}\frac{\|\tilde{v}_{\varepsilon}\|^p}
{\big(\int_{\Omega}|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx
\big)^{p/p^{\ast}}} = 0.
$$
\end{lemma}

\begin{proof}
By a straightforward computation we obtain
\begin{gather}\label{estimativa1}
\|\tilde{v}_{\varepsilon}\|^p
\leq [k_{a,p}(\varepsilon)]^{-p}(\tilde{S}_{a,p,R})^{\frac{p^{\ast}}{p^{\ast}-p}}
+ C, \\
\label{estimativa2}
\int_{\Omega}|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx
= \varepsilon^{-\frac{N-dp}{dp}p^{\ast}} \cdot C, \quad
\forall \varepsilon \in (0,1),
\end{gather}
where $C$ denotes a positive constant. Therefore, for all
$\varepsilon \in (0,1)$, from \eqref{estimativa1} and \eqref{estimativa2}
we obtain
\begin{align*}
\frac{\|\tilde{v}_{\varepsilon}\|^p}
{\big(\int_{\Omega}|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx
\big)^{p/p^{\ast}}}
&\leq  \frac{[k_{a,p}(\varepsilon)]^{-p}(\tilde{S}_{a,p,R})^{\frac{p^{\ast}}{p^{\ast}-p}} + C}{\left( \varepsilon^{-\frac{N-dp}{dp}p^{\ast}} \cdot C \right)^{p/p^{\ast}}}\\
&= \frac{c^{-p}(\tilde{S}_{a,p,R})^{\frac{p^{\ast}}{p^{\ast}-p}}
\varepsilon^{\frac{N-dp}{dp}(p-1)} + C\varepsilon^{\frac{N-dp}{dp}p}}{C}\,.
\end{align*}
Since $p>1$, we have
$$
\lim_{\varepsilon \to 0^{+}} \frac{\|\tilde{v}_{\varepsilon}\|^p}
{\big(\int_{\Omega}|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx
\big)^{p/p^{\ast}}} = 0.
$$
\end{proof}

\begin{lemma}\label{nivelabaixop=q}
Let $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$. Assume that
{\rm (H1)--(H5)}  hold. Set
\[
l^{\ast} = \min\Big\{\Big(\frac{1}{p}m_0 - \frac{1}{\xi}M_0(t_0) \Big)t_0,
\big( \frac{1}{\xi}-\frac{1}{p^{\ast}} \big)( m_0C_{a,p}^*
)^{\frac{p^{\ast}}{p^{\ast}-p}}\Big\}.
\]
Then, there exists $\varepsilon_1 \in (0,1)$ such that
$$
\sup_{t \geq 0}J(t\tilde{v}_{\varepsilon})  < l^{\ast},
$$
for all $\varepsilon \leq \varepsilon_1$.
\end{lemma}

\begin{proof}
 Let $0 < \varepsilon < 1$ and $\tilde{v}_{\varepsilon}$ be as in \eqref{vtil}.
Since from Lemmas \ref{mpg1 r=p} and \ref{mpg2 r=p} the functional $J$ satisfies
the Mountain Pass geometry, for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$,
there exists $t_{\varepsilon}$ such that
$$
\sup_{t \geq 0}J(t\tilde{v}_{\varepsilon})
= J(t_{\varepsilon}\tilde{v}_{\varepsilon}),
$$
for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$. So, we have
\begin{align*}
\sup_{t \geq 0}J(t\tilde{v}_{\varepsilon})
&= \frac{1}{p}\widehat{M}(\|t_{\varepsilon}\tilde{v}_{\varepsilon}\|^p)
 -\lambda \int_{\Omega}|x|^{-\delta}F(x,t_{\varepsilon}\tilde{v}_{\varepsilon})dx
 - \frac{1}{p^{\ast}}\int_{\Omega}|x|^{-bp^{\ast}}t_{\varepsilon}^{p^{\ast}}
 |\tilde{v}_{\varepsilon}|^{p^{\ast}}dx\\
&\leq \frac{\xi}{p^{2}}m_0t_{\varepsilon}^p\|\tilde{v}_{\varepsilon}\|^p
  -\frac{1}{p^{\ast}} t_{\varepsilon}^{p^{\ast}}\int_{\Omega}|x|^{-bp^{\ast}}
 |\tilde{v}_{\varepsilon}|^{p^{\ast}}dx,
\end{align*}
for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$. Now we consider the
function $g: \mathbb{R}^{+} \cup \{0\} \to \mathbb{R}^{+} \cup \{0\}$, given by
$$
g(s) = \Big(\frac{\xi}{p^{2}}m_0\|\tilde{v}_{\varepsilon}\|^p\Big)s^p
-\Big(\frac{1}{p^{\ast}}\int_{\Omega}|x|^{-bp^{\ast}}
 |\tilde{v}_{\varepsilon}|^{p^{\ast}}dx\Big) s^{p^{\ast}}.
$$
It is easy to see that
\[
{\bar{s} = \Big( \frac{\frac{\xi}{p}m_0\|\tilde{v}_{\varepsilon}\|^p}{\int_{\Omega}
|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx} \Big)^{\frac{1}{p^{\ast}-p}}}
\]
 is a maximum of $g$ and we have
$$
g(\bar{s}) = \big(\frac{1}{p} - \frac{1}{p^{\ast}}\big)
\Big(\frac{\xi}{p}m_0\Big)^{\frac{p^{\ast}}{p^{\ast}-p}}
\Big(\frac{\|\tilde{v}_{\varepsilon}\|^p}
{\big(\int_{\Omega}|x|^{-bp^{\ast}}|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx
\big)^{p/p^{\ast}}}\Big)^{\frac{p^{\ast}}{p^{\ast}-p}}.
$$
So, we have
$$
\sup_{t \geq 0}J(t\tilde{v}_{\varepsilon})
\leq \big(\frac{1}{p} - \frac{1}{p^{\ast}}\big)
\Big(\frac{\xi}{p}m_0\Big)^{\frac{p^{\ast}}{p^{\ast}-p}}
\Big(\frac{\|\tilde{v}_{\varepsilon}\|^p}{\big(\int_{\Omega}|x|^{-bp^{\ast}}
|\tilde{v}_{\varepsilon}|^{p^{\ast}}dx\big)^{p/p^{\ast}}}
\Big)^{\frac{p^{\ast}}{p^{\ast}-p}},
$$
for each $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$.

It follows from Lemma \ref{limitesistema} that there exists
$0 < \varepsilon_1 <1$ such that
$$
\sup_{t \geq 0}J(t\tilde{v}_{\varepsilon})  < l^{\ast},
$$
for all $\varepsilon \leq \varepsilon_1$ and for each
$\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$.
\end{proof}

\begin{remark}\label{remark1 r=p} \rm
 Let $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$ and let us consider the
path $\gamma_{\ast}(t) = t(\bar{t}v_{\varepsilon_1}), $ for $t \in [0,1]$,
which belongs to $\Gamma$. It follows from Lemma \ref{nivelabaixop=q}
that we obtain the following estimate
\[
0 < c_{\lambda} = \inf_{\gamma \in \Gamma} \max_{t \in [0,1]} J(\gamma(t))
\leq  \sup_{s \geq 0}J(s\tilde{v}_{\varepsilon_1}) < l^{\ast},
\]
for all $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$.
\end{remark}

\begin{lemma}\label{lema4 r=p}
Suppose that $r=p$, $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$, and
{\rm(H1), (H2), (H4), (H5)} hold.
Let $(u_n ) \in \mathcal{D}_{a}^{1,p}$  be a sequence such that
$$
J(u_{n}) \to c_{\lambda} \quad\text{and}\quad
J'(u_{n}) \to 0 \quad \text{in } (\mathcal{D}_{a}^{1,p})^{-1}, \quad
\text{as } n\to +\infty.
$$
Then, for all $n \in \mathbb{N}$, we have
$\|u_{n}\|^p \leq t_0$.
\end{lemma}

\begin{proof}
Suppose by contradiction that for some $n \in \mathbb{N}$
we have $\|u_{n}\|^p > t_0$. From the definition of $M_0(t)$, (H5),
and \eqref{trunc 2} we have that $(u_{n})$ bounded. Thus, we obtain
$$
|J'(u_{n})\cdot(u_{n})| \leq |J'(u_{n})|\,\|(u_{n})\| \to 0,
$$
as $n \to +\infty$. Which implies
\begin{equation}\label{nivel acima}
\begin{aligned}
c_{\lambda}
&=  J(u_{n}) - \frac{1}{\xi}J'(u_{n})(u_{n}) + o_{n}(1)\\
& \geq \frac{1}{p}\widehat{M_0}(\|u_{n}\|^p)
  -\frac{1}{\xi}  M_0(t_0)\|u_{n}\|^p +o_{n}(1)\\
& \geq \Big( \frac{1}{p}m_0 - \frac{1}{\xi}  M_0(t_0) \Big)\|u_{n}\|^p +o_{n}(1).
\end{aligned}
\end{equation}
Since $m_0 < M(t_0) < \frac{\xi}{p}m_0$ we have
$\frac{1}{p}m_0 - \frac{1}{\xi}  M_0(t_0) >0$. So we obtain
\[
c_{\lambda} \geq \Big(\frac{1}{p}m_0 - \frac{1}{\xi}  M_0(t_0)\Big)t_0 >0.
\]
 Since $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$, this contradicts the
Remark \ref{remark1 r=p}. Hence we conclude that $\|u_{n}\|^p \leq t_0$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1}]
Let $\lambda \in (0, \frac{m_0}{C_2}\lambda_1)$. It follows from Remark
\ref{remark1 r=p} that
\begin{equation}\label{nivel abaixo r=p}
c_{\lambda} < \big( \frac{1}{\xi}-\frac{1}{p^{\ast}} \big)
( m_0C_{a,p}^*)^{\frac{p^{\ast}}{p^{\ast}-p}}.
\end{equation}
From Lemmas \ref{mpg1 r=p} and \ref{mpg2 r=p}, there exists a bounded
sequence $(u_{n}) \subset \mathcal{D}_{a}^{1,p}$ such that
$J(u_{n}) \to c_{\lambda}$ and $J'(u_{n}) \to 0$,
$(\mathcal{D}_{a}^{1,p})^{-1}$, as $n \to \infty$.
Since \eqref{nivel abaixo r=p} holds, it follows from Lemma \ref{nivelabaixo} that,
up to a subsequence, $u_{n} \to u_{\lambda}$ strongly in $\mathcal{D}_{a}^{1,p}$.
Thus $u_{\lambda}$ is a weak solution of problem \eqref{problema auxiliar}.
By Lemma \ref{lema4 r=p}, we conclude that $u_{\lambda}$  is a weak solution
of problem \eqref{problema1}.
\end{proof}

\section{Proof of Theorem \ref{thm2}}\label{provateo2}

 Here we consider the case  $p<r<p^{\ast}$. The main idea of the proof is essentially
the same as in Theorem \ref{thm2}, we apply the mountain pass theorem
and use Lemma \ref{nivelabaixo}.
The next two lemmas show that the functional $J$ has the Mountain Pass geometry.

\begin{lemma}\label{mpg1 r>p}
Suppose that $p<r<p^{\ast}$. Assume that the conditions {\rm (H1)--(H4)} hold.
 There exist positive numbers $\rho$ and $\alpha$ such that
$$
J(u) \geq \alpha >0, \forall u \in \mathcal{D}_{a}^{1,p}, \quad \text{with }
 \|u\| = \rho.
$$
\end{lemma}

\begin{proof}
From (H1), (H3), (H4), and Caffarelli-Kohn-Nirenberg inequality, we obtain
$$
J(u) \geq \frac{m_0}{p}\|u\|^p - \lambda\tilde{C}_2\|u\|^{r}
- \frac{1}{p^{\ast}}\tilde{C}\|u\|^{p^{\ast}}.
$$
Since $p<r<p^{\ast}$, the result follows by choosing $\rho >0$ small enough.
\end{proof}

\begin{lemma}\label{mpg2 r>p}
Suppose that $p<r<p^{\ast}$. For all $\lambda >0$, there exists
$e \in \mathcal{D}_{a}^{1,p}$ with $J(e)<0$ and $\|e\|>\rho$.
\end{lemma}

\begin{proof}
Fix $v_0 \in \mathcal{D}_{a}^{1,p}\backslash \{0\}$,
with $v_0 > 0$ in $\Omega$. Using \eqref{trunc 2} and (H4) we obtain
\begin{equation*}
J(tv_0) \leq \frac{\xi}{p^{2}}m_0t^p\|v_0\|^p
-\frac{\lambda C_1}{r}t^{r}\int_{\Omega}\frac{|v_0|^{r}}{|x|^{\delta}}dx
- \frac{t^{p^{\ast}}}{p^{\ast}}
\int_{\Omega}\frac{|v_0|^{p^{\ast}}}{|x|^{bp^{\ast}}}dx.
\end{equation*}
Since $p<r<p^{\ast}$, we have ${\lim_{t\to +\infty}J(tv_0)} = -\infty$.
Thus, there exists $\bar{t}>0$ large enough, such that $\bar{t}\|v_0\|>\rho$
and $J(\bar{t}v_0) <0$.
The result follows by considering $e = \bar{t}v_0$.
\end{proof}

Using a version of the mountain pass theorem without the (PS) condition
(see \cite{willem}),
there exists a sequence $(u_{n}) \in \mathcal{D}_{a}^{1,p}$, satisfying
$$
J(u_{n}) \to c_{\lambda} \quad\text{and}\quad J'(u_{n}) \to 0 \quad\text{in }
(\mathcal{D}_{a}^{1,p})^{-1},
$$
where
\begin{gather*}
0< c_{\lambda} = \inf_{\gamma \in \Gamma}\max_{t\in [0,1]}J(\gamma(t)), \\
\Gamma := \big\{\gamma \in C([0,1],\mathcal{D}_{a}^{1,p}): \gamma(0) = 0,
\gamma(1)=\bar{t}v_0 \big\},
\end{gather*}
and $v_0 \in \mathcal{D}_{a}^{1,p}$ is such that $v_0 >0$.


\begin{remark}\label{remark1 r>p} \rm
 From  Lemmas \ref{mpg1 r>p} and \ref{mpg2 r>p},  Lemma \ref{nivelabaixop=q}
holds for all $\lambda>0$, when $p<r<p^{\ast}$. So, if we consider the path
$\gamma_{\ast}(t) = t(\bar{t}v_{\varepsilon_1}), $ for $t \in [0,1]$,
 which belongs to $\Gamma$, we obtain the estimate
\[
0 < c_{\lambda}
= \inf_{\gamma \in \Gamma} \max_{t \in [0,1]} J(\gamma(t))
\leq  \sup_{s \geq 0}J(s\tilde{v}_{\varepsilon_1}) < l^{\ast}
\]
for all $\lambda>0$.
\end{remark}

The next Lemma is a version of the Lemma \ref{lema4 r=p} when $p<r<p^{\ast}$.
By hypothesis (H5) and  Remark \ref{remark1 r>p}, its proof is similar to
the proof of Lemma \ref{lema4 r=p}.

\begin{lemma}\label{lema4 r>p}
Suppose that $p<r<p^{\ast}$, and {\rm (H1), (H2), (H4), (H5)} hold.
Let $(u_n ) \in \mathcal{D}_{a}^{1,p}$  be a sequence such that
$$
J(u_{n}) \to c_{\lambda} \quad \text{and}\quad
 J'(u_{n}) \to 0 \quad\text{in } (\mathcal{D}_{a}^{1,p})^{-1}, \quad
\text{as } n\to +\infty.
$$
Then, for all $n \in \mathbb{N}$, we have
$\|u_{n}\|^p \leq t_0$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm2}]
 It follows from Remark \ref{remark1 r>p} that
\begin{equation}\label{nivel abaixo r>p}
c_{\lambda} < \big( \frac{1}{\xi}-\frac{1}{p^{\ast}} \big)
( m_0C_{a,p}^*)^{\frac{p^{\ast}}{p^{\ast}-p}}.
\end{equation}
From Lemmas \ref{mpg1 r>p} and \ref{mpg2 r>p}, there exists a bounded
sequence $(u_{n}) \subset \mathcal{D}_{a}^{1,p}$ such that
$J(u_{n}) \to c_{\lambda}$ and $J'(u_{n}) \to 0$,
$(\mathcal{D}_{a}^{1,p})^{-1}$, as $n \to \infty$.
Since \eqref{nivel abaixo r>p} holds, it follows from Lemma \ref{nivelabaixo} that,
up to a subsequence, $u_{n} \to u_{\lambda}$ strongly in $\mathcal{D}_{a}^{1,p}$.
Thus $u_{\lambda}$ is a weak solution of problem \eqref{problema auxiliar}.
Moreover, by Lemma \ref{lema4 r=p} we conclude that $u_{\lambda}$  is a weak
solution of problem \eqref{problema1}.
\end{proof}

\subsection*{Acknowledgements}
This research was supported by
Grant \#2015/11912-6 from the  S\~ao Paulo Research Foundation (FAPESP).

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\end{thebibliography}

\end{document}