\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 $N\geq 3$, $1
0$. 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 $0\in \Omega$, $N\geq 3$, $1
0$ 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 $p
0$ 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 $p
0$ 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