\documentclass[12pt,reqno,a4paper]{article} \usepackage[latin1]{inputenc} % accents \usepackage{a4,latexsym,amssymb,amsfonts} \usepackage{amsmath} \usepackage{amsthm} \usepackage[dvips]{epsfig} \usepackage{enumerate} \usepackage{hyperref} \usepackage[english]{babel} \usepackage{graphicx} \textheight240mm \textwidth170mm \hoffset=-9mm \pagestyle{plain} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\contentsname}{Table des mati\`eres} \renewcommand{\cleardoublepage}{\clearpage} \renewcommand{\partname}{Partie} \renewcommand{\listtablename}{Liste des tables} \newcommand{\bibname}{Bibliography} \newcommand{\cqfd}{\nolinebreak \hspace*{5mm} \rule{2mm}{2mm}} \newcommand{\f}{\nolinebreak \hspace*{5mm} \rule{1mm}{1mm}} \newcommand{\ov}{\overline} \newcommand{\wi}{\widetilde} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\thesubsection {\thesection.\arabic{subsection}} \numberwithin{equation}{section} \newtheorem{thm}{\bf Theorem}[section] \newtheorem{lem}[thm]{\bf Lemma} \newtheorem{pre}{Proof} \newtheorem{prop}[thm]{\bf Proposition} \newtheorem{cor}[thm]{\bf Corollary} \newtheorem{rks}[thm]{\bf Remarks} \newtheorem{rk}[thm]{\bf Remark} \newtheorem{df}[thm]{\bf Definition} \newtheorem{example}{Example}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Existence and non-existence of a positive solution \\for $(p,q)$-Laplacian with singular weights } \author{{\sc Abdellah ZEROUALI\thanks{abdellahzerouali@yahoo.fr}} \\ Centre Régional des Métiers de l'\'{E}ducation et de la Formation, Fès, Maroc\\ {\sc Belhadj KARIM\thanks{karembelf@gmail.com}}\\ Faculté des Sciences et Techniques, Erraachidia, Maroc} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem $$ -\triangle_{p}u-\mu\triangle_{q}u = \lambda[ m_{p}(x)|u|^{p-2}u+\mu m_{q}(x)|u|^{q-2}u]\hbox{ in }\Omega$$ driven by nonhomogeneous operator $(p,q)$-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where $\mu>0$ and with $10$ there exists an interval of eigenvalues for our eigenvalue problem. \end{abstract} \medskip \noindent\textbf{Keywords:} Nonlinear eigenvalue problem; $(p,q)$-Laplacian; singular weight; indefinite weight the Hardy-Sobolev inequality; Harnack inequality . \medskip \noindent\textbf{2000 Mathematics Subject Classification:} 35J20, 35J62, 35J70, 35P05, 35P30. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section {Introduction} \hspace{1cm}Consider the $(p,q)$-Laplacian eigenvalue problem \begin{equation*} (P_{\lambda, \mu})\left\{ \begin{array}{rclclll} \hbox{To find } (u, \lambda)&\in& (W_{0}^{1, p}(\Omega)\setminus\{0\})\times\mathbb{R} &\hbox{ such that}\\ -\triangle_{p}u-\mu\triangle_{q}u &=& \lambda[ m_{p}(x)|u|^{p-2}u+\mu m_{q}(x)|u|^{q-2}u] &\hbox{ in }\Omega,\\ u &=& 0 &\hbox{ on } \partial \Omega \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with piecewise $C^{1}$ boundary $\partial \Omega$, $\lambda\in\mathbb{R}$, $\mu\geq 0$ and $1\tau r$; \item[(H2)] $\partial\Omega$ is piecewise $C^{1}$, $0<\tau <1$, $r< \frac{N}{1-\tau}\leq a$; \item[(H3):] $\partial\Omega$ is piecewise $C^{1}$, $\tau =1$ and $a=\infty$; \item[(H4):] $\Omega$ is any bounded domain, $\tau =0$ and $a=\infty$. \end{itemize} The problem $(P_{\lambda, \mu})$ comes, for example, from a general reaction diffusion system \begin{equation} \label{P2} u_{t}= div(D(u)\nabla u) + c(x,u), \end{equation} where $D(u)= (|\nabla u|^{p-2}+\mu|\nabla u|^{q-2})$. This system has a wide range of applications in physics and related sciences like chemical reaction design \cite{Aris}, biophysics \cite{Fife} and plasma physics \cite{Struwe}. In such applications, the function u describes a concentration, the first term on the right-hand side of \eqref{P2} corresponds to the diffusion with a diffusion coefficient $D(u)$; whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term $c(x; u)$ has a polynomial form with respect to the concentration. Our problem was addressed in \cite{MT} for domains with boundary $C^{2}$ and bounded weights, when only the condition (H4) holds true. These work proved that in the case where $\mu>0$, there exists an interval of eigenvalues. The authors proved the existence of positive solutions in resonant cases. A non-existence result is also given. Here we will assume that the boundary $\partial\Omega$ is a piecewise $C^{1}$ and singular weights $m_{r}$ $(r=p,q)$ which satisfy one of the conditions (H1), (H2), (H3) or (H4). Our work represent developments of the study performed in \cite{MT} because we prove all results of this paper by considering others conditions that represent the singularity of the domain and the weights. Our main tool is the Hardy-Sobolev inequality, see Lemma \ref{L1} in preliminary section.\\ Many authors have studied the nonhomogeneous operator $(p,q)$-Laplacian (see \cite{MP, Si, Y, Z}). However, there are few results one the eigenvalue problems for the $(p,q)$-Laplacian. In \cite{BB, BBe}, the authors established the existence of the principal eigenvalue and of a continuous family of eigenvalues for problem $$-\triangle_{p}u -\triangle_{q}u= \lambda g(x)|u|^{p-2}u \hbox{ in } \mathbb{R}^{N}.$$ where $g$ is a bounded positive weight. Eigenvalue problem for a $(p,2)$-Laplacian was studied in \cite{B}. The existence of non trivial solution for the following Dirichlet equation is proved in \cite{CD} $$-\triangle_{p}u -\mu\triangle u= \lambda |u|^{p-2}u+g(u) \hbox{ in } \Omega,~~u=0 \mbox{ on }\partial\Omega,$$ in the case where $p>2$, $g\in C^{1}$ and $\lambda\not\in \sigma(-\Delta_{p})$, where $\sigma(-\Delta_{p})$ is the spectrum of $(-\Delta_{p})$. Under the Neumann boundary condition, \cite{Mih} determined the set of eigenvalues for the equation $$-\triangle_{p}u -\triangle u= \lambda u \hbox{ in } \Omega,$$ where $p>2$. In \cite{Ta}, M. Tanaka has completely described the generalized eigenvalue $\lambda$ for which the following equation $$-\triangle_{r}u -\mu\triangle u_{r*}= \lambda m_{r} |u|^{r-2}u \hbox{ in } \Omega, ~~u=0 \mbox{ on }\partial\Omega.$$ has a positive solution, where $10$. We recall that a value $\lambda\in \mathbb{R}$ is an eigenvalue of problem $(P_{\lambda, \mu})$ if and only if there exists $u \in W_{0}^{1, p}(\Omega)\backslash \{0\}$ such that \begin{equation}\label{P5} \int_{\Omega}(|\nabla u|^{p-2}+ \mu|\nabla u|^{q-2})\nabla u \nabla \varphi dx = \lambda\left[\int_{\Omega}(m_{p}(x)|u|^{p-2} + \mu m_{q}(x)|u|^{q-2}) u\varphi dx\right] \end{equation} for all $\varphi\in W_{0}^{1, p}(\Omega)$. $u$ is then called an eigenfunction of $\lambda.$ Letting $\mu \rightarrow 0^{+}$, our problem $(P_{\lambda, \mu})$ turns into the $(p-1)$-homogeneous problem known as the usual weighted eigenvalue problem for the $p$-Laplacian with singular weight $m_{p}$: \begin{equation*} (P_{\lambda, m_{p}})\left\{ \begin{array}{rclclll} -\triangle_{p}u &=& \lambda m_{p}(x)|u|^{p-2}u &\hbox{ in }\Omega,\\ u &=& 0 &\hbox{ on } \partial \Omega \end{array} \right. \end{equation*} Moreover, after multiplying our equation $(P_{\lambda, \mu})$ by $1/\mu$ and then letting $\mu\rightarrow +\infty$, we obtain the $(q-1)$-homogeneous equation: \begin{equation*} (P_{\lambda, m_{q}})\left\{ \begin{array}{rclclll} -\triangle_{q}u &=& \lambda m_{q}(x)|u|^{q-2}u &\hbox{ in }\Omega,\\ u &=& 0 &\hbox{ on } \partial \Omega \end{array} \right. \end{equation*} Nonlinear eigenvalue problem $(P_{\lambda, m_{r}})$, where $r=p,q$ and with bounded weight have been studied by several authors, for example (see \cite{A, C, G, O, S, Struwe}). These works proved that there exists a first eigenvalue $\lambda_{1}(r,m_{r})>0,$ where \begin{equation} \label{equ0} \lambda_{1}(r, m_{r}):=\inf\left\{\frac{1}{r}\int_{\Omega}|\nabla u|^{r}dx; u\in W^{1,r}_{0}(\Omega) \mbox{ and }\frac{1}{r}\int_{\Omega}m_{r}(x)|u|^{r}dx=1\right\}, \end{equation} which is simple in the sense that two eigenfunctions corresponding to it are proportional. Moreover, the corresponding first eigenfunction $\phi_{1}(r, m_{r})$ can be assumed to be positive. It was also shown (see \cite{A}) that $\lambda_{1}(r, m_{r})$ is simple and isolated. Recently, the problem $(P_{\lambda, m_{r}})$ with singular weight $m_{r}$ satisfying the conditions (H1), (H2), (H3) or (H4), was studied in \cite{M}. The authors use the Hardy-Sobolev inequality to characterize the first eigenvalue. In some cases it is shown that $\lambda_{1}(r, m_{r})>0$ is positive simple, isolated and has a nonnegative corresponding eigenfunction $\phi_{1}(r, m_{r})\in L^{\infty}(\Omega)$. Higher eigenvalues, in particular the second one, are also determined. The plan of this paper is the following. In Section 2, which has a preliminary character, we collect some results concerning the first eigenvalue $\lambda_{1}(r, m_{r})$ of problem $(P_{\lambda, m_{r}})$, where $r=p,q$. In Section 3, we study Rayleigh quotient for our problem $(P_{\lambda, \mu})$. In contrast to homogeneous case, we prove that if $\lambda_{1}(p, m_{p})\neq \lambda_{1}(q, m_{q})$ or $\phi_{1}(p, m_{p})\neq k\phi_{1}(q, m_{q})$ for every $k>0$, then the infimum in Rayleigh quotient is not attained. We also show nonexistence results for positive solutions of the eigenvalue problem $(P_{\lambda, \mu})$ formulated as Theorem \ref{thm2}. Our existence results for positive solutions of the eigenvalue problem $(P_{\lambda, \mu})$ are presented in Section 4. After studying the non-resonant cases (Theorem \ref{thm21}) which prove that when $\mu>0$ there exists an interval of positive eigenvalues for the problem $(P_{\lambda, \mu})$, we present the resonant cases in Theorem \ref{thm3}. \section{Preliminaries} Throughout this paper $\Omega$ will be a bounded domain of $\mathbb{R}^{N}$ with piecewise $C^{1}$ boundary,\\ $1 0 $ is a constant. In the case $s=r=p,q$, no regularity on $\partial\Omega$ is needed. \end{lem} We give now an example of the weight $m_{r}$ such that $m_{r}\delta^{\tau}\in L^{a}(\Omega)$ with $m_{r}^{+}\not\equiv 0$, where $a$, $\tau$ and $r$ satisfying the condition (H2). \begin{example}\label{Ex1} \rm The weight $m_{r}(x)=\delta(x)^{-\beta}= (1 - |x|)^{-\beta}$ is admissible in the open unit ball of $\mathbb{R}^{N}$ (i.e. $\Omega=B_{1}(0)$). For $1/2<\beta<25/42,$ $p=3/2,$ $N=3,$ $\tau=1/2$ and $a=21/2,$ we have $m_{r}\not\in L^{N/r}(\Omega)=L^{2}(\Omega),$ but $m_{r}\delta^{\tau}\in L^{a}(\Omega)= L^{21/2}(\Omega)$. \end{example} To use Harnack inequality as in \cite{M} and \cite{Tr}, we make now the following definitions involving locally integrable weights. Let $\epsilon(\rho)$ be a smooth function defined for $\rho > 0$ such that \begin{equation}\label{equ2} \lim_{\rho \rightarrow 0^{+}}\epsilon(\rho)=0 \mbox{ and } \int_{0}^{\rho*}\frac{\epsilon(\rho)}{\rho}d\rho<\infty, \end{equation} for some $\rho*>0.$ We denote by $K_{x_{0}}(\rho)$ an $N$-dimensional cube contained in $\Omega$ whose edges are of length $\rho$ and are parallel to the coordinate axes. We define \begin{equation}\label{equ3} L^{t}_{\epsilon(\rho)}(\Omega)= \{u\in L^{t}(\Omega): \|u\|_{t,\epsilon(\rho),\Omega}<\infty\}, \mbox{ where } \|u\|_{t,\epsilon(\rho),\Omega}= \sup_{x_{0}\in \Omega, \rho>0}\frac{\|u\|_{L^{t}(K_{x_{0}}(\rho)\cap \Omega)}}{\epsilon(\rho)}. \end{equation} \begin{rk} \rm The weight $m_{r}$ in Example \ref{Ex1} is such that $m_{r}\in L^{N/r}_{\epsilon(\rho)}(\Omega)$, but $m_{r} \not\in L^{s}(\Omega)$ for $s>N/r$ if $1N$ or $m_{r}\in L^{N/p}_{\epsilon(\rho)}(\Omega)$ for $10$ we deduce that $v=su$ is a solution for problem $(P_{\lambda, \mu =s^{p-q}})$.\\ Conversely, let $u$ be a solution of problem $(P_{\lambda, \mu})$. Then it follows that $v=\mu^{1/q-p}u$ is a solution of $(P_{\lambda, 1})$. \end{rk} We introduce now the functionals $A$ and $B$ on $W^{1,p}_{0}(\Omega)$ by \begin{equation}\label{equ11} A(u):=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx+\frac{1}{q}\int_{\Omega}|\nabla u|^{q}dx \end{equation} \begin{equation}\label{equ12} B(u):=\frac{1}{p}\int_{\Omega}m_{p}(x)|u|^{p}dx+\frac{1}{q}\int_{\Omega}m_{q}(x)|u|^{q}dx \end{equation} for all $u\in W^{1,p}_{0}(\Omega)$. \begin{prop}\label{prop1} (i) The functional $A$ is well defined and sequently weakly lower semi-continuous.\\ (ii) If $m_{r}\delta^{\tau}\in L^{a}(\Omega)$ and $m_{r}^{+}\not \equiv 0$ ($r=p,q$), where $a$, $r$ and $\tau$ satisfy one of the conditions (H1), (H2), (H3) or (H4), then the functional $B$ are also well defined and weakly continuous. \end{prop} \begin{proof} (i) The functional $A$ is well defined. indeed, since $\Omega$ bounded and $q0$ is a constant that may differ in each case except if $a=\infty$, $C=C(N,r)>0$.\\ Let us now show that $B$ is weakly continuous. If $u_{n}\rightarrow u$ weakly in $W^{1, p}_{0}(\Omega)$, up to a subsequence, $u_{n}\rightarrow u$ strongly in $L^{r}(\Omega)$ and $|u_{n}|^{r-1}\rightarrow |u|^{r-1}$ strongly in $L^{r/r-1}(\Omega)$ with $r=p,q$, because $\|\nabla u_{n}\|_{p}$ is bounded and the embedding $W^{1, p}_{0}(\Omega)\subset L^{r}(\Omega)$ is compact. Hence by H\"{o}lder's inequality, we have \begin{align*} |B(u_{n})-B(u)|&\leq\frac{1}{p}\left|\int_{\Omega}m_{p}(x)(|u_{n}|^{p}-|u|^{p})dx\right|+\frac{1}{q}\left|\int_{\Omega}m_{q}(x)(|u_{n}|^{q}-|u|^{q})dx\right|\\ &\leq C_{p}\|m_{p}\delta^{\tau}\|_{a}\left\|\frac{|u_{n}|+|u|}{\delta^{\tau}}\right\|_{b(p)}\left\||u_{n}|^{p-1}-|u|^{p-1}\right\|_{p/p-1}\\&+ C_{q}\|m_{q}\delta^{\tau}\|_{a}\left\|\frac{|u_{n}|+|u|}{\delta^{\tau}}\right\|_{b(q)}\left\||u_{n}|^{q-1}-|u|^{q-1}\right\|_{q/q-1}\\& \rightarrow 0. \end{align*} because $\left\||u_{n}|^{r-1}-|u|^{r-1}\right\|_{r/r-1}\rightarrow 0$ and under (H1), (H2), (H3) or (H4) the norms $\|m_{r}\delta^{\tau}\|_{a}$ and $\left\|\frac{|u_{n}|+|u|}{\delta^{\tau}}\right\|_{b(r)}$ are bounded. The constant $C(r)=C/r>0$, where $C$ comes from the inequality $|\alpha^{r}-\beta^{r}|\leq C(\alpha +\beta)|\alpha^{r-1}-\beta^{r-1}|$ for positive numbers $\alpha$ and $\beta$. To be precise, $C=1$ if $r\geq 2$ and $C>r/(r-1)$ if $10\right\}. \end{equation} \begin{prop}\label{prop2} One assumes the same conditions as for Theorem \ref{thm1}. \\If $\lambda_{1}(p, m_{p})\neq \lambda_{1}(q, m_{q})$ or $\phi_{1}(p, m_{p})\neq k\phi_{1}(q, m_{q})$, for every $k>0$. Then the infimum in \ref{equ} is not attained. \end{prop} For the proof of Proposition \ref{prop2}, we will need to use the following lemma. \begin{lem} \label{L2} The infimum in \ref{equ} verifies $$\lambda^{*}=\min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}$$ \end{lem} \begin{proof} For sufficiently large $k>0$, using \eqref{equ11} et \eqref{equ12}, we have $$B(k\phi_{1}(p,m_{p}))= k^{q}\left(k^{p-q}+\frac{1}{q}\int_{\Omega}m_{q}(x)\phi_{1}^{q}(p,m_{p})dx\right)>0.$$ and $$A(k\phi_{1}(p,m_{p}))=k^{p}\left(\lambda_{1}(p, m_{p})+\frac{1}{q}k^{q-p}\int_{\Omega}|\nabla \phi_{1}(p,m_{p})|^{q}dx\right).$$ By \eqref{equ}, we find \begin{align*} \lambda^{*}&\leq\frac{A(k\phi_{1}(p,m_{p}))}{B(k\phi_{1}(p,m_{p}))}\\&=\frac{\lambda_{1}(p, m_{p})+\frac{1}{q}k^{q-p}\int_{\Omega}|\nabla \phi_{1}(p,m_{p})|^{q}dx}{1+\frac{1}{q}k^{q-p}\int_{\Omega}m_{q}(x)\phi_{1}^{q}(p,m_{p})dx}\\ &\rightarrow\lambda_{1}(p, m_{p}) \mbox{ as }k\rightarrow +\infty, \mbox{ because } q0$ and $$\frac{A(u)}{B(u)}<\min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}.$$ We distinguish three cases.\\ \textbf{Case (i)}: Suppose that $\int_{\Omega}m_{p}|u|^{p}dx>0$ and $\int_{\Omega}m_{q}|u|^{q}dx\leq 0.$ There hold $pB(u)\leq \int_{\Omega}m_{p}|u|^{p}dx$ and $pA(u)\geq\|\nabla u\|_{p}^{p}$. Using the definition of $\lambda_{1}(p, m_{p})$, we arrive at the contradiction. \begin{equation}\label{equ13} \min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}>\frac{A(u)}{B(u)}\geq\frac{\|\nabla u\|_{p}^{p}}{\int_{\Omega}m_{p}|u|^{p}dx}\geq \lambda_{1}(p, m_{p}). \end{equation} \textbf{Case (ii)}: Suppose that $\int_{\Omega}m_{p}|u|^{p}dx\leq0$ and $\int_{\Omega}m_{q}|u|^{q}dx >0.$ Using the definition of $\lambda_{1}(q, m_{q})$, we also arrive at contradiction \begin{equation}\label{equ14} \min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}>\frac{A(u)}{B(u)}\geq\frac{\|\nabla u\|_{q}^{q}}{\int_{\Omega}m_{q}|u|^{q}dx}\geq \lambda_{1}(q, m_{q}). \end{equation} \textbf{Case (iii)}: Suppose now that $\int_{\Omega}m_{p}|u|^{p}dx>0$ and $\int_{\Omega}m_{q}|u|^{q}dx > 0.$ It follows from the definition of $\lambda_{1}(r, m_{r}),$ where $r=p,q$ that $$\|\nabla u\|_{r}^{r}\geq \lambda_{1}(r, m_{r})\int_{\Omega}m_{r}|u|^{r}dx.$$ Hence we get \begin{equation}\label{equ15} \begin{aligned} A(u)&\geq \frac{\lambda_{1}(p, m_{p})}{p}\int_{\Omega}m_{p}|u|^{p}dx + \frac{\lambda_{1}(q, m_{q})}{q}\int_{\Omega}m_{q}|u|^{q}dx\\&\geq\min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}B(u). \end{aligned} \end{equation} Against the assumption in our reasoning by contradiction. \end{proof} \begin{proof}[Proof of Proposition \ref{prop2}.] By contradiction, we suppose that there exists $ u\in W^{1,p}_{0}(\Omega)$ such that $B(u)>0$ and $\frac{A(u)}{B(u)}=\lambda^{*}$. Using Lemma \ref{L2}, we give \begin{equation}\label{equ16} \frac{A(u)}{B(u)}=\lambda^{*}=\min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}. \end{equation} We argue by considering the three cases in the proof of Lemma \ref{L2}.\\ \textbf{Case (i)}: By \eqref{equ13}, \eqref{equ16} and $\int_{\Omega}m_{q}|u|^{q}dx \leq 0,$ we have $$\lambda^{*}=\frac{A(u)}{B(u)}\geq \frac{\|\nabla u\|_{p}^{p}+\frac{p}{q}\|\nabla u\|_{q}^{q}}{\int_{\Omega}m_{p}|u|^{p}dx}\geq\frac{\|\nabla u\|_{p}^{p}}{\int_{\Omega}m_{p}|u|^{p}dx}\geq \lambda_{1}(p, m_{p})\geq\lambda^{*}.$$ We deduce that $$\|\nabla u\|_{p}^{p}=\lambda_{1}(p, m_{p})\int_{\Omega}m_{p}|u|^{p}dx \mbox{ and } \|\nabla u\|_{q}=0.$$ This contradicts the fact that $u\neq 0$.\\ \textbf{Case (ii):} similarly, By \eqref{equ14}, \eqref{equ16} and $\int_{\Omega}m_{p}|u|^{p}dx \leq 0,$ we get $$\|\nabla u\|_{q}^{q}=\lambda_{1}(q, m_{q})\int_{\Omega}m_{q}|u|^{q}dx \mbox{ and } \|\nabla u\|_{p}=0.$$ Which contradicts $u\neq 0$.\\ \textbf{Case (iii):} In this case, using \eqref{equ15} and \eqref{equ16}, we find $$A(u)= \lambda^{*}B(u)= \frac{\lambda_{1}(p, m_{p})}{p}\int_{\Omega}m_{p}|u|^{p}dx + \frac{\lambda_{1}(q, m_{q})}{q}\int_{\Omega}m_{q}|u|^{q}dx. $$ It follows $$[\lambda_{1}(p, m_{p})-\lambda^{*}]\int_{\Omega}m_{p}|u|^{p}dx + [\lambda_{1}(q, m_{q})-\lambda^{*}]\int_{\Omega}m_{q}|u|^{q}dx=0. $$ Since $\int_{\Omega}m_{p}|u|^{p}dx> 0$, $\int_{\Omega}m_{q}|u|^{q}dx>0$ and $\lambda^{*}=\min\{\lambda_{1}(p, m_{p}),\lambda_{1}(q, m_{q})\}$, we have $$\lambda^{*}=\lambda_{1}(p, m_{p})=\lambda_{1}(q, m_{q}).$$ We deduce that $$\frac{\|\nabla u\|_{p}^{p} }{\int_{\Omega}m_{p}|u|^{p}dx}=\lambda_{1}(p, m_{p})=\lambda_{1}(q, m_{q})=\frac{\|\nabla u\|_{q}^{q} }{\int_{\Omega}m_{q}|u|^{q}dx}.$$ Hence, the simplicity of eigenvalue $\lambda_{1}(r, m_{r})$ (for $r=p,q$), given by Theorem \ref{thm1}, guarantees that $u=t\phi_{1}(p, m_{p})=s\phi_{1}(q, m_{q})$ for some $t\neq 0 $ and $s\neq 0$. The hypothesis of proposition is thus contradicted. \end{proof} \subsection{Non-existence results} This subsection studies a non-existence results for the problem $(P_{\lambda, 1})$ , so for the problem $(P_{\lambda, \mu})$. This work is inspired from \cite{MT}. The following theorem is the main result of this section. \begin{thm} \label{thm2} One assumes the same conditions as for Theorem \ref{thm1}. \begin{enumerate} \item If it holds $0<\lambda<\lambda^{*}$, then the problem $(P_{\lambda, 1})$ has no non-trivial solutions. \item Moreover, if one of the following conditions holds \begin{itemize} \item [(i)] $\lambda_{1}(p, m_{p})\neq \lambda_{1}(q, m_{q})$; \item [(ii)] $\phi_{1}(p, m_{p})\neq k\phi_{1}(q, m_{q})$, for every $k>0,$ \end{itemize} then the problem $(P_{\lambda, 1})$, with $\lambda=\lambda^{*}$ has no non-trivial solutions. \end{enumerate} \end{thm} \begin{rk}\rm It is easy to see that if $\lambda_{1}(p, m_{p})=\lambda_{1}(q, m_{q})$ and $\phi_{1}(p, m_{p})= k\phi_{1}(q, m_{q})$, for some $k>0,$ then $\phi_{1}(p, m_{p})$ and $\phi_{1}(q, m_{q})$ are positive solutions of problem $(P_{\lambda, 1})$, with $\lambda=\lambda_{1}(p, m_{p})= \lambda_{1}(q, m_{q})$. \end{rk} \begin{proof}[Proof of Theorem \ref{thm2}.] Assume by contradiction that there exists a non-trivial solution $u$ of problem $(P_{\lambda, 1}).$ Then, for every $s>0$, we have that $v=su$ is a non-trivial solution of problem $(P_{\lambda, s^{p-q}})$ (see Remark \ref{rk1}). Choose $s^{p-q}=p/q$ and then act with $su$ as test function on the problem $(P_{\lambda, s^{p-q}})$. We arrive at \begin{equation}\label{equ17} 00$). \begin{thm} \label{thm21} In addition to the hypotheses of Theorem \ref{thm1} one supposes that $\lambda_{1}(p, m_{p})\neq \lambda_{1}(q, m_{q})$. If $$\min\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\}<\lambda <\max\{\lambda_{1}(p, m_{p}), \lambda_{1}(q, m_{q})\},$$ then the problem $(P_{\lambda, 1})$ has at least one positive solution. \end{thm} \begin{rk} \label{rk20}\rm The proof of Theorem \ref{thm21} reduces to provide a non-trivial critical point of the functional $I_{\lambda, m_{p}, m_{q}}$ defined for all $ u\in W^{1,p}_{0}(\Omega)$ by $$I_{\lambda, m_{p}, m_{q}}(u):=A(u)-\lambda B(u^{+}),$$ where $u^{+}=\max\{u,0\}$ and $A,$ $B$ are the functionals defined by \eqref{equ11} and \eqref{equ12}. This non-trivial critical point $u$ of $I_{\lambda, m_{p}, m_{q}}$ is a non-negative solution of the problem $(P_{\lambda, 1})$. We can check that $u\in L^{\infty}(\Omega)$ (see Remark 1.7 in \cite{M}). Then the Harnack inequality of \cite{Tr} can be applied to ensure positiveness of $u$. The argument will be separately developed in two cases: \\ (a) $\lambda_{1}(q, m_{q})<\lambda <\lambda_{1}(p, m_{p})$.\\ (b)$\lambda_{1}(p, m_{p})<\lambda <\lambda_{1}(q, m_{q})$.\\ In case (a), we apply the minimum principle and in case (b), we use the mountain pass theorem. \end{rk} \textbf{Proof of case (a).} By Proposition \ref{prop1}, $A$ is sequently weakly lower semi-continuous and $B$ is weakly continuous. It follows that $I_{\lambda, m_{p}, m_{q}}$ is sequently weakly lower semi-continuous. It is remains to show that $I_{\lambda, m_{p}, m_{q}}$ is coercive and bounded from below.\\ We distinguish two cases:\\ (i) For $u\in W_{0}^{1,p}(\Omega)$ such that $\int_{\Omega}m_{p}(u^{+})^{p}dx\leq 0$. A calculation similar to that in the proof of Proposition \ref{prop1} for $r=q$ gives \begin{equation} \label{equ175} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq \frac{1}{p}\|\nabla u\|^{p}_{p}-\frac{\lambda}{q}\|m_{q}\delta^{\tau}\|_{a}\left\|\frac{u^{+}}{\delta^{\tau}}\right\|_{b(q)}\left\|(u^{+})^{q-1}\right\|_{q/q-1}\\&\geq \frac{1}{p}\|\nabla u\|^{p}_{p}-\frac{C\lambda}{q}\|m_{q}\delta^{\tau}\|_{a}\|\nabla u\|_{q}\|u\|_{q}^{q-1}\\&\geq \frac{1}{p}\|\nabla u\|^{p}_{p}-\frac{CC'\lambda}{q}\|m_{q}\delta^{\tau}\|_{a}\|\nabla u\|_{q}^{q}\\&\geq \frac{1}{p}\|\nabla u\|^{p}_{p}-\frac{CC'C''\lambda}{q }\|m_{q}\delta^{\tau}\|_{a}\|\nabla u\|^{q}_{p}, \end{aligned} \end{equation} where $C, C', C''>0$ are the constants given respectively by the Hardy-Sobolev inequality (see Lemma \ref{L1}), the compact embedding $W^{1, q}_{0}(\Omega)\subset L^{q}(\Omega)$ and the continuous embedding $W^{1, p}_{0}(\Omega)\subset W^{1, q}_{0}(\Omega)$.\\ (ii) For $u\in W_{0}^{1,p}(\Omega)$ such that $\int_{\Omega}m_{p}(u^{+})^{p}dx> 0$. Fix $\epsilon >0$ such that \begin{equation} \label{equ18} (1-\epsilon)\lambda_{1}(p,m_{p})>\lambda, \end{equation} which is possible due to the assumption in case (a). By the definition of $\lambda_{1}(p,m_{p})$ we have $$\|\nabla u^{+}\|^{p}_{p}\geq \lambda_{1}(p,m_{p})\int_{\Omega}m_{p}(u^{+})^{p}dx.$$ Then taking into account \eqref{equ18}, we derive \begin{equation}\label{equ19} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq \frac{\epsilon}{p}\|\nabla u\|^{p}_{p}+ \frac{(1-\epsilon)\lambda_{1}(p,m_{p})-\lambda}{p}\int_{\Omega}m_{p}(u^{+})^{p}dx\\& -\frac{C\lambda}{q}\|m_{q}\delta^{\tau}\|_{a}\|\nabla u\|_{q}\|u\|_{q}^{q-1}\\&\geq \frac{1}{p}\|\nabla u\|^{p}_{p}-\frac{CC'C''\lambda}{q }\|m_{q}\delta^{\tau}\|_{a}\|\nabla u\|^{q}_{p}. \end{aligned} \end{equation} Since $q0$, we have $$I_{\lambda, m_{p}, m_{q}}(k\phi_{1}(q, m_{q}))= k^{q}\left (\frac{k^{p-q}}{p}\|\nabla \phi_{1}(q, m_{q}) \|^{p}_{p}-\frac{\lambda k^{p-q}}{p}\int_{\Omega}m_{p}\phi_{1}^{p}(q, m_{p})dx+\frac{\lambda_{1}(q, m_{q})-\lambda}{q}\right).$$ Then $I_{\lambda, m_{p}, m_{q}}(k\phi_{1}(q, m_{q}))<0$, because $\lambda_{1}(q, m_{q})<\lambda,$ which completes the proof of case (a).\\ \textbf{Proof of case (b).} We organize the proof of this case in several lemmas. In the sequel, we design by $o(1)$ a quantity tending to 0 as $n\rightarrow \infty$. \begin{lem}\label{lem1} Suppose that $m_{r}\delta^{\tau}\in L^{a}(\Omega)$ and $m_{r}^{+}\not \equiv 0$ ($r=p,q$), where $a$, $r$ and $\tau$ satisfy one of the conditions (H1), (H2), (H3) or (H4). In addition, we assume that $ m_{r}\in L^{1}(\Omega)$ for $r>N$ or $m_{r}\in L^{N/p}_{\epsilon(\rho)}(\Omega)$ for $11$, the inequality \eqref{EEE} implies the boundedness of $v_{n}$ in $W^{1,p}_{0}(\Omega)$. For a subsequence, $v_{n}\rightarrow v$ weakly in $W^{1,p}_{0}(\Omega)$. By the compact embedding $W^{1,p}_{0}(\Omega)\subset L^{r}(\Omega)$ ($r=p,q$), we have $v_{n}\rightarrow v$ strongly in $L^{r}(\Omega)$ ($r=p,q$). First we, observe that $v^{-}\equiv 0$ in $\Omega$. In fact, acting with $-u_{n}^{-}$ as test function, we have \begin{equation}\label{E} o(1)\|\nabla (u_{n}^{-})\|_{p}=\langle I'_{\lambda, m_{p}, m_{q}}(u_{n}), -u_{n}^{-}\rangle=\|\nabla (u_{n}^{-})\|_{p}^{p}+\|\nabla (u_{n}^{-})\|_{q}^{q}\geq \|\nabla (u_{n}^{-})\|_{p}^{p}. \end{equation} Because $p>1$, the inequality \eqref{E} guarantees the boundedness of $\|\nabla (u_{n}^{-})\|_{p}$ and so $\|\nabla v_{n}^{-}\|_{p}=\|\nabla (u_{n}^{-})\|_{p}/\|u_{n}\|_{p}\rightarrow 0$. Thus $v^{-}\equiv 0$ holds, hence $v\geq 0$ in $\Omega$.\\ Now, by taking $(v_{n}-v)/\|u_{n}\|_{p}^{p-1}$ as test function, we have \begin{equation}\label{E1} \begin{aligned} o(1)&= \left\langle I'_{\lambda, m_{p}, m_{q}}(u_{n}),\frac{ v_{n}-v}{\|u_{n}\|_{p}^{p-1}}\right \rangle\\&=\int_{\Omega}|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla (v_{n}- v)dx + \frac{1}{\|u_{n}\|_{p}^{p-q}}\int_{\Omega}|\nabla v_{n}|^{q-2}\nabla v_{n}\nabla (v_{n}-v)dx\\ &-\lambda\int_{\Omega}m_{p}|v_{n}|^{p-2}v_{n}(v_{n}-v)dx- \frac{\lambda}{\|u_{n}\|_{p}^{p-q}}\int_{\Omega}m_{q}|v_{n}|^{q-2}v_{n}(v_{n}-v)dx\\&=\int_{\Omega}|\nabla v_{n}|^{p-2}\nabla v_{n}\nabla (v_{n}- v)dx+ o(1), \end{aligned} \end{equation} because $q0$ in $\Omega$. This implies that $\lambda=\lambda_{1}(p, m_{p})$ because any positive eigenvalue other than $\lambda_{1}(p, m_{p})$ has no positive eigenfunctions (see Theorem \ref{thm1}). Therefore, we obtain a contradiction since we assumed $\lambda\neq\lambda_{1}(p, m_{p}).$ Hence $u_{n}$ is bounded in $W^{1,p}_{0}(\Omega)$. For a subsequence, $u_{n}\rightarrow u$ weakly in $W^{1,p}_{0}(\Omega)$ and $u_{n}\rightarrow u$ strongly in $L^{r}(\Omega)$ ($r=p,q$). We claim now that $u_{n}\rightarrow u$ strongly in $W^{1,p}_{0}(\Omega)$. It suffices to prove that $\|\nabla u_{n}\|_{p}\rightarrow\|\nabla u\|_{p}$, because $W^{1,p}_{0}(\Omega)$ is uniformly convex. It is clear that \begin{equation}\label{EPS5} \begin{aligned} o(1)=&\langle I'_{\lambda, m_{p}, m_{q}}(u_{n})-I'_{\lambda, m_{p}, m_{q}}(u),u_{n}-u\rangle\\&= \int_{\Omega}(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u|^{p-2}\nabla u)\nabla (u_{n}-u)dx\\& + \int_{\Omega}(|\nabla u_{n}|^{q-2}\nabla u_{n}-|\nabla u|^{q-2}\nabla u)\nabla (u_{n}-u)dx+o(1). \end{aligned} \end{equation} Using H\"{o}lder inequality and for $r=p,q$, we have \begin{equation}\label{EPS6} \begin{aligned} &\int_{\Omega}(|\nabla u_{n}|^{r-2}\nabla u_{n}-|\nabla u|^{r-2}\nabla u)\nabla (u_{n}-u)dx\\& = \int_{\Omega}|\nabla u_{n}|^{r}dx+\int_{\Omega}|\nabla u|^{r}dx-\int_{\Omega}|\nabla u_{n}|^{r-2}\nabla u_{n}\nabla udx-\int_{\Omega}|\nabla u|^{r-2}\nabla u\nabla u_{n}dx\\&\geq \int_{\Omega}|\nabla u_{n}|^{r}dx+\int_{\Omega}|\nabla u|^{r}dx-\left(\int_{\Omega}|\nabla u_{n}|^{r}dx\right)^{(r-1)/r}\left(\int_{\Omega}|\nabla u|^{r}dx\right)^{1/r}\\&-\left(\int_{\Omega}|\nabla u_{n}|^{r}dx\right)^{1/r}\left(\int_{\Omega}|\nabla u|^{r}dx\right)^{(r-1)/r}\\&=(\|\nabla u_{n}\|_{r}^{r-1} -\|\nabla u\|_{r}^{r-1})(\|\nabla u_{n}\|_{r} -\|\nabla u\|_{r})\\&\geq 0. \end{aligned} \end{equation} Moreover, \eqref{EPS5} and \eqref{EPS6} imply that $\|\nabla v_{n}\|_{r}\rightarrow\|\nabla v\|_{r}$ (for $r=p,q$). Thus $u_{n}\rightarrow u$ strongly in $W^{1,p}_{0}(\Omega)$. \end{proof} \begin{lem}\label{lem2} Suppose $m_{r}\delta^{\tau}\in L^{a}(\Omega)$ and $m_{r}^{+}\not \equiv 0$ ($r=p,q$), where $a$, $r$ and $\tau$ satisfy one of the conditions (H1), (H2), (H3) or (H4) If $\lambda <\lambda_{1}(q, m_{q})$, then there exist $\alpha >0$ and $\beta>0$ such that \begin{equation}\label{equ20} I_{\lambda, m_{p}, m_{q}}(u)\geq\alpha \mbox{ whenever } \|u\|_{q}=\beta. \end{equation} \end{lem} To prove the Lemma \ref{lem2}, we need the following lemma. \begin{lem} \label{lem3} Suppose $\tau$, $a$ and $p$ as in Lemma \ref{lem2} and let $b$ be such that $b=\frac{ap}{a-p}$ if $a<\infty$ and $b=p$ if $a=\infty$. Set $X(d):=\{u\in W^{1,p}_{0}(\Omega); \|\nabla u\|_{p}^{p}\leq d\|\frac{u}{\delta^{\tau}}\|_{b}\|u\|_{p}^{p-1}\}$, for $d>0$. Then there exists $\alpha(d)>0$ such that $\|\nabla u\|_{p}\leq \alpha(d)\|u\|_{q}$, for all $u\in X(d)$. \end{lem} \begin{rk}\rm Conditions (H4) implies that the Lemma \ref{lem3} includes the Lemma 11 of the previously cited paper \cite{MT}. \end{rk} \begin{proof} Suppose, by contradiction, that \begin{equation}\label{equ22} (\forall n\in \mathbb{N}^{*})~(\exists u_{n}\in X(d)):~ \frac{1}{n}\|\nabla u_{n}\|_{p}> \|u_{n}\|_{q}. \end{equation} Because of $\|\nabla u_{n}\|_{p}\neq 0$, we set $v_{n}:=\frac{u_{n}}{\|\nabla u_{n}\|_{p}}$. Thus, by \eqref{equ22}, $v_{n}\rightarrow 0$ strongly in $L^{q}(\Omega)$. Since $\|\nabla v_{n}\|_{p}=1$, the sequence $v_{n}$ is bounded in $W^{1,p}_{0}(\Omega)$. For a subsequence, $v_{n}\rightarrow v$ weakly in $W^{1,p}_{0}(\Omega)$. By the compact embedding $W^{1,p}_{0}(\Omega)\subset L^{r}(\Omega)$ ($r=p,q$), we have $v_{n}\rightarrow v$ strongly in $L^{r}(\Omega)$ ($r=p,q$). Hence, $\|v_{n}\|_{q}\rightarrow \|v\|_{q}$ and $\|v_{n}\|_{p}\rightarrow \|v\|_{q}$. By uniqueness of limit, we deduce that $\|v\|_{r}=0$. It follows that $v=0$. As $u_{n}\in X(d)$, we obtain \begin{equation}\label{equ23} \frac{1}{d}\leq\frac{\|\frac{u_{n}}{\delta^{\tau}}\|_{b}}{\|\nabla u_{n}\|_{p}}\|v_{n}\|_{p}^{p-1}. \end{equation} By Lemma \ref{L1} and under one of the properties (H1), (H2), (H3) or (H4), the sequence $\frac{\|\frac{u_{n}}{\delta^{\tau}}\|_{b}}{\|\nabla u_{n}\|_{p}}$ is bounded (see the proof of Proposition \ref{prop1}). Passing to the limit in \eqref{equ23}, we get $\frac{1}{d}\leq 0$. This contradiction completes the proof of Lemma \ref{lem3}. \end{proof} \begin{proof}[Proof of Lemma \ref{lem2}.] According to Lemma \ref{lem3}, choose \begin{equation}\label{dequ} d > \max\{1, \lambda\|m_{p}\delta^{\tau}\|_{a}, C\lambda\|m_{p}\delta^{\tau}\|_{a}\}, \end{equation} where $C>0$ is a constant that may differ in each case (H1), (H2), (H3) or (H4) (see the proof of Proposition \ref{prop1}).\\ For any $u\in X(d)$ satisfying $\int_{\Omega}m_{q}(u^{+})^{q}dx \leq 0$, there exists $\alpha(d)>0$ such that $\|\nabla u\|_{p}\leq \alpha(d)\|u\|_{q}$ and we have \begin{equation}\label{equ27} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq\frac{1-d}{p}\|\nabla u\|_{p}^{p}+\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}+\frac{d}{p}\|\nabla u\|_{p}^{p}-\frac{C\lambda\|m_{p}\delta^{\tau}\|_{a}}{p}\|\nabla u\|_{p}^{p}\\& \geq\frac{1-d}{p}\|\nabla u\|_{p}^{p}+\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}+\frac{1}{p}\|\nabla u\|_{p}^{p}\left(d-C\lambda\|m_{p}\delta^{\tau}\|_{a}\right)\\ &\geq\frac{(1-d)[\alpha(d)]^{p}}{p}\|u\|_{q}^{p}+\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}. \end{aligned} \end{equation} For any $u\not\in X(d)$ satisfying $\int_{\Omega}m_{q}(u^{+})^{q}dx \leq 0$, we have $\|\nabla u\|_{p}^{p}> d\|\frac{u}{\delta^{\tau}}\|_{b}\|u\|_{p}^{p-1}.$ It follows that \begin{equation}\label{equ29} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq\left(\frac{d-\lambda\|m_{p}\delta^{\tau}\|_{a}}{p}\right)\left\|\frac{u}{\delta^{\tau}}\right\|_{b}\|u\|_{p}^{p-1} +\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}\\&\geq\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}. \end{aligned} \end{equation} If $u\in W^{1,p}_{0}(\Omega)$ satisfying $\int_{\Omega}m_{q}(u^{+})^{q}dx >0$, by the definition of $\lambda_{1}(q, m_{q})$, we get \begin{equation}\label{equ24} \|\nabla u\|_{q}^{q}\geq\|\nabla u^{+}\|_{q}^{q}\geq\lambda_{1}(q, m_{q})\int_{\Omega}m_{q}(u^{+})^{q}dx. \end{equation} Our assumption in $\lambda$ enables us to fix $0<\epsilon <1$ with \begin{equation}\label{equ25} (1-\epsilon)\lambda_{1}(q, m_{q})> \lambda. \end{equation} If in addition $u\not\in X(d)$, then due to \eqref{dequ}, \eqref{equ24} and \eqref{equ25} we have \begin{equation}\label{equ26} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq\left(\frac{d-\lambda\|m_{p}\delta^{\tau}\|_{a}}{p}\right)\left\|\frac{u}{\delta^{\tau}}\right\|_{b}\|u\|_{p}^{p-1} +\frac{\epsilon}{q}\|\nabla u\|_{q}^{q}+\frac{1}{q}[(1-\epsilon)\lambda_{1}(q, m_{q})-\lambda]\int_{\Omega}m_{q}(u^{+})^{q}dx\\&\geq\frac{\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}. \end{aligned} \end{equation} Finally, if $u\in X(d)$ and $\int_{\Omega}m_{q}(u^{+})^{q}dx >0,$ then \eqref{dequ}, \eqref{equ24} and \eqref{equ25} imply \begin{equation}\label{equ28} \begin{aligned} I_{\lambda, m_{p}, m_{q}}(u)&\geq\frac{1-d}{p}\|\nabla u\|_{p}^{p} +\frac{\epsilon}{q}\|\nabla u\|_{q}^{q}+\frac{1}{q}[(1-\epsilon)\lambda_{1}(q, m_{q})-\lambda]\int_{\Omega}m_{q}(u^{+})^{q}dx\\&+\frac{d}{p}\|\nabla u\|_{p}^{p}-\frac{\lambda\|m_{p}\delta^{\tau}\|_{a}}{p}\left\|\frac{u}{\delta^{\tau}}\right\|_{b}\|u\|_{p}^{p-1}\\& \geq\frac{(1-d)[\alpha(p)]^{p}}{p}\|u\|_{q}^{p}+\frac{\epsilon\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}+\frac{d-C\lambda\|m_{p}\delta^{\tau}\|_{a}}{p}\|\nabla u\|_{p}^{p}\\&\geq\frac{(1-d)[\alpha(p)]^{p}}{p}\|u\|_{q}^{p}+\frac{\epsilon\lambda_{1}(q,1)}{q}\|u\|_{q}^{q}. \end{aligned} \end{equation} Using that $q0$ such that \begin{equation}\label{equ21} \|R\phi_{1}(p, m_{p})\|_{q}>\beta \mbox{ and } I_{\lambda, m_{p}, m_{q}}(R\phi_{1}(p, m_{p}))<0, \end{equation} where $\beta > 0$ is the constant in \eqref{equ20}. \end{lem} \begin{proof} For sufficiently large $R>0$, taking into account that $q 0$ in \eqref{equ20}, namely $c:=\inf\limits_{\gamma \in \Sigma}\max\limits_{t\in [0,1]}I_{\lambda, m_{p}, m_{q}}(\gamma(t)),$ where $\Sigma :=\{\gamma\in C([0,1], W^{1,p}_{0}(\Omega)); \gamma(0)=0, \gamma(1)=R\phi_{1}(p, m_{p})\}$. This completes the proof of Theorem \ref{thm21}. \subsection{Resonant cases} In this section, we study the existence result for problem $(P_{\lambda, 1})$ (or $(P_{\lambda, \mu})$) in the resonant cases. The following theorem is our main result in this section. \begin{thm}\label{thm3} One assumes the same conditions as for Theorem \ref{thm1}. If one of the following assertions holds \begin{itemize} \item [(i)] $\lambda=\lambda_{1}(p,m_{p})>\lambda_{1}(q,m_{q})$ and $\int_{\Omega}|\nabla \phi_{1}(p,m_{p})|^{q}dx-\lambda\int_{\Omega}m_{q}| \phi_{1}(p,m_{p})|^{q}dx>0$; \item [(ii)] $\lambda=\lambda_{1}(q,m_{q})>\lambda_{1}(p,m_{p})$ and $\int_{\Omega}|\nabla \phi_{1}(q,m_{q})|^{p}dx-\lambda\int_{\Omega}m_{p}| \phi_{1}(q,m_{q})|^{p}dx>0,$ \end{itemize} then the problem $(P_{\lambda, 1})$ (or $(P_{\lambda, \mu})$) has at least one positive solution. \end{thm} \begin{proof} Proof of case (i): Since $m_{p}^{+}\not\equiv 0$, the Lebesgue mesure of $\{x\in \Omega; m_{p}(x)>0\}$ is positive. Thus there exists an $n_{0}\in\mathbb{N}$ such that $(m_{p}-1/n_{0})\not\equiv 0$. For $n\geq n_{0}$, we define, as in \cite{MT}, the functional $I_{n}$ on $W^{1,p}_{0}(\Omega)$ by $$I_{n}(u)=I_{\lambda, m_{p}, m_{q}}(u)+\frac{\lambda}{pn}\|u^{+}\|_{p}^{p}=I_{\lambda, m_{p}-1/n, m_{q}}(u).$$ Using the strict monotonicity of the first eigenvalue of problem $(P_{\lambda, m_{p}})$ (see \cite{M}), we obtain $\lambda_{1}(p,m_{p}-1/n)>\lambda_{1}(p,m_{p})=\lambda.$ Thus we are able to apply Theorem \ref{thm2} obtaining a positive solution $u_{n}$ of the problem \begin{equation*} \left\{ \begin{array}{rclclll} -\triangle_{p}u-\triangle_{q}u &=& \lambda[ (m_{p}(x)-1/n)|u|^{p-2}u+ m_{q}(x)|u|^{q-2}u] &\hbox{ in }\Omega,\\ u &=& 0 &\hbox{ on } \partial \Omega. \end{array} \right. \end{equation*} We may assume that $u_{n}$ is a global minimizer of $I_{n}$ and $I_{n}(u_{n})<0$ (see the case (a) in the proof of Theorem \ref{thm2}). In addition, observing that $I_{n}\leq I_{n_{0}}$ provided $n\geq n_{0}$, we infer that for all $n\geq n_{0}$, $$I_{n}(u_{n})=\min_{W^{1,p}_{0}(\Omega)}I_{n}\leq I_{n}(u_{n_{0}})\leq I_{n_{0}}(u_{n_{0}})<0.$$ We claim that if $u_{n}$ is bounded in $W^{1,p}_{0}(\Omega)$, then $u_{n}$ is a bounded Palais-Smale sequence of $I_{\lambda, m_{p}, m_{q}}$. Indeed, there exists $c\in\mathbb{R}$ such that $I_{\lambda, m_{p}, m_{q}}(u_{n})\rightarrow c$ because $I_{n}(u_{n})$ is a convergent sequence and $\|\nabla u_{n}\|_{p}$ is bounded. On the other hand, since $I'_{n}(u_{n})=0$, we have $$\|I'_{\lambda, m_{p}, m_{q}}(u_{n})\|_{(W^{1,p}_{0}(\Omega))^{*}}=\|I'_{\lambda, m_{p}, m_{q}}(u_{n})-I'_{n}(u_{n})\|_{(W^{1,p}_{0}(\Omega))^{*}} \leq \frac{\lambda[\lambda_{1}(p,1)]^{-1/p}}{n}\|\nabla (u_{n})^{+}\|^{p-1}_{p}.$$ As $\|\nabla u_{n}\|_{p}$ is bounded, we obtain $\|I'_{\lambda}(u_{n}\|_{(W^{1,p}_{0}(\Omega))^{*}}\rightarrow 0$. This completes the proof of our claim.\\ We prove now the boundedness of $u_{n}$ in $W^{1,p}_{0}(\Omega)$ by way of contradiction. Suppose that $\|\nabla u_{n}\|_{p}\rightarrow \infty$ and let $v_{n}:=u_{n}/\|\nabla u_{n}\|_{p}$. For a subsequence, $v_{n}\rightarrow v$ weakly in $W^{1,p}_{0}(\Omega)$ and $v_{n}\rightarrow v$ strongly in $L^{p}(\Omega)$. Following the same steps of the proof of Lemma \ref{lem1}, we show that $v$ is a positive solution of problem $(P_{\lambda,m_{p}})$. This entails $v$ is a positive eigenfunction corresponding to $\lambda_{1}(p, m_{p})$. Thus the simplicity of $\lambda_{1}(p, m_{p})$ implies that $v=\phi_{1}(p, m_{p})$. The facts that $I_{n}(u_{n})<0$ for all $n\geq n_{0}$ and $u_{n}$ is a critical point of $I_{n}$ result in $$ 0>\frac{I_{n}(u_{n})}{\|\nabla u_{n}\|_{p}^{q}}=\left(\frac{1}{q}-\frac{1}{p}\right)\left (\|\nabla v_{n}\|_{q}^{q}-\lambda\int_{\Omega}m_{q}v_{n}^{q}dx\right). $$ Passing to the limit, we obtain $\int_{\Omega}|\nabla \phi_{1}(p,m_{p})|^{q}dx-\lambda\int_{\Omega}m_{q}| \phi_{1}(p,m_{p})|^{q}dx\leq 0$ which contradicts second point of assertion (i). Thus $u_{n}$ is a bounded Palais-Smale sequence of $I_{\lambda, m_{p}, m_{q}}$. Since $\lambda\neq\lambda_{1}(p, m_{p})$, $I_{\lambda, m_{p}, m_{q}}$ satisfies Pais-Smale condition (see Lemma \ref{lem1}). It follows that $u_{n}$ has a subsequence converging to some critical point $u_{0}$ of $I_{\lambda, m_{p}, m_{q}}$.\\ We note that $u_{0}\neq 0$ because $I_{\lambda, m_{p}, m_{q}}(u_{0})=\lim\limits_{n}I_{n}(u_{n})\leq I_{n_{0}}(u_{0})<0.$ Therefore $u_{0}$ is a positive solution of problem $(P_{\lambda, 1})$ (see Remark \ref{rk20}). Proof of case (ii): As in the proof of case (i), we can choose $n_{0}\in \mathbb{N}$ such that $(m_{q}-1/n_{0})\neq 0$. For $n\geq n_{0}$, we define the functional $J_{n}$ on $W^{1,p}_{0}(\Omega)$ by $$J_{n}(u)=I_{\lambda, m_{p}, m_{q}}(u)+\frac{\lambda}{pn}\|u^{+}\|_{q}^{q}=I_{\lambda, m_{p}, m_{q}-1/n}(u).$$ Using the strict monotonicity of the first eigenvalue of problem $(P_{\lambda, m_{q}})$, we obtain $\lambda_{1}(q,m_{q}-1/n)>\lambda_{1}(q,m_{q})=\lambda,$ for any $n\geq n_{0}$. Thus we may apply Theorem \ref{thm2} obtaining a positive solution $u_{n}$ of the problem \begin{equation*} \left\{ \begin{array}{rclclll} -\triangle_{p}u-\triangle_{q}u &=& \lambda[ m_{p}(x)|u|^{p-2}u+ (m_{q}(x)-1/n)|u|^{q-2}u] &\hbox{ in }\Omega,\\ u &=& 0 &\hbox{ on } \partial \Omega. \end{array} \right. \end{equation*} Following the pattern of the proof of case (i), this time proceeding as in case (b) in the proof of Theorem \ref{thm2}, we deduce that $J_{n}(u_{n})>0$ for all $n\geq n_{0}$. For the boundedness of $u_{n}$ in $W^{1,p}_{0}(\Omega)$, proceeding as in the proof of case (i) and the contradiction follows from the condition $\lambda=\lambda_{1}(q,m_{q})>\lambda_{1}(p,m_{p})$. The bounded sequence $u_{n}$ is a Palais-Smale sequence for the functional $I_{\lambda, m_{p}, m_{q}}$ as can be seen from the estimate $$\|I'_{\lambda, m_{p}, m_{q}}(u_{n})\|_{(W^{1,p}_{0}(\Omega))^{*}}=\|I'_{\lambda, m_{p}, m_{q}}(u_{n})-J'_{n}(u_{n})\|_{(W^{1,p}_{0}(\Omega))^{*}} \leq \frac{c}{n}\|u_{n}\|^{q-1}_{q},$$ where $c$ is a positive constant independent of $n$. It follows that $u_{n}$ has a subsequence converging to some critical point $u_{0}$ of $I_{\lambda, m_{p}, m_{q}}$. In order to complete the proof, due to the Harnack inequality, it suffices to justify that $u_{0}\neq 0$. We assume, by contradiction, that $u_{n}\rightarrow 0$ strongly in $W^{1,p}_{0}(\Omega)$. Set $v_{n}:=\frac{u_{n}}{\|\nabla u_{n}\|_{p}}$. Then, for a subsequence, $v_{n}\rightarrow v$ weakly in $W^{1,p}_{0}(\Omega),$ weakly in $W^{1,q}_{0}(\Omega)$ and strongly in $L^{p}(\Omega)$. It is easy to see that $v\geq 0$ in $\Omega$. Using $\frac{v_{n}-v}{\|\nabla u_{n}\|_{p}^{q-1}}$ as test function, we obtain $$0=\left\langle J'_{n}(u_{n}), \frac{v_{n}-v}{\|u_{n}\|^{q-1}}\right\rangle=\int_{\Omega}|\nabla v_{n}|^{q-2}\nabla v_{n}\nabla(v_{n}-v)dx + o(1).$$ By $(S_{+})$ property for $-\Delta_{q}$ on $W^{1,q}_{0}(\Omega)$, we have $v_{n}\rightarrow v$ strongly in $W^{1,q}_{0}(\Omega).$ We claim that $v\neq 0$ in $\Omega$. Using $u_{n}$ as test function, we may write \begin{equation}\label{EEE1} \begin{aligned} 0&=\langle J'_{n}(u_{n}), u_{n}\rangle\\&=\|\nabla u_{n}\|_{p}^{p}-\lambda\int_{\Omega}m_{p}u_{n}^{p}dx+\|\nabla u_{n}\|_{q}^{q}-\lambda\int_{\Omega}(m_{q}-\frac{1}{n})u_{n}^{q}dx\\&\geq \|\nabla u_{n}\|_{p}^{p}-\lambda\int_{\Omega}m_{p}u_{n}^{p}dx, \end{aligned} \end{equation} because $\lambda=\lambda_{1}(q, m_{q})$ and by the definition of $\lambda_{1}(q, m_{q})$, we have $\|\nabla u_{n}\|_{q}^{q}-\lambda\int_{\Omega}m_{q}u_{n}^{q}dx\geq 0$. Thus according (H1), (H2), (H3) or (H4), we have $\|\nabla u\|_{p}^{p}\leq \lambda\|m_{p}\delta^{\tau}\|_{a}\|\frac{u}{\delta^{\tau}}\|_{b}\|u\|_{p}^{p-1}$, where $\tau$, $a$ and $b$ as in Lemma \ref{lem3}. Whence $u_{n}\in X(d)$ with $d=\lambda\|m_{p}\delta^{\tau}\|_{a}$. Therefore, Lemma \ref{lem3} guarantees the existence of a constant $\alpha(d)>0$ such that $$\|\nabla u\|_{p}\leq\alpha(d)\|u_{n}\|_{q}\leq\alpha(d)[\lambda_{1}(q, 1)]^{-1/q}\|\nabla u_{n}\|_{q}, \mbox{ for all } n\geq n_{0}.$$ Consequently, recalling that $v_{n}\rightarrow v$ strongly in $W^{1,q}_{0}(\Omega),$ we have $$\|\nabla v\|_{q}=\lim \|\nabla v_{n}\|_{q}=\lim\frac{\|\nabla u_{n}\|_{q}}{\|\nabla u_{n}\|_{p}}\geq\frac{1}{\alpha(d)[\lambda_{1}(q, 1)]^{-1/q}}>0, $$ thus proving our claim.\\ For any $\varphi\in W^{1,p}(\Omega)$, by using $\frac{\varphi}{\|\nabla u_{n}\|_{p}^{q-1}}$ as test function we show, as in proof of case (i), that $v$ is a nonnegative nontrivial solution of $(P_{\lambda, m_{q}})$. The simplicity of $\lambda_{1}(q, m_{q})$ guarantees that $v=\phi_{1}(q, m_{q})$ is a positive eigenfunction corresponding to $\lambda_{1}(q, m_{q})$.\\ Using that $J_{n}(u_{n})>0$ for all $n\geq n_{0}$ in conjunction with \eqref{EEE1}, we have $$0\leq\frac{J_{n}(u_{n})}{\|\nabla u_{n}\|_{p}^{p}}=\left(\frac{1}{p}-\frac{1}{q}\right)\left(\|\nabla v_{n}\|_{p}^{p}-\lambda\int_{\Omega}m_{p}v_{n}^{p}dx\right).$$ Since $q