\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 291, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/291\hfil A singular problem with variable potentials] {Existence and non-existence of solutions for a singular problem with variable potentials} \author[K. Saoudi \hfil EJDE-2017/291\hfilneg] {Kamel Saoudi} \address{Kamel Saoudi \newline College of sciences at Dammam, University of Imam Abdulrahman Bin Faisal\\ 31441 Dammam, Saudi Arabia} \email{kasaoudi@gmail.com} \dedicatory{Communicated by Claudianor O. Alves} \thanks{Submitted March 14, 2017. Published November 21, 2017.} \subjclass[2010]{35J35, 35R05} \keywords{Singular elliptic equation; variational methods; existence; \hfill\break\indent non-existence} \begin{abstract} The purpose of this article is to prove some existence and nonexistence theorems for the inhomogeneous singular Dirichlet problem $$ - \Delta_p u = \frac{\lambda k(x)}{u^\delta}\pm h(x) u^q. $$ For proving our results we use the sub and super solution method, and monotonicity arguments. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} \label{S1} In this paper we are interested in the following quasilinear and singular problem with variable potentials: \begin{equation} \label{103} \begin{gathered} - \Delta_p u = \lambda k(x) u^{-\delta} \pm h(x) u^q \quad \text{in } \Omega,\\ u|_{\partial\Omega} = 0 ,\quad u > 0\quad\text{in }\Omega, \end{gathered} \end{equation} where $\Omega\subset\mathbb{R}^N$, $(N\geq 2)$ is a bounded domain with smooth boundary, $\lambda$ is a positive parameter, $1 < p < \infty$, $p-1 < q\leq p^{*} - 1$, and $0 < \delta < 1$. As usual, $p^{*} = \frac{Np}{N-p}$ if $1 < p < N$, $p^{*}\in (p,\infty)$ is arbitrarily large if $p = N$, and $p^{*} = \infty$ if $p > N$, and the variable weight functions $h,\, k \in L^\infty(\Omega)$ satisfy \begin{equation} \label{104} \operatorname{ess\,inf}_{ x\in\Omega} k(x) >0 \quad \text{and}\quad \operatorname{ess\,inf}_{ x\in\Omega} h(x) >0. \end{equation} Associated with problem \eqref{103} we have the singular functional $E_\lambda:$ $W^{1,p}_0(\Omega)\to\mathbb{R}$ defined by \begin{equation}\label{energy} E_{\lambda}(u) = \frac{1}{p}\int_{\Omega} |\nabla u|^p \,\mathrm{d}x - \frac{\lambda}{1-\delta}\int_{\Omega} k(x)u^{1-\delta} \,\mathrm{d}x \pm \frac{1}{q+1}\int_{\Omega}h(x) u^{q+1} \,\mathrm{d}x \end{equation} in the Sobolev space $W^{1,p}_0(\Omega)$. \begin{definition} \rm $u\in W^{1,p}_0 (\Omega)$ is called a \emph{weak} solution (or solution, for short) of problem~\eqref{103}, that is, for functions $u\in W_0^{1,p}(\Omega)$ satisfying $\operatorname{ess\,inf}_K u > 0$ over every compact set $K\subset\Omega$ and \begin{equation} \label{e1} \int_{\Omega} |\nabla u|^{p-2}\nabla u\cdot \nabla\phi \,{\rm d}x = \lambda\int_{\Omega}k(x) u^{-\delta} \phi \,{\rm d}x \pm \int_{\Omega} h(x)u^q\phi \,{\rm d}x \end{equation} for all $\phi\in C^{\infty}_\mathrm{c}(\Omega)$. As usual, $C^{\infty}_\mathrm{c}(\Omega)$ denotes the space of all $C^{\infty}$ functions $\phi\colon \Omega\to \mathbb{R}$ with compact support. \end{definition} Obviously, every critical point of $E_\lambda$ is a weak solution of the problem \eqref{103}. $\Delta_{p}u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$, where $p > 1$ is a real constant is called the $p-$Laplacian or the $p$-Laplace operator. The $p$-Laplacian is an elliptic partial differential equation, which is degenerate if $p > 2$ and singular if $p < 2$. If $p = 2$, then the $p$-Laplacian reduces to the simpler classical linear Laplace equation $\Delta u := \nabla.\nabla u$ and in the case of one spatial dimension, we have $\Delta_{p}u =(| u'|^{p-2} u')'$. The class of problems \eqref{103} appears in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mappings (for this see \cite{Mi,Uh}), in the generalized reaction-diffusion theory \cite{HeVa}, in the turbulent flow of a gas in a porous medium and in the non-Newtonian fluid theory \cite{EsVa}. In the non-Newtonian fluid theory, the quantity $p$ is the characteristic of the medium. If $p < 2$, the fluids are called pseudo-plastics, if $p = 2$, the fluids are called Newtonian, and if $p > 2$, the fluids are called dilatants. This kind of problems with convex and concave nonlinearities have been extensively studied by many authors. We refer the reader to the celebrate paper of Ambrosetti-Brezis-Cerami \cite{AmBrCe}, Saoudi \cite{Sa2}, Santos \cite{Sa} with their references therein. For $p = 2$, we refer the reader to \cite{RaRe,BrKa} and references therein. The basic work in our direction is the paper \cite{CP} where Coclite-Palmieri have been considered the nonlinear elliptic equation containing singular term \begin{equation}\label{e1.2} \begin{gathered} -\Delta u=u^{p}+\lambda u^{-\gamma}, \quad\text{in }\Omega,\\ u>0, \quad\text{in }\Omega,\\ u=0, \quad\text{on }\partial \Omega, \end{gathered} \end{equation} where $\Omega\subset R^{N}$ $(N\geq 3)$ is a bounded domain with smooth boundary $\partial \Omega$ and $\lambda$ is a positive parameter. The exponent $p$ of the sublinear satisfies $00$ small enough, and has no solution when $\lambda$ is large. We mention that in the work \cite{CP} the authors have been extended the results of Crandall-Rabinowitz-Tartar \cite{CrRaTa}. Problem \eqref{e1.2} have been also studied with different elliptic operators. We refer the reader to \cite{CP,CrRaTa,GaSa, GaSa1, GiSa1,GiScTa,LM,AA,Sa5, SaKr} and references therein. The aim of this work is to extend the results obtained in \cite{CP} to the more general problems \eqref{103}. Precisely, the main goal of this paper is to prove some existence and non-existence theorems for the non-linear singular elliptic problem \eqref{103}. Firstly, we state the following definitions. \begin{definition} \rm A function $ \underline{u}\in W^{1,p}_0(\Omega)$ is called a weak sub-solution to \eqref{103}$_+$ if $\underline{u}\in C^2(\Omega)\cap C(\overline{\Omega})$ and \begin{gather*} - \Delta_p \underline{u} \leq \lambda k(x) \underline{u}^{-\delta} + h(x) \underline{u}^q \quad \text{in } \Omega,\\ \underline{u}|_{\partial\Omega} = 0 ,\quad \underline{u} > 0\quad\text{ in }\Omega, \end{gather*} A function $ \overline{u}\in W^{1,p}_0(\Omega)$ is called a weak super-solution to \eqref{103}$_+$ if $\overline{u}\in C^2(\Omega)\cap C(\overline{\Omega})$ and \begin{gather*} - \Delta_p \overline{u} \leq \lambda k(x) \overline{u}^{-\delta} + h(x) \overline{u}^q \quad \text{in } \Omega,\\ \overline{u}|_{\partial\Omega} = 0 ,\quad \overline{u} > 0\quad\text{ in }\Omega, \end{gather*} \end{definition} \begin{definition} \rm A solution $u_\lambda$ of problem \eqref{103}$_+$ is called minimal if $u_\lambda\leq v$ almost everywhere in $\Omega$ for any further solution $v$ of problem \eqref{103}$_+$. \end{definition} We state below the results that we will prove. \begin{theorem} \label{T101} Assume $0< \delta<1$, $p-1 \Lambda^*$. \end{enumerate} \end{theorem} \begin{theorem} \label{T102} Assume $0< \delta<1$, $p-1\Lambda_*$, then problem \eqref{103}$_-$ has at least one solution; \item If $\lambda < \Lambda_*$, then problem \eqref{103}$_-$ does not have any solution. \end{enumerate} \end{theorem} A comparison between our main result (Theorems \ref{T101} and \ref{T102}) and some of those the previously cited ones, is now in order: in the present paper, we extended the main result of Giacomoni-Schindler-Tak\'{a}\v{c} \cite[Theorem 2.1]{GiScTa} to a class of perturbed singular functionals, this feature gains a remarkable importance in the applications. Moreover, it is worth noticing that, since parameter $k(x)$ and $h(x)$ in problem \eqref{103}$_\pm$, is variable, causes that the quasilinear singular problem is investigate in a complete form. On the other hand, the main difference between Theorems \ref{T101} and \ref{T102} above and the main result of R\v{a}dulescu-Repov\v{s} \cite[Theorems 1.1 and 1.2]{RaRe} in applications consists in different from two directions: one is the operator considered in this work is more general than in \cite{RaRe}, the other is with considering singular term instead of R\v{a}dulescu and Repov\v{s} in \cite{RaRe}. \section{Proof of Theorem \ref{T101}} The proof is organized in several steps. \smallskip \noindent\textbf{Step 1:} Existence of minimal solution for $0<\lambda<\Lambda^*$. Let us define \begin{equation}\label{def:Lambda1} \Lambda^*= \sup\{ \lambda > 0\colon \eqref{103}_+\text{ has a weak solution} \} \end{equation} and let $\lambda_1(\Omega, m)\equiv \lambda_1$ be the first (principal) eigenvalue of $-\Delta_p$ and let $\Phi_m$ denote an eigenfunction of $-\Delta_p$ associated to $\lambda_1$ i.e., $\Phi_m$ solves \begin{gather*} -\Delta_p \Phi_m = \lambda_1 m(x) |\Phi_m|^{p-2}\Phi_m \quad \text{in } \Omega\\ \Phi_m > 0 \quad \text{in } \Omega\\ \Phi_m = 0 \quad \text{in } \partial\Omega. \end{gather*} It is well-known that $\Phi_m$ belongs to $C^1(\overline{\Omega})$, that $\Phi_m$ may be chosen positive in $\Omega$ and that $|\nabla\Phi|$ is positive on a neighborhood of $\partial\Omega$. To show the existence of a solution to the problem \eqref{103}$_+$, we construct a well ordered pair of sub-solution $\underline{u}_\lambda$, and a super-solution $\overline{u}_\lambda$, such that $\underline{u}_\lambda \leq \overline{u}_\lambda$. To find a sub-solution, we assume that $m(x)=\min\{ k(x),h(x)\}$ and $\lambda_1\leq \lambda$. Define $\psi_c=c\Phi_m^{\frac{p}{p-1+\delta}}$. By a straightforward calculation, we have $$ \nabla \psi_c= c\Big(\frac{p}{p-1+\delta}\Big) \Phi_m ^{\frac{1-\delta}{p-1+\delta}}\nabla\Phi_m $$ and \begin{align*} &-\Delta_p(\psi_c)\\ &= -\operatorname{div}(|\nabla \psi_c|^{p-2}\nabla\psi_c)\\ & =\frac{(pc)^{p-1}(\delta-1)(p-1)}{(p-1+\delta)^p}|\nabla \Phi_m|^p \Phi_m^{\frac{-\delta p}{p-1+\delta}} +\lambda_1 \Big(\frac{pc}{p-1+\delta}\Big)^{p-1} m(x)\phi_m^p \Phi_m^{\frac{-\delta p}{p-1+\delta}} \end{align*} Thus, \begin{align*} &-\Delta_p(\psi_c) \\ &=\frac{(pc)^{p-1}(\delta-1)(p-1)}{(p-1+\delta)^p}|\nabla \Phi_m|^p\Phi_m^{\frac{-\delta p}{p-1+\delta}} +\lambda_1 m(x) \Big(\frac{pc}{p-1+\delta}\Big)^{p-1} \phi_m^p \Phi_m^{\frac{-\delta p}{p-1+\delta}} \\ & \leq m(x)\Big( \Big(\frac{p}{(p-1+\delta)^p}\Big)^p \frac{c^{p-1+\delta}(\delta-1)(p-1)}{p}|\nabla \Phi_m|^p\psi_c^{-\delta} \\ &\quad + \lambda_1 \Big(\frac{p}{p-1+\delta}\Big)^{p-1}c^{p-1-q} c^q \Phi_m^{\frac{p(p-1)}{p-1+\delta}}\Big)\\ & \leq m(x)\Big( \Big(\frac{p}{(p-1+\delta)^p}\Big)^p \frac{c^{p-1+\delta} (\delta-1)(p-1)}{p}|\nabla \Phi_m|^p\psi_c^{-\delta} \\ &\quad +\lambda_1 \Big(\frac{p}{p-1+\delta}\Big)^{p-1}c^{p-1-q} c^q \Phi_m^{\frac{pq}{p-1+\delta}}\Big)\\ & \leq m(x)\Big( \Big(\frac{p}{(p-1+\delta)^p}\Big)^p \frac{c^{p-1+\delta} (\delta-1)(p-1)}{p}|\nabla \Phi_m|^p\psi_c^{-\delta} \\ &\quad +\lambda_1 \Big(\frac{p}{p-1+\delta}\Big)^{p-1}c^{p-1-q} \psi_c^{q}\Big) \end{align*} Therefore, for $c>0$ small enough, we have \begin{align*} -\Delta_p(\psi_c)\leq m(x)\left(\lambda\psi_c^{-\delta} + \psi_c^{q}\right)\leq \lambda k(x)\psi_c^{-\delta} + h(x) \psi_c^{q} \end{align*} This shows that $\psi_c$ is a sub-solution of the problem \eqref{103}$_+$. Let us now show that problem \eqref{103}$_+$ has a super-solution. Now, we put $m(x)=\max\{ k(x),h(x)\}$ and $\lambda_1\geq \lambda$. Define $\psi_M= M \Phi_m^{\frac{p}{p-1+\delta}}$ for $M > c$ large enough. Using similar arguments as above we have $$ \nabla \psi_M= M\Big(\frac{p}{p-1+\delta}\Big) \Phi_m ^{\frac{1-\delta}{p-1+\delta}}\nabla\Phi_m $$ and \begin{align*} &-\Delta_p(\psi_M) \\ &= -\operatorname{div}(|\nabla \psi_M|^{p-2}\nabla\psi_M)\\ &=\frac{(pM)^{p-1}(\delta-1)(p-1)}{(p-1+\delta)^p}| \nabla \Phi_m|^p\Phi_m^{\frac{-\delta p}{p-1+\delta}} + \lambda_1 m(x) \Big(\frac{pM}{p-1+\delta}\Big)^{p-1}\phi_m^p \Phi_m^{\frac{-\delta p}{p-1+\delta}} \end{align*} Thus, \begin{align*} & -\Delta_p(\psi_M) \\ &=\Big(\frac{pM}{p-1+\delta}\Big)^{p-1}\Phi_m^{\frac{-\delta p}{p-1+\delta}} \Big[ \frac{(\delta-1)(p-1)}{p-1+\delta}|\nabla \Phi_m|^p+\lambda_1 m(x)\phi_m^p\Big] \\ & =\Big(\frac{pM}{p-1+\delta}\Big)^{p-1}\Phi_m^{\frac{-\delta p}{p-1+\delta}} \Big[ \frac{(\delta-1)(p-1)}{p-1+\delta}|\nabla \Phi_m|^p +\frac{\lambda_1 m(x)}{2}\phi_m^p\Big]\\ &\quad + \frac{\lambda_1 m(x)}{2}\Big(\frac{pM}{p-1+\delta}\Big)^{p-1} \Phi_m^{\frac{-\delta p}{p-1+\delta}}\phi_m^p\\ &=\Big(\frac{p}{p-1+\delta}\Big)^{p-1}M^{p-1+\delta}\left[ \frac{(\delta-1)(p-1)}{p-1+\delta}|\nabla \Phi_m|^p +\frac{\lambda_1(m) m(x)}{2}\phi_m^p\right]\psi_M^{-\delta}\\ &\quad + \frac{\lambda_1 m(x)}{2}\Big(\frac{p}{p-1+\delta}\Big)^{p-1} M^{p-1-q}\Phi_m^{\frac{p(p-1-q)}{p-1+\delta}}\psi_M^q \end{align*} Therefore, for $M>0 $ may be chosen arbitrarily large, we have \begin{align*} -\Delta_p(\psi_M)\geq m(x)\left(\lambda\psi_M^{-\delta} + \psi_M^{q}\right) \geq \lambda k(x)\psi_M^{-\delta} + h(x) \psi_M^{q} \end{align*} This shows that $\psi_M$ is a super-solution of the problem \eqref{103}$_+$. It remains to show that $\psi_c=\underline{u}_\lambda\leq \psi_M= \overline{u}_\lambda$. Therefore, for $c>0$ small enough and $M>0$ large enough, we obtain \begin{align*} &-\Delta_p(\underline{u}_\lambda) \\ &= \Big(\frac{pc}{p-1+\delta}\Big)^{p-1}\Phi_m^{\frac{-\delta p}{p-1+\delta}} \Big[ \frac{(\delta-1)(p-1)}{p-1+\delta}|\nabla \Phi_m|^p+\lambda_1 m(x)\phi_m^p\Big]\\ &\leq \Big(\frac{pM}{p-1+\delta}\Big)^{p-1}\Phi_m^{\frac{-\delta p}{p-1+\delta}} \Big[ \frac{(\delta-1)(p-1)}{p-1+\delta}|\nabla \Phi_m|^p+\lambda_1 m(x)\phi_m^p\Big] = -\Delta_p(\overline{u}_\lambda). \end{align*} Consequently, we may apply the weak comparison principle (see in \cite[Theorem 2.3]{GiScTa}) in order to conclude that $\underline{u}_\lambda\leq \overline{u}_\lambda$. Thus, By the classical iteration method \eqref{103}+ has a solution between the sub-solution and the super-solution. Let us now prove that $u_\lambda$ is a minimal weak solution of \eqref{103}$_+$. We use here the weak comparison principle (see Proposition 2.3 in Cuesta and Tak\'a\v{c} \cite{CuTa}) and the following monotone iterative scheme: \begin{equation}\label{scheme} \begin{gathered} -\Delta_pu_n-\lambda k(x) u_{n}^{-\delta}= h(x)u_{n-1}^q\quad\text{in }\Omega;\\ u_n|_{\partial\Omega}=0, \end{gathered} \end{equation} where $u_0=\underline{u}_\lambda$, according to Giacomoni, Schindler and Tak\'{a}\v{c} \cite{GiScTa}, is the unique solution to the following purely singular problem \begin{gather*} -\Delta_p u=\lambda k(x)u^{-\delta}\quad \text{ in }\Omega ,\\ u|_{\partial\Omega}=0, \quad u>0 \quad \text{ in }\Omega. \end{gather*} Note that $u_0$ is a weak subsolution to \eqref{103}$_+$ and $u_0\leq U$ where $U$ is any weak solution to \eqref{103}$_+$. Then, from the weak comparison principle, we obtain easily that $u_0 \leq u_1$ and $\{u_n\}_{n=1}^\infty$ is a nondecreasing sequence. Furthermore, $u_n \leq U$ and $\{u_n\}_{n=1}^\infty$ is uniformly bounded in $W^{1,p}_0(\Omega)$. Hence, it is easy to prove that $\{u_n\}$ converges weakly in $W^{1,p}_0(\Omega)$ and pointwise to $u_\lambda$, a weak solution to the problem \eqref{103}$_+$. Let us show that $u_\lambda$ is the minimal solution to \eqref{103}$_+$ for any $0<\lambda<\Lambda^*$. Let $v_\lambda$ a weak solution to \eqref{103}$_+$ for any $0<\lambda<\Lambda^*$. Then, $u_0 = \underline{u}_\lambda \leq v_\lambda$. From the weak comparison principle, $u_n \leq v_\lambda$ for any $n \geq 0$. Letting $n\to \infty$, we obtain $u_\lambda \leq v_\lambda$. This completes the proof of the Step 1. \smallskip \noindent\textbf{Step 2:} \eqref{103}$_+$ has no positive solution for $\lambda > \Lambda^*$. Firstly, from Step 1 we have that $\Lambda^*>0$. Now, let us show that $\Lambda^*<\infty$. We argue by contradiction: suppose there exists a sequence $\lambda_n \to \infty$ such that \eqref{103}$_+$ admits a solution $u_n$. Denote $$ m := \min \{\operatorname{ess\,inf}_{x\in\Omega} k(x), \operatorname{ess\,inf}_{x\in\Omega}\inf h(x)\} > 0. $$ There exists $\lambda^{*}>0$ such that \[ m\left(\lambda t^{-\delta} + t^q\right) \ge (\lambda_1+\epsilon) t^{p-1} \quad \text{for all } t>0,\; \epsilon\in(0,1),\; \lambda > \lambda^{*} \] where $\lambda_1$ is the first Dirichlet eigenvalue of $- \Delta_p$ is positive and is given by \begin{equation} \label{201} \lambda_1 = \min_{u \ne 0} \frac{ \int_\Omega |\nabla u|^p}{ \int_\Omega |u|^p} \end{equation} (see Lindqvist \cite{Lin1}). Choose $\lambda_n > \lambda^{*}$. Clearly $u_n$ is a supersolution of the problem \begin{equation}\label{problem-max-Lambda} \begin{gathered} -\Delta_p u = (\lambda_1 + \epsilon ) u^{p-1}\quad \text{in }\Omega;\\ u>0,\quad u|_{\partial\Omega}=0. \end{gathered} \end{equation} for all $\epsilon\in (0,1)$. We now use the \cite[Lemma 3.1]{GiScTa} to choose $\mu< \lambda_1 + \epsilon$ small enough so that $\mu \phi_1(x) < u_n (x)$ and $\mu \phi_1$ is a subsolution to problem \eqref{problem-max-Lambda}. By a monotone interation procedure we obtain a solution to \eqref{problem-max-Lambda} for any $\epsilon \in (0,1)$, contradicting the fact that $\lambda_1$ is an isolated point in the spectrum of $- \Delta_p$ in $W^{1,p}_0(\Omega)$ (see Anane \cite{An}). This proves the claim and completes the proof of the step 2. \smallskip \noindent\textbf{Step 3:} Existence of at least one positive weak solution for $\lambda=\Lambda^*$ to \eqref{103}$_+$. Let $\{\lambda_k\}_{k\in\mathbb{N}}$ such that $\lambda_k\uparrow \Lambda^*$ as $k\to \infty$. Then, from Step 1, there exists $u_k=u_{\lambda_k}\geq \underline{u}_{\lambda_k}$ to a weak positive solution to \eqref{103}$_+$ for $\lambda=\lambda_k$. Therefore, for any $\phi\in C^{\infty}_c(\Omega)$, we have: \begin{equation}\label{eq30} \int_\Omega|\nabla u_k|^{p-2}\nabla u_k\nabla\phi\,\mathrm{d}x =\lambda_k \int_\Omega k(x)u_k^{-\delta}\phi\,\mathrm{d}x +\int_\Omega h(x) u_k^q\phi\,\mathrm{d}x. \end{equation} Since $u_k\in W^{1,p}_0(\Omega)$ and $u_k\geq\underline{u}_{\lambda_k}$ it is easy to see that \eqref{eq30} holds also for $\phi\in W^{1,p}_0(\Omega)$. Moreover, from above \begin{equation}\label{eq31} E_{\lambda_k}(u_k)\leq E_{\lambda_k}(\underline{u}_{\lambda_k}) <\frac{1}{p}\int_\Omega|\nabla\underline{u}_{\lambda_k}|^p\,\mathrm{d}x- \frac{\lambda_k } {1-\delta} \int_\Omega k(x){\underline{u}_{\lambda_k}}^{1-\delta}\,\mathrm{d}x<0, \end{equation} Thus, by Sobolev imbedding and using the fact that $k,h\in L^\infty (\Omega )$ it follows that \begin{equation}\label{eq31b} \sup_k\|u_k\|_p<\infty. \end{equation} Hence, there exists $u_{\Lambda^*}\geq\underline{u}_{\lambda_k}$ such that $u_k\rightharpoonup u_{\Lambda^*}$ in $W^{1,p}_0(\Omega)$ as $k\to \infty$ and \begin{equation}\label{eq32} u_k\rightharpoonup u_{\Lambda^*}\text{ in $L^q(\Omega)$ since $p-10$ \begin{equation}\label{coerci} J_\lambda(u) \geq \frac{1}{p}\|u\|^p - C_1\|u\|^{1-\delta} + C_2\|u\|_{q+1}^{q+1} \end{equation} where $C_1 =\lambda |\Omega|^{D+E(1-\delta)}S^{\frac{\delta-1}{p}} \frac{ ||k||_{L^\infty}}{( 1-\delta)}$ with $D=\frac{q+\delta}{q+1}$, $E=\frac{p^*-q-1}{p^*(q+1)}$ and $S > 0$ is the best Sobolev constant and $C_2 = (q + 1)^{-1} \operatorname{ess\,inf}_{x\in\Omega} h(x)$ are positive constants. It follows from \eqref{coerci} that \begin{equation}\label{coerci1} J_\lambda(u) \geq \frac{1}{p}\|u\|^p - C_1\|u\|^{1-\delta} . \end{equation} and hence $ J_\lambda(u) \to +\infty$ as $\|u\| \to \infty$. This completes the proof of our Claim. Now, let $n \mapsto u_n$ be a minimizing sequence of $J_\lambda$ in $W^{1,p}_0(\Omega)$. The coercivity of $J_\lambda$ implies the boundedness of $u_n$ in $W^{1,p}_0(\Omega)$. Since $J_\lambda(u)=J_\lambda(|u|)$, without loss of generality, we may assume that $(u_n)_n$ is non-negative, converges weakly to some $u$ in $W^{1,p}_0(\Omega)$ and converges also pointwise. Moreover, by the weak lower semicontinuity of the norm $\|\cdot\|$ and the boundedness of $(u_n)_n$ in $W^{1,p}_0(\Omega)$ we obtain $$ J_\lambda(u)\leq\underset{n\to \infty}{\lim}\inf J_\lambda(u_n). $$ Hence $u$ is a global minimizer of $J_\lambda$ in $W^{1,p}_0(\Omega)$. Which completes the proof of the Step 1. \smallskip \noindent\textbf{Step 2:} The weak limit $u$ is a non-negative weak solution of problem \eqref{103}$_-$ if $\lambda > 0$ is sufficiently large. Firstly, observe that $J_\lambda(0)=0$. So, to prove that the non-negative solution is non-trivial, it suffices to prove that there exists $\lambda_\ast > 0$ such that \begin{equation}\label{nega} \inf_{u\in W^{1,p}_0(\Omega)} J_\lambda(u)<0\quad \text{for all } \lambda>0. \end{equation} For this purpose, take any positive $u$ and consider $\epsilon u$. Then, for a fixed $\lambda > 0$, $J_\lambda(\epsilon u) < 0$ if $\epsilon > 0$ is small enough. Therefore the minimum is negative for all $\lambda > 0$. Now, we consider the variational problem with constraints, \begin{equation}\label{mini} \begin{aligned} \lambda_\ast & = \inf\Big\{\frac{1}{p}\int_{\Omega} |\nabla w|^p \,\mathrm{d}x + \frac{1}{q+1}\int_{\Omega} h(x)|w|^{q+1} \,\mathrm{d}x: w\in W^{1,p}_0(\Omega) \text{ and }\\ & \quad \frac{1}{1-\delta}\int_{\Omega}k(x) |w|^{1-\delta} \,\mathrm{d}x =1\Big\}. \end{aligned} \end{equation} and define \begin{equation} \label{def:Lambda2} \Lambda_*= \inf\{ \lambda > 0\colon \eqref{103}_-\text{ admits a nontrivial weak solution} \}. \end{equation} From above, we have $$ J_\lambda(u) = \lambda_\ast - \lambda < 0\quad \text{for any }\lambda > \lambda_\ast. $$ Therefore, the above remarks show that $\lambda_\ast \geq \Lambda_*$ and that problem \eqref{103}$_-$ has a solution for all $\lambda> \lambda_\ast$. We now argue that problem \eqref{103}$_-$ has a solution for all $\lambda > \Lambda_*$. Fixed $\lambda>\Lambda_*$, by the definition of $\Lambda_*$, we can take $\mu \in (\Lambda_* , \lambda)$ such that that $J_\mu$ has a non-trivial critical point $u_\mu \in W^{1,p}_0(\Omega)$. Since $\mu < \lambda$, $u_\mu$ is a sub-solution of the problem \eqref{103}$_-$. In order to find a super-solution of the problem \eqref{103}$_-$ which dominates $u_\mu$. For this purpose we consider the constrained minimization problem \begin{equation}\label{mini1} \inf \{J_\lambda(w):\; w \in W^{1,p}_0(\Omega) \text{ and } w \geq u_\mu.\} \end{equation} Arguments similar to those used to treat \eqref{mini} show that the above minimization problem has a solution $u_\lambda > u_\mu$. Moreover, $u_\lambda$ is also a weak solution of problem \eqref{103}$_-$ for all $\lambda > \Lambda_*$. With the arguments developed in \cite{GiScTa} we deduce that problem \eqref{103}$_-$ has a solution if $\lambda=\Lambda_*$. Thus, one applies \cite[Theorem A.1]{An}, based on the Moser iteration, shows that $u \in L^\infty_{\rm loc}$. Next, again by a bootstrap regularity due to Giacomoni-Schindler-Tak\'{a}\v{c} \cite[Theorem B.1]{GiScTa} shows that the weak solution $u\in C^{1,\alpha}(\Omega)$ where $\alpha\in(0,1)$. Finally, the non-negative follows immediately by the strong maximum principle (see \cite[Theorem 2.3]{GiScTa}) since $u$ is a $C^1$ non-negative weak solution of the differential inequality $$ -\nabla(|\nabla u|^{p-2} \nabla u) + h(x) u^q \geq 0\;\text{ in} \; \Omega. $$ We deduce that $u$ is positive everywhere in $\Omega$. The proof of the step 2 is now complete. \smallskip \noindent\textbf{Step 3:} Non-existence for $\lambda > 0$ small. The same monotonicity arguments as in Step 2 show that \eqref{103}$_-$ does not have any solution if $\lambda < \Lambda_*$. 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