Existence of compact support solutions for a quasilinear and singular problem

Abstract

Let $\Omega$ be a $\mathcal{C}^{2}$ bounded domain of ${{\mathbb R}}^N$, $N\geq 2$. We consider the following quasilinear elliptic problem: $$ ({ P}_{\lambda})\left\lbrace \begin{array}{l} -\Delta_p u = K(x)(\lambda u^q-u^r),\quad \ \ \mbox{ in }\Omega, \\ \quad \;\;\;u= 0 \quad\mbox{ on }\partial\Omega, \quad u\geq 0\quad\mbox{ in }\Omega, \end{array}\right. $$ where $p>1$ and $\Delta_p u{\stackrel{{\rm {def}}}{=}} \mathrm{div} \left(\vert \nabla u\vert ^{p-2}\nabla u\right)$ denotes the $p$-Laplacian operator. In this paper, $\lambda>0$ is a real parameter, the exponents $q$ and $r$ satisfy $-1<r<q<p-1$, and $K:\Omega\longrightarrow {{\mathbb R}}$ is a positive function having a singular behaviour near the boundary $\partial \Omega$. Precisely, $K(x)=d(x)^{-k}L(d(x))$ in $\Omega$, with $0<k<p$, $L$ a positive perturbation function, and $d(x)$ the distance of $x\in\Omega$ to $\partial\Omega$. By using a sub- and supersolution technique, we discuss the existence of positive solutions or compact support solutions of $({ P}_{\lambda})$ in respect to the blow-up rate $k$. Precisely, we prove that if $k<1+r$, $({ P}_{\lambda})$ has at least one positive solution for $\lambda>0$ large enough, whereas it has only compact support solutions if $k\geq 1+r$.