Existence of Weak Solutions to a Class of Singular Elliptic Equations

Abstract

This paper is concerned with the existence of solutions to the following singular elliptic boundary value problem involving ${p}$-Laplace operator

$$-{\rm div}(|\nabla u|^{p-2} \nabla u)=\frac{h}{u^\gamma} \text{in } \Omega, \quad u > 0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega.$$

Here, ${\Omega\subset \mathbb{R}^N(N\geq3)}$ is a bounded domain with smooth boundary, and ${h}$ is a positive ${L^1}$ function on ${\Omega}$. A “compatibility condition” on the couple ${(h(x),\gamma)}$ is given for the problem to have at least one solution. More precisely, it is shown that the problem admits at least one solution if and only if there exists a ${u_0\in W_0^{1,p}(\Omega)}$ such that ${\int_\Omega hu_0^{1-\gamma} \mathrm{d}x < \infty}$. This generalizes a previous result obtained by Sun and Zhang (Calc Var Partial Differ Equ 49:909–922, 2014) who considered the case ${p=2}$.