Existence of Large Solutions for Quasilinear Elliptic Equation

Abstract

In this paper, we consider the problem \begin{align*}\begin{cases}-\mbox{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda u-b(x)h(u),&x\in \Omega\\u=+\infty, &\mbox{on}\ \partial\Omega.\end{cases}\end{align*}where $\Omega$ is a smooth bounded domain in ${\bf R^N}$. The weight function $b(x)$ is a non-negative continuous function in the domain, $h(u)$ is locally Lipschitz continuous, $h(u)/u^{p-1}$ is increasing on $(0,\infty)$ and $h(u)\sim Hu^{m(p-1)}$ for sufficiently large $u$ with $H>0$ and $m>1$. We establish conditions sufficient to ensure the existence of positive large solutions of the equation.