A singular problem with nonlinearities of exponential growth

Abstract

We obtain a nonnegative and nontrivial solution of the singular problem $-\Delta u=-u^{-\beta }\chi _{\{u>0\}}+\lambda u^p+\mu f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $\lambda$ and $\mu$ are nonnegative parameters, $\Omega \subset {\mathbb {R}}^2$ is a bounded smooth domain, $0<\beta <1$, $p>0$ and f is allowed to have exponential growth. We obtain a solution $u_{\epsilon }$ of the perturbed problem $-\Delta u+g_{\epsilon }(u)=\lambda u^p+\mu f(u)$ in $\Omega$, $u=0$ on $\partial \Omega$ where $g_{\epsilon }$ is smooth and converges pointwisely to $u^{-\beta }$. We show that $u_{\epsilon }$ converge uniformly to a nontrivial function u that solves the original problem.