Weak Solution of a Singular Semilinear Elliptic Equation in a Bounded Domain

Abstract

We study the singular semilinear elliptic equation $∆u+f(.,u)=0$ in ${\mathcal{D}}^{\prime }(\Omega )$, where $\Omega \subset R^n$ ($n\ge 1$) is a bounded domain of class $C^{1,1}.f:\Omega \times (0,\infty )\to [0,\infty )$ is such that $f(.,u)\in L^1(\Omega )$ for $u>0$ and $u\to f(x,u)$ is continuous and nonincreasing for a.e. $x$ in $\Omega$. We assume that there exists a subset ${\Omega }^{\prime }\subset \Omega$ with positive measure such that $f(x,u)>0$ for $x\in {\Omega }^{\prime }$ and $u>0$ and that ${\int }_{\Omega }f(x,cd(x,\partial \Omega ))dx<\infty$ for all $c>0$. Then we show that there exists a unique solution $u$ in $W_0^{1,1}(\Omega )$ such that $∆u\in L^1(\Omega )$, $u>0$ a.e. in $\Omega$.