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

### Robert Dalmasso

Equipe EDP, Grenoble, France

## Abstract

We study the singular semilinear elliptic equation $∆u+f(.,u)=0$ in $D_{′}(Ω)$, where $Ω⊂R_{n}$ ($n≥1$) is a bounded domain of class $C_{1,1}.f:Ω×(0,∞)→[0,∞)$ is such that $f(.,u)∈L_{1}(Ω)$ for $u>0$ and $u→f(x,u)$ is continuous and nonincreasing for a.e. $x$ in $Ω$. We assume that there exists a subset $Ω_{′}⊂Ω$ with positive measure such that $f(x,u)>0$ for $x∈Ω_{′}$ and $u>0$ and that $∫_{Ω}f(x,cd(x,∂Ω))dx<∞$ for all $c>0$. Then we show that there exists a unique solution $u$ in $W_{0}(Ω)$ such that $∆u∈L_{1}(Ω)$, $u>0$ a.e. in $Ω$.

## Cite this article

Robert Dalmasso, Weak Solution of a Singular Semilinear Elliptic Equation in a Bounded Domain. Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, pp. 793–798

DOI 10.2977/PRIMS/1145475230