Weak solution of a singular semilinear elliptic problem

Abstract

We study the singular semilinear elliptic equation $\Delta u + f(.,u)$ $= 0$ in ${\cal D}'({\mathbb R}^N)$, $N \geq 3$. $f: {\mathbb R}^N \times (0,\infty) \to [0,\infty)$ is such that $f(.,u) \in L^1({\mathbb R}^N)$ for $u > 0$ and $u \to f(x,u)$ is continuous and nonincreasing for a.e. $x$ in ${\mathbb R}^N$. We assume that there exists a subset $\Omega \subset {\mathbb R}^N$ with positive measure such that $f(x,u) > 0$ for $x \in \Omega$ and $u > 0$ and that $\int_{\rl^N}f(x,c|x|^{2-N})dx < \infty$ for some $c > 0$. Then we show that there exists a unique solution $u$ in the Marcinkiewicz space $M^{N/(N-2)}({\mathbb R}^N)$ such that $\Delta u \in L^1({\mathbb R}^N)$,$u > 0$ a.e. in ${\mathbb R}^N$.