Hokkaido Mathematical Journal

Hokkaido Mathematical Journal, 34 (2005) pp.149-157

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Abstract

We study the singular semilinear elliptic equation $\Delta u + f(.,u)$ $= 0$ in ${\cal D}'(\RL^N)$, $N \geq 3$. $f: \, \RL^N \times (0,\infty) \to [0,\infty)$ is such that $f(.,u) \in L^1(\RL^N)$ for $u > 0$ and $u \to f(x,u)$ is continuous and nonincreasing for a.e. $x$ in $\RL^N$. We assume that there exists a subset $\Omega \subset \RL^N$ 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 $M^{N/(N-2)}(\RL^N)$ such that $\Delta u \in L^1(\RL^N)$, $u > 0$ a.e. in $\RL^N$.

MSC(Primary) 35J60 singular elliptic equation, weak solution