Singular nonlinear elliptic equations in $\mathbf{R}^N$

Abstract

This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form $-\Delta u+a\left(x\right)u=h\left(x\right)u^{-\gamma }$ in ${\mathbf{R}}^n$ where $a,h$ are given, not necessarily continuous functions, and $\gamma$ is a positive number. We explore both situations where $a,h$ are radial functions, with $a$ being eventually identically zero, and cases where no symmetry is required from either $a$ or $h$. Schauder′s fixed point theorem, combined with penalty arguments, is exploited.