Existence and Regularity of Solutions for Unbounded Elliptic Equations with Singular Nonlinearities

Abstract

For $q,\gamma >0$ , we study existence and regularity of solutions for unbounded elliptic problems whose simplest model is $\left\{\begin{array}{cc}-\text{div}\left[\left(1+{\left|u\right|}^q\right)\nabla u\right]=\left(f/{\left|u\right|}^{\gamma }\right)& \text{in\hspace{0.17em}}\mathrm{\Omega }\\ u=0& \text{on\hspace{0.17em}}\partial \mathrm{\Omega }\end{array}\right)$ , where $f\in L^m\left(\mathrm{\Omega }\right)$ , $m\ge 1$ .