An existence of positive solutions to singular elliptic equations

Abstract

In this paper we study the existence of solutions to the following semilinear elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -{{\mathrm{div}}}(M(x)\nabla u) - \mu \frac{u}{|x|^2} = \frac{f(x)}{u^{\theta }}\quad \text { in } \Omega , \\ u>0\quad \text { in }\Omega , \\ u =0\quad \text { on }\partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is an open bounded subset of ${\mathbb R}^N,\,N\ge 3,\,\,0\in \Omega$ and $\theta >0,\,0\le \,f\in L^{m}(\Omega ),1< m<\frac{N}{2},\,\,0<\mu < (\frac{N-2}{2})^2.$ The special feature of this problem is that it has singularity at the origin as well as on the boundary of $\Omega .$