On an Elliptic Equation Involving a Kirchhoff Term and a Singular Perturbation

Abstract

In this paper we consider the existence of positive solutions for the following class of singular elliptic nonlocal problems of Kirchhoff-type $$ \left\{\begin{array}{rclcc} -M(\|u\|^{2})\Delta u = \frac{h(x)}{u^{\gamma}}+k(x)u^{\alpha} \mbox{in} \Omega ,\\ u > 0 \mbox{in} \Omega ,\\ u = 0 \mbox{on} \partial\Omega ,\\ \end{array} \right. $$ where $\Omega \subset \mathbb R^{N}, N \geq 2,$ is a bounded smooth domain, $M:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $\|u\|^{2}=\int_{\Omega}|\nabla u|^{2}$ is the usual norm in $H^{1}_{0}(\Omega )$. The main tools used are the Galerkin method and a Hardy-Sobolev inequality.