Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

Abstract

In this paper we study existence and multiplicity of weak solutions of the homogenous Dirichlet problem for a singular semilinear elliptic equation with a quadratic gradient term. The proofs for the main results are based on a priori estimates of solutions of approximate problems.

Multiplicity of positive solutions for singular elliptic problems

“…The research concerning the existence of positive (nonnegative) solutions of the elliptic singular problems has been very active and enjoying increasing interest for many years ([2, 3, 8–10, 18, 19, 26]). Problems modeled by elliptic PDEs with singularities arise in various areas of applied mathematics, in biological, chemical or physical phenomena, for example in fluid mechanics (see e.g.…”

“…It appeared that this approach could be applied also in the singular case. In the paper [26] the existence and multiplicity of positive weak solutions for the following singular PDE on bounded domain $$\Omega \subset \mathbb {R}^{n}$$, $$\begin{equation*} -\triangle u+\lambda u^{-m}\Vert \nabla u\Vert ^{2}=f(x) \ \ \text{for} \ \ x\in \Omega , \ \ u>0, \ \ u=0 \ \ \text{on} \ \ \partial \Omega , \end{equation*}$$was investigated, in the case when $$m>1$$, $$\lambda \ne 0$$ and f was a nonnegative measurable function. The authors applied also the sub and supersolution method.…”