Positive solutions for elliptic equations with singular nonlinearity

Abstract

We study an elliptic boundary-value problem with singular nonlinearity via the method of monotone iteration scheme: $$displaylines{ -Delta u(x)=f(x,u(x)),quad x in Omega,cr u(x)=phi(x),quad x in partial Omega , }$$ where $Delta$ is the Laplacian operator, $Omega$ is a bounded domain in $mathbb{R}^{N}$, $N geq 2$, $phi geq 0$ may take the value 0 on $partialOmega$, and $f(x,s)$ is possibly singular near $s=0$. We prove the existence and the uniqueness of positive solutions under a set of hypotheses that do not make neither monotonicity nor strict positivity assumption on $f(x,s)$ which improvements of some previous results.

Keywords

• Singular nonlineararity
• elliptic equation
• positive solution
• monotonic iteration.