Existence of nonnegative solutions for singular elliptic problems

• Main Authors: Tomas Godoy, Alfredo J. Guerin
• Format: Article
• Language: English
• Published: Texas State University 2016-07-01
• Series: Electronic Journal of Differential Equations
• Subjects: Singular elliptic problem; variational problems; nonnegative solution; positive solution; sub-supersolution
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/191/abstr.html

We prove the existence of nonnegative nontrivial weak solutions to the problem $$\displaylines{ -\Delta u=au^{-\alpha}\chi_{\{ u>0\} }-bu^p\quad\text{in }\Omega, \cr u=0\quad\text{on }\partial\Omega, }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$. A sufficient condition for the existence of a continuous and strictly positive weak solution is also given, and the uniqueness of such a solution is proved. We also prove a maximality property for solutions that are positive a.e. in $\Omega$.