Existence of nonnegative solutions for singular elliptic problems

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 existe...

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Bibliographic Details
Published in:Electronic journal of differential equations 2016-07, Vol.2016 (191), p.1-16
Main Authors: Tomas Godoy, Alfredo J. Guerin
Format: Article
Language:English
Subjects:
nonnegative solution
positive solution
Singular elliptic problem
sub-supersolution
variational problems
Online Access:Get full text
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Summary: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 $\Omega$.
ISSN:1072-6691