Electron. J. Diff. Equ., Vol. 2016 (2016), No. 191, pp. 1-16.

Existence of nonnegative solutions for singular elliptic problems

Tomas Godoy, Alfredo J. Guerin

Abstract:
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 $\Omega$.

Submitted March 8, 2016. Published July 13, 2016.
Math Subject Classifications: 35J75, 35D30, 35J20.
Key Words: Singular elliptic problem; variational problems; nonnegative solution; positive solution; sub-supersolution.

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Tomas Godoy
FaMAF, Universidad Nacional de Córdoba
Ciudad Universitaria, 5000
Córdoba, Argentina
email: tomasgodo@gmail.com
  Alfredo J. Guerin
FaMAF, Universidad Nacional de Córdoba
Ciudad Universitaria, 5000
Córdoba, Argentina
email: guerin.alfredojose@gmail.com

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