# American Institute of Mathematical Sciences

March  2018, 38(3): 1505-1525. doi: 10.3934/dcds.2018062

## Existence of nonnegative solutions to singular elliptic problems, a variational approach

* Corresponding author: Tomas Godoy

Received  May 2017 Published  December 2017

We consider the problem $-Δ u = χ_{\{ u>0\} }g( .,u) +f( .,u)$ in $Ω,$ $u = 0$ on $\partialΩ,$ $u≥0$ in $Ω,$ where $Ω$ is a bounded domain in $\mathbb{R}^{n}$, $f:Ω×[ 0,∞) →\mathbb{R}$ and $g:Ω×( 0,∞) →[ 0,∞)$ are Carathéodory functions, with $g( x,.)$ nonnegative, nonincreasing, and singular at the origin. We establish sufficient conditions for the existence of a nonnegative weak solution $0\not \equiv u∈ H_{0}^{1}( Ω)$ to the stated problem. We also provide conditions that guarantee that the found solution is positive $a.e.$ in $Ω$. The problem with a parameter $Δ u = χ_{\{ u>0\} }g( .,u) +λ f( .,u)$ in $Ω,$ $u = 0$ on $\partialΩ,$ $u≥0$ in $Ω$ is also studied. For both problems, the special case when $g( x,s) : = a( x) s^{-α( x) },$ i.e., a singularity with variable exponent, is also considered.

Citation: Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062
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