Elliptic problems with singular nonlinearities of indefinite sign

Tomas Godoy; Tomas Godoy

doi:10.3934/math.2020120

AIMS Mathematics

2020, Volume 5, Issue 3: 1779-1798. doi: 10.3934/math.2020120

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Research article

Elliptic problems with singular nonlinearities of indefinite sign

Tomas Godoy ^,

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

Received: 02 November 2019 Accepted: 06 February 2020 Published: 14 February 2020
MSC : Primary: 35J75; Secondary: 35D30, 35J20

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1, 1}$ boundary. We consider problems of the form $-\Delta u = \chi_{\left\{ u>0\right\}}\left(au^{-\alpha}-g\left(., u\right) \right) $ in $\Omega, $ $u = 0$ on $\partial\Omega, $ $u\geq0$ in $\Omega, $ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left(\Omega\right), $ $\alpha\in\left(0, 1\right), $ and $g:\Omega\times\left[ 0, \infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g, $ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega, $ of the found solution $u$, is also proved.
- singular elliptic problems,
- nonnegative solutions,
- sub and supersolutions
Citation: Tomas Godoy. Elliptic problems with singular nonlinearities of indefinite sign[J]. AIMS Mathematics, 2020, 5(3): 1779-1798. doi: 10.3934/math.2020120

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Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1, 1}$ boundary. We consider problems of the form $-\Delta u = \chi_{\left\{ u>0\right\}}\left(au^{-\alpha}-g\left(., u\right) \right) $ in $\Omega, $ $u = 0$ on $\partial\Omega, $ $u\geq0$ in $\Omega, $ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left(\Omega\right), $ $\alpha\in\left(0, 1\right), $ and $g:\Omega\times\left[ 0, \infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g, $ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega, $ of the found solution $u$, is also proved.

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