AIMS Mathematics, 2020, 5(3): 1779-1798. doi: 10.3934/math.2020120

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Elliptic problems with singular nonlinearities of indefinite sign

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

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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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