#### Tomas Godoy, Alfredo Guerin, Nonnegative solutions to some singular semilinear elliptic problems, Vol. 2017 (2017), Article ID 11, pp. 1-23

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DOI: 10.23952/jnfa.2017.11

Received  July 30, 2016; Accepted December 5, 2016

Abstract. We prove the existence of a nonnegative weak solution $0\not \equiv u\in H_0^1(\Omega)$ to the singular semilinear elliptic problem $-\Delta u=\chi_{\{u>0\}}au^{-\alpha}+f(.,u)$ in $\Omega,$ $u=0$ on $\partial\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0<\alpha<3,$ $a\in L^{\infty}(\Omega),$ $0\not \equiv a\geq0,$ and $f:\Omega\times [0,\infty)\rightarrow\mathbb{R}$ is a Caratheodory function which satisfies some suitable hypothesis. We also obtain results about the problem with a parameter $-\Delta u=\chi_{\{u>0\}}au^{-\alpha}+\lambda f(.,u)$ in $\Omega,$ $u\geq0$ in $\Omega,$ $u=0$ on $\partial\Omega$.