Elliptic problems with singular nonlinearities of indefinite sign

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 Caratheodory 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.