Elliptic problems with singular nonlinearities of indefinite sign

AIMS Mathematics

Tomas Godoy¹ 

[1] Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina;

关键词: singular elliptic problems;    nonnegative solutions;    sub and supersolutions;   

DOI  :  10.3934/math.2020120

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown