Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Abstract

Let $\Omega$ be a $C^2$ bounded domain in $\mathbb{R}^{n}$ such that $\partial\Omega=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_1$ and $\Gamma_2$ are disjoint closed subsets of $\partial \Omega$, and consider the problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\tau$ on $\Gamma_1$, $\frac{\partial u}{\partial\nu}=\eta$ on $\Gamma_2$, where $0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})$, $\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}$, and $g:\Omega \times(0,\infty)\rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. Under suitable assumptions on $g$ and $\eta$ we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow $g$ to be singular at $s=0$ and also at $x \in S$ for some suitable subsets $S\subset\overline{\Omega}$. The Dirichlet problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\sigma$ on $\partial \Omega$ is also studied in the case when $0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)$.