Strong solutions for singular Dirichlet elliptic problems - Tomas Godoy (Electronic Journal of Qualitative Theory of Differential Equations, 2022)
We prove an existence result for strong solutions $u\in W^{2,q}\left(\Omega\right) $ of singular semilinear elliptic problems of the form $-\Delta u=g\left( \cdot,u\right) $ in $\Omega,$ $u=\tau$ on $\partial\Omega,$ where $1<q<\infty,$ $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{2}$ boundary, $0\leq\tau\in W^{2-\frac{1}{q},q}\left( \partial\Omega\right) ,$ and with $g:\Omega\times\left( 0,\infty\right) \rightarrow\left[ 0,\infty\right)$ belonging to a class of nonnegative Carathéodory functions, which may be singular at $s=0$ and also at $x\in S$ for some suitable subsets $S\subset\overline{\Omega}.$ In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.