Existence and regularity of weak solutions for singular elliptic problems
B. Bougherara, Jacques Giacomoni, Jesús Hernández · DOAJ (DOAJ: Directory of Open Access Journals) · 2015
In this article we study the semilinear singular elliptic problem $$\displaylines{ -\Delta u = \frac{p(x)}{u^{\alpha}}\quad \text{in } \Omega \cr u = 0\quad \text{on } \partial\Omega,\quad u>0 \text{ in } \Omega, }$$ where $\Omega$ is a regular bounded domain of $\mathbb R^{N}$, $\alpha\in\mathbb R$, $p\in C(\Omega)$ which behaves as $d(x)^{-\beta}$ as $x\to\partial\Omega$ with $d$ the distance function up to the boundary and $0\leq \beta <2$. We discuss the existence, uniqueness and stability of the weak solution. We also prove accurate estimates on the gradient of the solution near the boundary. Consequently, we can prove that the solution belongs to $W^{1,q}_0(\Omega)$ for $1 1$.