2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,
Electron. J. Diff. Eqns., Conference 22 (2015), pp. 19-30.

Existence and regularity of weak solutions for singular elliptic problems

Brahim Bougherara, Jacques Giacomoni, Jesus Hernandez

Abstract:
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 $1<q<\frac{1+\alpha}{\alpha+\beta-1}$ which is optimal if $\alpha+\beta>1$.

Published November 20, 2015.
Math Subject Classifications: 35B65.
Key Words: Semilinear elliptic and singular problems; comparison principle; regularity of the gradient of solutions; Hardy inequalities.

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Brahim Bougherara
Département de mathématiques
ENS de Kouba
16308--Alger, Algérie
email: brahim.bougherara@univ-pau.fr
Jacques Giacomoni
LMAP (UMR CNRS 5142) Bat. IPRA
Avenue de l'Université F-64013 Pau, France
email: jacques.giacomoni@univ-pau.fr
Jesus Hernández
Departemento de Matemáticas
Universidad Autónoma de Madrid
28049 Madrid, Spain
email: jesus.hernandez@uam.es

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