A priori estimates for elliptic problems with a strongly singular gradient term and a general datum

Abstract

In this paper we show approximation procedures for studying singular elliptic problems whose model is $$ \begin{cases} - \Delta u= b(u)|\nabla u|^2+f(x), & \text{in } \; \Omega;\\ u = 0, & \text{on } \; \partial \Omega; \end{cases} $$ where $b(u)$ is singular in the $u$-variable at $u=0$, and $f \in L^m (\Omega)$, with $m>\frac N2$, is a function that does not have a constant sign. We will give an overview of the landscape that occurs when different problems (classified according to the sign of $b(s)$) are considered. So, in each case and using different methods, we will obtain a priori estimates, prove the convergence of the approximate solutions, and show some regularity properties of the limit.