A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. (English) Zbl 1299.35139

Giachetti, Daniela; Petitta, Francesco; Segura de León, Sergio

Differ. Integral Equ. 26, No. 9-10, 913-948 (2013).

Summary: 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 \(m>N/2\) 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.

MSC:

35J75	Singular elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

singular elliptic problem; a priori estimate; convergence of approximate solutions