Stable solutions of equations with a quadratic gradient term

Abstract

We study existence and regularity properties of stable positive solutions to the nonvariational problem - Delta u - b(x)|nabla u|^2 = lambda g(u) in a bounded smooth domain. In the case where b is constant, by means of a Hopf-Cole transformation, the problem can be taken to a variational form, for which there are classical results of Crandall-Rabinowitz, Mignot-Puel and Brezis-Vazquez. In this paper we obtain results for a general bounded function b=b(x) which coincide with the classical ones in the constant b case.