Stable solutions of equations with a quadratic gradient term

TL;DR

This paper investigates the existence and regularity of stable positive solutions to a nonlinear PDE with a quadratic gradient term, extending classical results to variable coefficient cases.

Contribution

It generalizes classical results by analyzing stable solutions for equations with a variable coefficient quadratic gradient term, beyond the constant coefficient case.

Findings

Established existence and regularity results for variable b(x)

Extended classical results to non-constant coefficient scenarios

Provided conditions under which solutions are stable and regular

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.