Stable solutions of equations with a quadratic gradient term

Abstract

We consider positive solutions to a non-variational family of equations of the form−∆u − b(x)|∇u| 2 = λg(u) in Ω,where λ ≥ 0, b(x) is a given function, g is an increasing nonlinearity with g(0) > 0 and Ω ∈ R n isa bounded smooth domain. We introduce the definition of stability for nonvariational problemsand establish existence and regularity results for stable solutions. These results generalize theclasical results obtained when b(x) = b is a constant function making the problem variationalafter a suitable transformation.