Gelfand type quasilinear elliptic problems with quadratic gradient terms

Abstract

In this paper, for 0<m1⩽m(x)⩽m2 and positive parameters λ and p, we study the existence of positive solution for the quasilinear model problem{−Δu+m(x)|∇u|21+u=λ(1+u)pin Ω,u=0on ∂Ω. We prove that the maximal set of λ for which the problem has at least one positive solution is an interval (0,λ⁎], with λ⁎>0, and there exists a minimal regular positive solution for every λ∈(0,λ⁎). We also prove, under suitable conditions depending on the dimension N and the parameters p, m1, m2, that for λ=λ⁎ there exists a minimal regular positive solution. Moreover we characterize minimal solutions as those solutions satisfying a stability condition in the case m1=m2.