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 ⎧ ⎨ ⎩ −u + m(x)|∇u| 2 1 + u = λ(1 + u)p in Ω, u = 0 on ∂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