The Gelfand problem for the 1-homogeneous p-Laplacian

Abstract

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain Ω ⊂ ℝN, that is, we deal with − 1 p − 1|∇u|2−p div(|∇u|p−2∇u) = λeu in Ω with u = 0 on ∂Ω. For this problem we show that, for p ∈ [2, ∞], there exists a positive critical value λ∗ = λ∗(Ω, N, p) such that the following holds: ∙ If λ < λ∗, the problem admits a minimal positive solution wλ. ∙ If λ > λ∗, the problem admits no solution. Moreover, the branch of minimal solutions {wλ} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 with λ = 0, and for a small fixed λ we also obtain a continuum of solutions with p ∈ [2, ∞].