The Gelfand problem for the 1-homogeneous p-Laplacian

Abstract

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 {\Omega\subset\mathbb{R}^{N}} , that is, we deal with -\frac{1}{p-1}\lvert\nabla u\rvert^{2-p}\operatorname{div}(\lvert\nabla u% \rvert^{p-2}\nabla u)=\lambda e^{u} in Ω with {u=0} on {\partial\Omega} . For this problem we show that, for {p\in[2,\infty]} , there exists a positive critical value {\lambda^{*}=\lambda^{*}(\Omega,N,p)} such that the following holds: • If {\lambda<\lambda^{*}} , the problem admits a minimal positive solution {w_{\lambda}} . • If {\lambda>\lambda^{*}} , the problem admits no solution. Moreover, the branch of minimal solutions {\{w_{\lambda}\}} 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 {\lambda=0} , and for a small fixed λ we also obtain a continuum of solutions with {p\in[2,\infty]} .