A concave-convex problem with a variable operator

Abstract

We study the following elliptic problem -A(u) = λ uq with Dirichlet boundary conditions, where A(u) (x) = u (x) D1 (x)+ p u(x) D2(x) is the Laplacian in one part of the domain, D1, and the p-Laplacian (with p>2) in the rest of the domain, D2 . We show that this problem exhibits a concave-convex nature for 1<q<p-1. In fact, we prove that there exists a positive value λ* such that the problem has no positive solution for λ > λ* and a minimal positive solution for 0<λ < λ*. If in addition we assume that p is subcritical, that is, p<2N/(N-2) then there are at least two positive solutions for almost every 0<λ < λ*, the first one (that exists for all 0<λ < λ*) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every 0<λ < λ*) comes from an appropriate (and delicate) mountain pass argument.