A concave-convex problem with a variable operator

Abstract

We study the following elliptic problem $-A(u)=\lambda u^q$ with Dirichlet boundary conditions, where $A(u)(x)=\Delta u(x){\chi }_{D_1}(x)+{\Delta }_{p}u(x){\chi }_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2$. 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 ${\lambda }^{\ast }$ such that the problem has no positive solution for $\lambda >{\lambda }^{\ast }$ and a minimal positive solution for $0<\lambda <{\lambda }^{\ast }$. 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 $0<\lambda <{\lambda }^{\ast }$, the first one (that exists for all $0<\lambda <{\lambda }^{\ast }$) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $0<\lambda <{\lambda }^{\ast }$) comes from an appropriate (and delicate) mountain pass argument.