A concave–convex problem with a variable operator

We study the following elliptic problem - A ( u ) = λ u q with Dirichlet boundary conditions, where A ( u ) ( x ) = Δ u ( x ) χ D 1 ( x ) + Δ p u ( x ) χ 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 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 < 2 N / ( 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.