An indefinite concave-convex equation under a Neumann boundary condition I

Abstract

We investigate the problem (P _λ) −Δu = λb(x)|u|^q−2 u + a(x)|u|^p−2 u in Ω, ∂u/∂n = 0 on ∂Ω, where Ω is a bounded smooth domain in R^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ ${C^\alpha }\left( {\overline \Omega } \right)$ with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).