Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

Abstract

We consider the following singularly perturbed elliptic problem: where Ω is a bounded domain in ℝ2 with smooth boundary, ϵ > 0 is a small parameter, n denotes the outward normal of ∂Ω, and a, b are smooth functions that do not depend on ϵ. We assume that the zero set of a − b is a simple closed curve Γ, contained in Ω, and ∇(a − b) ≠ 0 on Γ. We will construct solutions u ϵ that converge in the Hölder sense to max {a, b} in Ω, and their Morse index tends to infinity, as ϵ → 0, provided that ϵ stays away from certain critical numbers. Even in the case of stable solutions, whose existence is well established for all small ϵ > 0, our estimates improve previous results.