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.

Keywords:

• Exchange of stability
• Infinite-dimensional Lyapunov-Schmidt reduction
• Resonance phenomena
• Semilinear elliptic equations
• Singular perturbations

Mathematics Subject Classification:

• 34D15
• 37F15
• 35J61
• 35J66

Acknowledgments

We would like to thank M. Kowalczyk for stimulating discussions on [17, 18], and M. Ward for bringing reference [24] to our attention. We would also like to thank M. del Pino for drawing our attention to [16] and [31]. The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7-REGPOT-2009-1) under grant agreement n^o245749.