Abstract
We consider the following singularly perturbed elliptic problem:
where Ω is a bounded domain in ℝ
2 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.
Mathematics Subject Classification:
Acknowledgments
We would like to thank M. Kowalczyk for stimulating discussions on [Citation17, Citation18], and M. Ward for bringing reference [Citation24] to our attention. We would also like to thank M. del Pino for drawing our attention to [Citation16] and [Citation31]. The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7-REGPOT-2009-1) under grant agreement no245749.