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

摘要

We consider the following singularly perturbed elliptic problem: where Ω is a bounded domain in 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.View full textDownload full textKeywordsExchange of stability, Infinite-dimensional Lyapunov-Schmidt reduction, Resonance phenomena, Semilinear elliptic equations, Singular perturbationsMathematics Subject Classification34D15, 37F15, 35J61, 35J66Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/03605302.2012.681333