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

Abstract

Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities. GEORGIA KARALI AND CHRISTOS SOURDISAbstract. 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. 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 uε 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.

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…In contrast, in the problem at hand the corresponding Equation (65) is non-autonomous, as was the case in [47,71,73,134], and [135]. The interested reader can verify that similar situations also occur in the singularly perturbed Fisher's equation [118,Chpt.…”

“…Actually, we will prefer to work with the equivalent (for ε > 0) problem in stretched variables y = ε (i) Firstly, we construct a "good" smooth approximate solution for the (stretched) problem which we call u ap . The function u ap is carefully built, along the lines set in [135], throughout Sections 3.1-3.3 in the following steps: Starting from the Hastings-McLeod solution V , described above, we construct an inner approximation u in that is valid only in a tubular neighborhood of the (stretched) curve ε…”

“…Actually, in one-dimensional or radially symmetric cases, for Proposition 5 to hold, it suffices to know that zero is not in the kernel of M . However, this may not be true in general higher dimensional problems due to a possible resonance phenomenon (see [78,135]). …”

“…Keeping in mind (7) which implies that (70); we refer to [135] for a related problem. Actually, we have computed that in the radially symmetric case, in N 1 dimensions, we have …”

“…The construction of arbitrary order approximations is especially important in the treatment of singularly perturbed elliptic problems involving resonance, where the order of accuracy of the approximation is dictated by the space dimension, see for instance [160]. Problems of these type which feature the presence of a corner layer (similar to the problem at hand) have been studied recently in [135] (in two dimensions), see also Remark [114] herein. It should be noticed that in Allen-Cahn or (focusing) Schrödinger type equations, where its possible to construct arbitrary order approximations (see [160] and the references therein), the phenomenon is exponentially localized, i.e, the corresponding terms approach certain constants exponentially fast.…”