Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains

Abstract

We prove new concentration phenomena for the equation −ε 2 ∆u + u = u p in a smooth bounded domain Ω ⊆ R 3 and with Neumann boundary conditions. Here p > 1 and ε > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ε → 0.

Bubbling along boundary geodesics near the second critical exponent

“…We point out that the resonance phenomenon has already been found to arise in the analysis of higher dimensional concentration in other elliptic boundary value problems, in particular for a Neumann singular perturbation problem in [17,18,16,15] and in Schrödinger equations in the plane in [12]. Theorem 1.1 seems to be the first result on higher dimensional concentration phenomena associated to critical exponents.…”

Triple Junction Solutions for a Singularly Perturbed Neumann Problem

“…[36].) Progress in this direction, although still limited, has also been made in [2,30,31,32,33]. In particular, we mention the results of Malchiodi and Montenegro [31,32] on the existence of solutions concentrating on the whole boundary provided that the sequence ε satisfies some gap condition.…”

“…In the papers [27]- [30]- [32], the higher dimensional concentration set is on the boundary. A natural question is whether there are solutions with higher dimensional concentration set inside the domain.…”

Bubbling solutions for supercritical problems on manifolds

“…Solutions concentrating in higher dimensional sets and the gap condition have been found in elliptic problems in the Euclidean setting. We mention among, among many results, [11][12][13][14] for a Neumann singular perturbation problem and [3] for a Schödinger equation in the plane.…”