Layered Solutions With Concentration on Lines in Three-Dimensional Domains

Abstract

We consider the following singularly perturbed elliptic problem ε 2 ∆u − u + u p = 0, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω,where Ω is a bounded domain in R 3 with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uε concentrating along a curveΓ near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curveΓ will collapse to Γ as ε → 0.Problem (1.1) is known as the stationary equation of the Keller-Segel system in chemotaxes [21]. It can also be viewed as a limiting stationary equation for the Geirer-Meinhardt system in biological pattern formation [12]. Even though simplelooking, problem (1.1) has a rich and interesting structure of solutions. For the last 15 years, it has received considerable attention. In particular, various concentration phenomena exhibited by the solutions of (1.1) seem both mathematically intriguing and scientifically useful. We refer to three survey articles [30,31,37] for backgrounds and references.In the pioneering papers [32,33], Ni and Takagi proved the existence of least energy solutions to (1.1), that is, a solution u with minimal energy. Furthermore, they showed in [32,33] that, for each > 0 sufficiently small, u has a spike at the most curved part of the boundary, i.e. the region where the mean curvature attains maximum value.Since the publication of [32, 33], problem (1.1) has received a great deal of attention and significant progress has been made. More specifically, solutions with multiple boundary peaks as well as multiple interior peaks have been established. (13)(14)(15)19,20,34,38] and the references therein.) In particular, it was established in Gui and Wei [15] that for any two given integers k ≥ 0, l ≥ 0 and k + l > 0, problem (1.1) has a solution with exactly k interior spikes and l boundary spikes for every ε sufficiently small. Furthermore, Lin, Ni and Wei [22] showed that there are at least C N (ε|log ε|) −N number of interior spikes. Recently, the first author, Wei and Zeng [2] improve the upper bound to O(ε −N ) which is optimal.It seems natural to ask if problem (1.1) has solutions which "concentrate" on higher dimensional sets, e.g., curves, or surfaces. In this regard, we mention that it has been conjectured for a long time that problem (1.1) actually possesses solutions which have k-dimensional concentration sets for every 0 ≤ k ≤ N − 1. This is often called Ni's conjecture, see [30] or [31]. Progress in this direction, although still limited, has also been made in [26][27][28][29]. In particular, we mention the results of Malchiodi and Montenegro [27, 28] on the existence of solutions concentrating on the whole boundary provided that the sequence ε satisfies some gap c...

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

“…There are also some other results [6,7,24,25,26] to exhibit concentration phenomena on interior line segments connecting the boundary of Ω. For higher dimensional extension, the reader can refer to [8,16,35,40]. In [16], Dancer and Yan constructed solutions of (1.2) concentrating on higher dimensional subsets inside the domain and on the boundary of the domain separately.…”