On the Effect of Domain Topology in a Singular Perturbation Problem

where Ω ⊂ R (N ≥ 2) is a smooth bounded domain, 1 < p < (N + 2)/(N − 2) for N ≥ 3, 1 < p < ∞ for N = 2 and ε > 0 is a positive small parameter. Our interest in (1.1) arises from two aspects. First, (1.1) is a typical singular perturbation problem. Singular perturbation problems have received much attention lately due to their significances in applications such as chemotaxis (see [18] and [19]), population dynamics (see [1], [16]) and chemical reaction theory (see [1]), etc. Secondly, we are interested in the effect of the properties of the domain, such as geometry, topology on the solutions of nonlinear elliptic problems. Problem (1.1) can be a prototype. Recently, the geometry of the domain on the solutions of (1.1) has been a subject of study. Beginning in [20], Ni and Wei studied the “least-energy solutions” of (1.1) and showed that for ε sufficiently small, the least-energy solution has only one local maximum point Pε and Pε must lie in the most centered part of Ω, namely, d(Pε, ∂Ω)→ maxP∈Ω d(P, ∂Ω), where d(P, ∂Ω) is the distance from