Multipeak Solutions for a Singularly Perturbed Neumann Problem

Problem (1.1) appears in applied mathematics. See for example [13, 14] and the references therein. For the interesting link between this problem and the modelling of activator-inhibitor systems, the authors can refer to [11]. In [13, 14], Ni and Takagi prove that the least energy solution of (1.1) has exactly one local maximum point xε which lies in ∂Ω, and xε tends to a point x0 which attains the maximum of H(x), where H(x) is the mean curvature function of ∂Ω. Later, Wei [21] proves that for a solution uε of (1.1) in a certain energy level, uε has only one local maximum point xε which is in ∂Ω, and xε tends to a critical point of H(x). He also gets a kind of converse, that is, for each nondegenerate critical point x0 of H(x), there exists a solution uε for (1.1), such that uε has only one local maximum point xε, and xε → x0 as ε → 0. In the recent paper [10], Li shows that the assumption of nondegenercy can be replaced by C1-stable (see definition 0.1 in [10]). Of course, nondegenerate critical point, strictly local maximum point and strictly local minimum point are C1-stable critical points. Thus, Li extends the results in [8, 21]. Gui [8] and Li [10] also consider the existence of multipeak solutions. But locally speaking, these solutions have one local maximum point. Other results on this problem can also be found in Bates, Dancer and Shi [5] and Wang [17] We mention here the works on the Neumann problem involving critical Sobolev exponent [1, 2, 12, 16, 18, 19, 20].