On the Existence of Multipeaked Solutions to a Semilinear Neumann Problem

∂ν = 0, x ∈ ∂ , where is the Laplace operator, is a smooth bounded domain in R(N ≥ 2), ν is the unit outward normal vector to ∂ , 2< p < 2N/(N−2) for N ≥ 3, 2< p <+∞ for N = 2, and ε > 0 is a constant. Problem (P)ε may be viewed as prototype of pattern formation in biology. Indeed, the steady-state problem for a chemotactic aggregation model with logarithmic sensitivity is reduced by Keller and Segel to (P)ε (see [20]). Moreover, in the study of activator-inhibitor systems modeling biological pattern formation, proposed by Gierer and Meinhardt in [17], (P)ε plays an important role when the diffusion rate of the inhibitor is sufficiently large. See, for example, [24], [33], and the references therein for more details. One of the motivations of this paper is the “point-condensation phenomena” of solutions to (P)ε expected from numerical simulations (to the Keller-Segel model as well as to the Gierer-Meinhardt model). That is, the solutions to (P)ε seem to tend to zero as ε −→ 0 except at a finite number of points. Here we show that these points are determined as local maximun and minimum points of the mean curvature of the boundary ∂ . The results are obtained by variational methods. Define an “energy”