An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem

We consider the following singularly perturbed Neumann problem epsilon(2) Delta u - u + u(p) = 0 in Omega, u > 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where p is subcritical and Omega is a smooth and bounded domain in R-N with its unit outward normal v. Lin, Ni and Wei (2007) [20] proved that there exists epsilon(0) such that for 0 < epsilon < so and for each integer k bounded by 1 <= k <= delta(Omega, N, p)/(epsilon vertical bar log epsilon vertical bar)(N) (0.1) where delta(Omega, N, p) is a constant depending only on Omega, p and N, there exists a solution with k interior spikes. We show that the bound on k can be improved to 1 <= k <= delta(Omega, N, p)/epsilon(N), (0.2) which is optimal. (C) 2013 Elsevier Inc. All rights reserved.