An Optimal Bound on the Number of Interior Spike Solutions for the Lin-ni-takagi Problem

We consider the following singularly perturbed Neumann problem ε∆u− u+ u = 0 in Ω, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where p is subcritical and Ω is a smooth and bounded domain in R with its unit outward normal ν. Lin-Ni-Wei [18] proved that there exists ε0 such that for 0 < ε < ε0 and for each integer k bounded by 1 ≤ k ≤ δ(Ω, n, p) (ε| log ε|)n (0.1) where δ(Ω, n, p) is a constant depending only on Ω, p and n, there exists a solution with k interior spikes. We show that the bound on k can be improved to 1 ≤ k ≤ δ(Ω, n, p) εn , (0.2) which is optimal.