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

Abstract We consider the following singularly perturbed Neumann problem e 2 Δ u − u + u p = 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 N with its unit outward normal ν . Lin, Ni and Wei (2007) [20] proved that there exists e 0 such that for 0 e e 0 and for each integer k bounded by (0.1) 1 ⩽ k ⩽ δ ( Ω , N , p ) ( e | log e | ) N 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 (0.2) 1 ⩽ k ⩽ δ ( Ω , N , p ) e N , which is optimal.