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

We consider the following singularly perturbed Neumann problemε2Δu−u+up=0in Ω,u>0in Ω,∂u∂ν=0on ∂Ω, where p is subcritical and Ω   is a smooth and bounded domain in RNRN with its unit outward normal ν. Lin, Ni and Wei (2007) [20] proved that there exists ε0ε0 such that for 0<ε<ε00<ε<ε0 and for each integer k bounded byequation(0.1)1⩽k⩽δ(Ω,N,p)(ε|logε|)N where δ(Ω,N,p)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 toequation(0.2)1⩽k⩽δ(Ω,N,p)εN, which is optimal.