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

We consider the following singularly perturbed Neumann problem {eqnarray*} \ve^2 Δu -u +u^p = 0 \quad {in} \quad Ω, \quad u>0 \quad {in} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {on} \quad \partial Ω, {eqnarray*} where $p$ is subcritical and $Ω$ is a smooth and bounded domain in $\R^n$ with its unit outward normal $ν$. Lin-Ni-Wei \cite{LNW} proved that there exists $\ve_0$ such that for $0<\ve<\ve_0$ and for each integer $k$ bounded by {equation} 1\leq k\leq \frac{δ(Ω,n,p)}{(\ve |\log \ve |)^n} {equation} 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 {equation} 1\leq k\leq \frac{δ(Ω,n,p)}{\ve^n}, {equation} which is optimal.