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 \Delta u -u +u^p = 0 \quad {{in}} \quad \Omega, \quad u>0 \quad {{in}} \quad \Omega, \quad {\partial u \over \partial \nu}=0 \quad {{on}} \quad \partial \Omega, {eqnarray*} where $p$ is subcritical and $\mathrm{\Omega }$ is a smooth and bounded domain in ${\mathbb{R}}^n$ with its unit outward normal $\nu$. Lin-Ni-Wei \cite{LNW} proved that there exists ${\text{ve}}_0$ such that for $0<\text{ve}<{\text{ve}}_0$ and for each integer $k$ bounded by {equation} 1\leq k\leq \frac{\delta(\Omega,n,p)}{(\ve |\log \ve |)^n} {equation} where $\delta (\mathrm{\Omega },n,p)$ is a constant depending only on $\mathrm{\Omega }$, $p$ 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{\delta(\Omega,n,p)}{\ve^n}, {equation} which is optimal.