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

TL;DR

This paper improves the known upper bound on the number of interior spike solutions for a singularly perturbed Neumann problem, establishing an optimal bound proportional to rac{1}{ ext{epsilon}^n}.

Contribution

It provides an improved, optimal bound on the maximum number of interior spike solutions for the Lin-Ni-Takagi problem, refining previous estimates involving logarithmic factors.

Findings

Bound on spike solutions is improved to rac{ ext{delta}( ext{Omega},n,p)}{ ext{epsilon}^n}.

The new bound is proven to be optimal.

The result refines previous estimates involving logarithmic factors.

Abstract

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 $\Omega$ is a smooth and bounded domain in $\R^n$ with its unit outward normal $\nu$. 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{\delta(\Omega,n,p)}{(\ve |\log \ve |)^n} {equation} where $\delta(\Omega,n,p)$ is a constant depending only on $\Omega$, $p$ 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.