A new minimax principle and application to the p-Laplace equation

Abstract

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation $$-{\epsilon }^p{\Delta }_{p}u=u^{q-1}-u^{p-1}\text{in}\Omega ,$$ where $\Om$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary, $1<p<n$ and $p<q<\frac{np}{n-p}$. The minimax principle will be applied to the set of peak functions, which is a subset of the Sobolev space $W^{1,p}(\Omega )$. The argument is based on a combination of variational method, topological degree theory, and gradient flow.