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

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation $$-{\epsilon }^p{\mathrm{\Delta }}_{p}u=u^{q-1}-u^{p-1}\text{ in }\mathrm{\Omega },$$ where $\text{Om}$ 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 $W^{1,p}(\mathrm{\Omega })$. The argument is based on a combination of variational method, topological degree theory, and gradient flow.