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arXiv · 2509.24268

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

Xu-Jia Wang·Xinyue Zhang
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Abstract

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation $$ -\varepsilon^p \Delta_p u = u^{q-1} - u^{p-1} \ \ \text{in}\ \Omega,$$ 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.

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Xu-Jia Wang, Xinyue Zhang. 2025-09-29. A new minimax principle and application to the p-Laplace equation. https://arxiv.org/abs/2509.24268

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