Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist

TL;DR

This paper proves the existence of non-Ljusternik--Schnirelman eigenvalues for the $p$-Laplacian in certain planar domains when $p$ is close to 2, addressing a longstanding open problem in critical point theory.

Contribution

It demonstrates that for $p$ near 2 and specific planar domains, non-LS eigenvalues of the $p$-Laplacian exist, expanding understanding of the spectrum beyond LS eigenvalues.

Findings

Existence of non-LS eigenvalues for $p>2$ close to 2.

Non-LS eigenvalues occur near simple Laplacian eigenvalues.

Continuity of LS eigenvalues with respect to $p$ implies non-LS eigenvalues must appear.

Abstract

An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $\Omega$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(\Omega)$-sphere in $W_0^{1,p}(\Omega)$ lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $\Omega$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues.