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

Vladimir Bobkov

Abstract

An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $Ω$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(Ω)$-sphere in $W_0^{1,p}(Ω)$ 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 $Ω$ 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.

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

Abstract

An old and well-known open problem in the critical point theory asks whether, for some and some bounded domain , there exists a critical value of the -Dirichlet energy over an -sphere in lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the -Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when is sufficiently close to and 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 -Laplacian as varies. Since LS eigenvalues are continuous with respect to and exhaust the whole spectrum when , we deduce that at least one of the branches must contain non-LS eigenvalues.

Paper Structure

This paper contains 9 sections, 15 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform regularity of eigenfunctions
  4. Regularity of eigenvalues
  5. Square
  6. Rectangles
  7. Uniform spectral gap
  8. Proof of Theorem \ref{['thm:main']}
  9. A bit more about the square

Key Result

Theorem 1.1

For any sufficiently small $a>1$, there exists a sufficiently small $p>2$ such that either $\lambda_{\hbox{$\boxbar$}}(p; \mathcal{R}_{a})$ or $\lambda_{\boxminus}(p; \mathcal{R}_{a})$ is a non-LS eigenvalue of the $p$-Laplacian in $\mathcal{R}_a$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Schematic behavior of branches of eigenvalues in the arguments.

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Corollary 2.6
  • ...and 18 more