Low eigenvalues of the $p-$Laplacian in general open sets

TL;DR

This paper investigates the properties of low eigenvalues of the $p$-Laplacian in general open sets, establishing conditions under which certain levels correspond to eigenvalues and analyzing eigenfunction decay.

Contribution

It demonstrates that minmax levels below a specific threshold are actual eigenvalues and provides decay estimates for eigenfunctions in unbounded domains.

Findings

Levels below the Poincaré threshold are eigenvalues.

Eigenfunctions exhibit exponential decay at infinity.

Applicable to certain unbounded open sets.

Abstract

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a \v{S}nol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.