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arXiv · 2602.21118

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

Lorenzo Brasco·Luca Briani·Francesca Prinari
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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.

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Lorenzo Brasco, Luca Briani, Francesca Prinari. 2026-02-24. Low eigenvalues of the $p-$Laplacian in general open sets. https://arxiv.org/abs/2602.21118

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