Preprint

Inserted: 24 feb 2026
Last Updated: 24 feb 2026

Pages: 38
Year: 2026

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 $L^p$ Poincaré 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 Shnol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

Keywords: Nonlinear eigenvalue problems, $p-$Laplacian, decay of eigenfunctions, essential spectrum, Persson's Theorem

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