Low eigenvalues of the p–Laplacian in general open sets

Abstract

We consider the minmax Ljusternik–Schnirelmann levels of the constrained –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 Poincaré constant “at infinity,” it actually defines an eigenvalue of the Dirichlet –Laplacian. We also prove exponential decay at infinity for the relevant eigenfunctions: this can be seen as a Šnol–Simon–type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.