On a non-smooth eigenvalue problem in Orlicz–Sobolev spaces

Abstract

This article studies a non-smooth eigenvalue problem for a Dirichlet boundary value inclusion on a bounded domain Ω which involves a φ-Laplacian and the generalized gradient in the sense of Clarke of a locally Lipschitz function depending also on the points in Ω. Specifically, the existence of a sequence of eigensolutions satisfying in addition certain asymptotic and locational properties is established. The approach relies on an approximation process in a suitable Orlicz–Sobolev space by eigenvalue problems in finite-dimensional spaces for which one can apply a finite-dimensional, non-smooth version of the Ljusternik–Schnirelman theorem. As a byproduct of our analysis, a version of Aubin–Clarke's theorem in Orlicz spaces is obtained.

Keywords:

• non-smooth eigenvalue problem
• Orlicz–Sobolev spaces
• finite-dimensional approximation
• Ljusternik–Schnirelman theory
• Krasnoselskii genus