[see pdf for notation]-Reularity for the Solution of Strongly Nonlinear Eigenvalue Problems on Orlicz-Sobolev Spaces

Abstract

**Please note that the full text is embargoed** ABSTRACT: We present a new method to prove the [see pdf for notation]-regularity of the eigenfunctions for Dirichlet problems with strictly convex Young functionnonlinearities in their principal part. The basic idea is threefold: we first invoke the topological methods of [12] to infer the existence of a countable infinity of [see pdf for notation]-eigensolutions; we then use Schauder's inversion technique to associate with each one of these eigensolutions a unique [see pdf for notation]-solution of an auxiliary Dirichlet problem; we finally prove the [see pdf for notation]-regularity of the original elgensolutions from the [see pdf for notation]-regularity of the auxiliary solutions, using distributional arguments and a new convexity inequality which characterizes the shape of the given nonlinearities. We give several examples and counter examples which illustrate the role of the various hypotheses, and which allow comparison with Pohozaev's celebrated existence and non-existence results [4].