Form

Abstract

. We consider the problem of establishing conditions on p ( x ) that ensure that the form associated with the p ( x )-Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that – unlike the p = constant case – this is not possible if p has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for p = 2 , involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in Ω ⊂ R n with n ≥ 1 , and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous p ( x ) . Our basic criteria involve restrictions on p ( x ) and its gradient.