P(x)-Laplacian-like Neumann problems in variable-exponent Sobolev spaces via topological degree methods

Abstract

In this paper, we investigate the existence of a ?weak solutions? for a Neumann problems of p(x)-Laplacian-like operators, originated from a capillary phenomena, of the following form {?div( |?u|p(x)?2?u + |?u|2p(x)?2?u /?1 + |?u|2p(x))= ?f (x, u,?u) in ?,(|?u|p(x)?2?u + |?u|2p(x)?2 ?u/ ?|?u|2p(x)) ?u/?? = 0 on ??, in the setting of the variable-exponent Sobolev spaces W1,p(x)(?), where ? is a smooth bounded domain in RN, p(x) ? C+(??) and ? is a real parameter. Based on the topological degree for a class of demicontinuous operators of generalized (S+) type and the theory of variable-exponent Sobolev spaces, we obtain a result on the existence of weak solutions to the considered problem.