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

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