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) ) = λ f (x,u,∇u) in Ω, ( |∇u|p(x)−2∇u + |∇u|2p(x)−2∇u√ 1+|∇u|2p(x) ) ∂u ∂η = 0 on ∂Ω, in the setting of the variable-exponent Sobolev spaces W1,p(x)(Ω), where Ω is a smooth bounded domain in RN, p(x) ∈ C+(Ω) and λ 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.