  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 \begin{equation*}
  \displaystyle\left\{\begin{array}{ll}
  \displaystyle-{\rm{div}}\Big(\vert\nabla u\vert^{p(x)-2}\nabla
u+\frac{\vert\nabla u\vert^{2p(x)-2}\nabla u}{\sqrt{1+\vert\nabla
u\vert^{2p(x)}}}\Big)=\lambda f(x, u, \nabla u) & \mathrm{i}\mathrm{n}\
\Omega,\\
  \Big(\vert\nabla u\vert^{p(x)-2}\nabla u+\frac{\vert\nabla
u\vert^{2p(x)-2}\nabla u}{\sqrt{1+\vert\nabla
u\vert^{2p(x)}}}\Big)\frac{\partial u}{\partial\eta}=0 & \mathrm{o}\mathrm{n}\
\partial\Omega,
  \end{array}\right. \end{equation*} in the setting of the variable-exponent
Sobolev spaces $W^{1,p(x)}(\Omega)$, where $\Omega$ is a smooth bounded domain
in $\mathbb{R}^{N}$, $p(x)\in C_{+}(\overline{\Omega})$ and $\lambda$ 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.
