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 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.