Existence result for Neumann problems with p(x)-Laplacian-like operators in generalized Sobolev spaces

The paper studies the existence of a weak solutions for Neumann problems with p(x)-Laplacian-like operators, originated from a capillary phenomena, of the following form -div|∇u|p(x)-2∇u+|∇u|2p(x)-2∇u1+|∇u|2p(x)=μ|u|α(x)-2u+λf(x,u,∇u)inΩ,|∇u|p(x)-2∇u+|∇u|2p(x)-2∇u1+|∇u|2p(x)∂u∂η=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -\mathrm{{div}}\left( \vert \nabla u\vert ^{p(x)-2}\nabla u+\frac{\vert \nabla u\vert ^{2p(x)-2}\nabla u}{\sqrt{1+\vert \nabla u\vert ^{2p(x)}}}\right) \\ =\mu \vert u\vert ^{\alpha (x)-2}u+\lambda f(x, u, \nabla u) &{} \mathrm {i}\mathrm {n}\ \Omega ,\\ \left( \vert \nabla u\vert ^{p(x)-2}\nabla u+\frac{\vert \nabla u\vert ^{2p(x)-2}\nabla u}{\sqrt{1+\vert \nabla u\vert ^{2p(x)}}}\right) \frac{\partial u}{\partial \eta }=0 &{} \mathrm {o}\mathrm {n}\ \partial \Omega , \end{array}\right. \end{aligned}$$\end{document}in the setting of the generalized Sobolev spaces W1,p(x)(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,p(x)}(\Omega )$$\end{document}, where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} is a smooth bounded domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{N}$$\end{document}, p(·),α(·)∈C+(Ω¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot ),\alpha (\cdot )\in C_{+}(\overline{\Omega })$$\end{document}, ∂u∂η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial u}{\partial \eta }$$\end{document} is the exterior normal derivative, μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} are two real parameters. Based on the topological degree for a class of demicontinuous operators of generalized (S+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S_{+})$$\end{document} type, under appropriate assumptions on f, we obtain a result on the existence of weak solutions to the considered problem.