In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation - d i v a ( x , u , ∇ u ) = b ( x ) | u | p - 2 u + λ H ( x , u , ∇ u ) , - div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right), where Ω is a bounded smooth domain of 𝕉 ^N .

This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: $$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -\Delta ^{l}_{p(x)}u+\delta \vert u\vert ^{\alpha (x)-2}u=\mu g(x, u)+\lambda f(x, u, \nabla u) &{} \mathrm {i}\mathrm {n}\ \Omega ,\\ \\ u=0 &{} \mathrm {o}\mathrm {n}\ \partial \Omega , \end{array}\right. \end{aligned}$$- Δ p ( x ) l u + δ | u | α ( x ) - 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) i n Ω , u = 0 o n ∂ Ω , where $$\Delta ^{l}_{p(x)}$$Δ p ( x ) l is the p(x)-Laplacian-like operator, $$\Omega $$Ω is a smooth bounded domain in $$\mathbb {R}^{N}$$R N , $$\delta ,\mu $$δ , μ , and $$\lambda $$λ are three real parameters, and $$p(\cdot ),\alpha (\cdot )\in C_{+}(\overline{\Omega })$$p ( · ) , α ( · ) ∈ C + ( Ω ¯ ) and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized $$(S_{+})$$( S + ) type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.