On $$\vec {p}(\cdot )$$-Laplacian Equations with Generalized Anisotropic Neumann Boundary Condition

Abstract

Abstract In this paper, we investigate the existence and multiplicity of solutions for an anisotropic variable exponent problem of the form $$\begin{aligned} -\sum _{i=1}^{N}\frac{\partial }{\partial x_{i}} \left( \left| \frac{\partial u}{\partial x_{i}}\right| ^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\right) +\sum _{i=1}^{N}| u|^{p_{i}(x)-2}u = \lambda k(x)| u|^{\alpha (x)-2}u \end{aligned}$$ - ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i ( x ) - 2 ∂ u ∂ x i + ∑ i = 1 N | u | p i ( x ) - 2 u = λ k ( x ) | u | α ( x ) - 2 u in $$\Omega $$ Ω , subject to the anisotropic Neumann boundary condition $$\begin{aligned} \sum _{i=1}^{N}\left| \frac{\partial u}{\partial x_{i}}\right| ^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\eta _{i} =h(x)| u|^{\beta (x)-2}u \quad {\text {on }} \partial \Omega , \end{aligned}$$ ∑ i = 1 N ∂ u