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

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