Solutions for a quasilinear elliptic p⃗(x)${\vec{p}(x)}$‐Kirchhoff type problem with weight and nonlinear Robin boundary conditions

This paper deals with the existence and multiplicity of weak solutions to a class of quasilinear elliptic p⃗(x)$\vec{p}(x)$‐Kirchhoff type problems with weight and a nonlinear Robin boundary condition such as a+bK∑i=1N∫Ω1pi(x)∂u∂xipi(x)dx(−Δp⃗(x)u)+∑i=1NVi(x)upi(x)−2u=θ(x)um(x)−2u+f(x,u)inΩ,∑i=1N∂u∂xipi(x)−2∂u∂xiυi=η|u|q(x)−2uon∂Ω,\begin{equation*}\hskip7pc {\begin{cases} \displaystyle {\left(a+b K{\left(\sum _{i=1}^{N}\int \nolimits _ {\Omega } \frac{1}{p_{i}(x)} {\left|\frac{\partial u}{\partial x_{i}}\right|}^{p_{i}(x)} \mathrm{d}x\right)} \right)} \big (-\Delta _{\vec{p}(x)} u\big )+\sum _{i=1}^{N}V_{i}(x){\left|u\right|}^{p_{i}(x)-2}u\\[12pt] \quad =\theta (x){\left|u\right|}^{m(x)-2}u+f(x,u) \text{ in } \Omega , \\[2pt] \displaystyle \sum _{i=1}^{N}{\left| \frac{\partial u}{\partial x_{i}}\right|}^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\upsilon _{i} =\eta | u|^{q(x)-2}u \quad \text{on } \partial \Omega , \end{cases}}\hskip-7pc \end{equation*}where Ω is a smooth bounded domain. Under suitable conditions on the data, we show the existence and multiplicity of weak solutions by means of a variational approach in the framework of anisotropic Sobolev spaces with variable exponents.