Existence of positive solutions for nonlocal p ⁢ ( x ) p(x) -Kirchhoff elliptic systems

Abstract In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following p ⁢ ( x ) {p(x)} -Kirchhoff system: { - M ⁢ ( I 0 ⁢ ( u ) ) ⁢ △ p ⁢ ( x ) ⁢ u = λ p ⁢ ( x ) ⁢ [ λ 1 ⁢ f ⁢ ( v ) + μ 1 ⁢ h ⁢ ( u ) ] in ⁢ Ω , - M ⁢ ( I 0 ⁢ ( v ) ) ⁢ △ p ⁢ ( x ) ⁢ v = λ p ⁢ ( x ) ⁢ [ λ 2 ⁢ g ⁢ ( u ) + μ 2 ⁢ τ ⁢ ( v ) ] in ⁢ Ω , u = v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with C 2 {C^{2}} boundary ∂ ⁡ Ω {\partial\Omega} , △ p ⁢ ( x ) ⁢ u = div ⁡ ( | ∇ ⁡ u | p ⁢ ( x ) - 2 ⁢ ∇ ⁡ u ) {\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)} , p ⁢ ( x ) ∈ C 1 ⁢ ( Ω ¯ ) {p(x)\in C^{1}(\overline{\Omega})} , with 1 < p ⁢ ( x ) {1<p(x)} , is a function satisfying 1 < p - = inf Ω ⁡ p ⁢ ( x ) ≤ p + = sup Ω ⁡ p ⁢ ( x ) < ∞ {1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty} , λ, λ 1 {\lambda_{1}} , λ 2 {\lambda_{2}} , μ 1 {\mu_{1}} and μ 2 {\mu_{2}} are positive parameters, I 0 ⁢ ( u ) = ∫ Ω 1 p ⁢ ( x ) ⁢ | ∇ ⁡ u | p ⁢ ( x ) ⁢ 𝑑 x {I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx} , and M ⁢ ( t ) {M(t)} is a continuous function.