Existence of periodic solutions for a class of ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian difference system with asymptotically ( p , q ) $(p,q)$ -linear conditions

Abstract

Abstract In this paper, we consider a ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system as follows: { Δ ϕ 1 ( Δ u ( t − 1 ) ) + ∇ u F ( t , u ( t ) , v ( t ) ) = 0 , Δ ϕ 2 ( Δ v ( t − 1 ) ) + ∇ v F ( t , u ( t ) , v ( t ) ) = 0 , $$\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}$$ where F ( t , u ( t ) , v ( t ) ) = − K ( t , u ( t ) , v ( t ) ) + W ( t , u ( t ) , v ( t ) ) $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$ is T-periodic in t. By using the mountain pass theorem, we obtain that the ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system has at least one periodic solution if W is asymptotically ( p , q ) $(p,q)$ -linear at infinity. Our results improve and extend some known works.

Keywords

• ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system
• Periodic solutions
• Asymptotically ( p , q ) $(p,q)$ -linear condition
• Mountain pass theorem