Existence and multiplicity of homoclinic solutions for difference systems involving classical ( ϕ 1 , ϕ 2 ) $(\phi_{1},\phi_{2})$ -Laplacian and a parameter

Abstract

Abstract In this paper, we investigate the existence and multiplicity of homoclinic solutions for a class of nonlinear difference systems involving classical ( ϕ 1 , ϕ 2 ) $(\phi_{1},\phi _{2})$ -Laplacian and a parameter: { Δ ( ρ 1 ( n − 1 ) ϕ 1 ( Δ u 1 ( n − 1 ) ) ) − ρ 3 ( n ) ϕ 3 ( u 1 ( n ) ) + λ ∇ u 1 F ( n , u 1 ( n ) , u 2 ( n ) ) = f 1 ( n ) , Δ ( ρ 2 ( n − 1 ) ϕ 2 ( Δ u 2 ( n − 1 ) ) ) − ρ 4 ( n ) ϕ 4 ( u 2 ( n ) ) + λ ∇ u 2 F ( n , u 1 ( n ) , u 2 ( n ) ) = f 2 ( n ) . $$ \textstyle\begin{cases} \Delta (\rho_{1}(n-1)\phi_{1} (\Delta u_{1}(n-1) ) )-\rho _{3}(n)\phi_{3}(u_{1}(n)) \\ \quad {}+\lambda\nabla_{u_{1}} F (n,u_{1}(n),u_{2}(n) )=f_{1}(n), \\ \Delta (\rho_{2}(n-1)\phi_{2} (\Delta u_{2}(n-1) ) )-\rho _{4}(n)\phi_{4}(u_{2}(n)) \\ \quad {}+\lambda\nabla_{u_{2}} F (n,u_{1}(n),u_{2}(n) )=f_{2}(n). \end{cases} $$ When F is not periodic in n and has ( p , q ) $(p,q)$ -sublinear growth or ( p , q ) $(p,q)$ -linear growth, by using the least action principle, we obtain that a system with classical ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi_{2})$ -Laplacian has at least one homoclinic solution and, by using Clark’s theorem, we see that a system with f 1 = f 2 ≡ 0 $f_{1}=f_{2}\equiv 0$ has at least m distinct pairs of homoclinic solutions.

Keywords

• difference systems
• classical ( ϕ 1 , ϕ 2 ) $(\phi_{1},\phi_{2})$ -Laplacian
• homoclinic solutions
• variational method