Infinitely many homoclinic solutions for the second-order discrete $p$-Laplacian systems

Abstract

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order discrete $p$-Laplacian systems $\triangle (\varphi_p(\Delta u(n-1)))-a(n)|u(n)|^{p-2}u(n)+\nabla W(n, u(n))=0$ has infinitely many homoclinic orbits, where $p>1, \ n\in {\mathbb{Z}},\ u\in {\mathbb{R}}^{N}$, $a:{\mathbb{Z}}\rightarrow{\mathbb{R}}$ and $W:{\mathbb{Z}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ are not periodic in $n$. Our conditions on the nonlinear term $W(n,u(n))$ are rather relaxed and we generalize some existing results in the literature.