EXISTENCE OF HOMOCLINIC SOLUTIONS FOR THE SECOND-ORDER DISCRETE P-LAPLACIAN SYSTEMS

Abstract

By using critical point theory, we establish some existence criteria to guarantee the second-order discrete $p$-Laplacian systems $\triangle (\varphi_p(\Delta u(n-1))) - a(n) |u(n)|^{p-2} u(n) + \nabla W(n, u(n)) = 0$ have at least one homoclinic orbit, where $p \gt 1$, $n \in \mathbb{Z}$, $u \in \mathbb{R}^{N}$, $a \in C(\mathbb{Z}, \mathbb{R})$ and $W \in C^{1}(\mathbb{Z} \times \mathbb{R}^{N}, \mathbb{R})$ are no periodic in $n$.