Existence of Homoclinic Orbits for a Class of Asymptotically p‐Linear Difference Systems with p‐Laplacian

Abstract

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically p‐linear difference system with p‐Laplacian Δ(|Δu(n − 1)|p−2Δu(n − 1))+∇[−K(n, u(n)) + W(n, u(n))] = 0, where p ∈ (1, +∞), n ∈ ℤ, u ∈ ℝN, K, W : ℤ × ℝN → ℝ are not periodic in n, and W is asymptotically p‐linear at infinity.