On sequences of large homoclinic solutions for a difference equation on the integers involving oscillatory nonlinearities - Robert Stegliński (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
In this paper, we determine a concrete interval of positive parameters , for which we prove the existence of infinitely many homoclinic solutions for a discrete problem $$Δ(a(k)φ_p(u(k-1)))+ b(k)φ_p(u(k)) = λ f(k;u(k)),\quad k \in Z,$$ where the nonlinear term $f : \mathbb{Z} × \mathbb{R} → \mathbb{R}$ has an appropriate oscillatory behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.