On sequences of large homoclinic solutions for a difference equation on the integers involving oscillatory nonlinearities

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 $$\Delta(a(k)\phi_p(u(k-1)))+ b(k)\phi_p(u(k)) = \lambda f(k;u(k)),\quad k \in Z,$$ where the nonlinear term $f : \mathbb{Z} \times \mathbb{R} \to \mathbb{R}$ has an appropriate oscillatory behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory

Existence of solutions for p-Laplacian discrete equations

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete p-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a positive real parameter