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

TL;DR

This paper establishes the existence of infinitely many homoclinic solutions for a class of discrete difference equations with oscillatory nonlinearities, using critical point theory to analyze parameter intervals.

Contribution

It identifies specific parameter ranges ensuring multiple solutions for difference equations with oscillatory nonlinearities, without symmetry assumptions.

Findings

Existence of infinitely many homoclinic solutions for certain parameter intervals.

Application of critical point theory to discrete problems with oscillatory nonlinearities.

No symmetry assumptions required for the nonlinear term.

Abstract

In this paper, we determine a concrete interval of positive parameters $\lambda$, for which we prove the existence of infinitely many homoclinic solutions for a discrete problem $-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi_{p}(u(k))=\lambda f(k,u(k))$, $k\in Z$; where the nonlinear term $f : Z \times R \rightarrow R$ has an appropriate oscillatory behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.