Homoclinic solutions for a second order difference equation with p-Laplacian

摘要

In this paper, we obtain new conditions under which the difference equation-Δa(k)ϕp(Δu(k-1))+b(k)ϕp(u(k))=λf(k,u(k)),k∈Z.has infinitely many homoclinic solutions, where p>1 is a real number, ϕp(t)=|t|p-2t for t∈R, λ>0 is a parameter, a,b:Z→(0,∞), and f:Z×R→R is continuous in the second variable. Some known results in the literature are extended and complemented. A variant of the fountain theorem is utilized in the proof of our theorem.