Existence of homoclinic constant sign solutions for a difference equation on the integers

We consider a difference equation involving the discrete p-Laplacian operator, depending on a positive real parameter lambda. We prove, under convenient assumptions, that for lambda big enough the equation admits at least one homoclinic constant sign solution in Z. Our method consists in two parts: first, we prove the existence of two Dirichlet-type solutions for the equation in the discrete interval [-n,n], for all n big enough; then, we show that such solutions converge to a homoclinic solution in Z, as n tends to infinity.