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

Abstract

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 equations 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 $n\in\N$ big enough; then, we show that such solutions converge to a homoclinic solution in $\Z$, as $n\to\infty$