  In this paper, we obtain conditions under which the difference equation
$-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi_{p}(u(k))=\lambda
f(k,u(k)),\quad k\in \mathbb{Z}$, has infinitely many homoclinic solutions. A
variant of the fountain theorem is utilized in the proof of our theorem. It
improves the results in [L.Kong, homoclinic solutions for a second order
difference equation with $p-$Laplacian, \textit{Appl. Math. Comput}.,
\textbf{247} (2014), 1113--1121], where the set of conditions imposed on
nonlinearity is inconsistent.
