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The mean curvature problem is an important class of problems in mathematics and physics. We consider the existence of homoclinic solutions to a discrete partial mean curvature problem, which is tied to the existence of discrete solitons. Under the assumptions that the potential function is unbounded and that the nonlinear term is superlinear at infinity, we obtain the existence of infinitely many homoclinic solutions to this problem by means of the fountain theorem in the critical point theory. In the end, an example is given to illustrate the applicability of our results.