We study a $p$-Laplacian difference equation on the set of integers, involving a coercive weight function and a reaction term satisfying the Ambrosetti--Rabinowitz condition. By means of critical-point theory and a discrete maximum principle, we prove the existence of a positive homoclinic solution.
Antonio Iannizzotto. Vicenţiu D. Rădulescu. "Positive homoclinic solutions for the discrete $p$-Laplacian with a coercive weight function." Differential Integral Equations 27 (1/2) 35 - 44, January/February 2014. https://doi.org/10.57262/die/1384282852