Ground states for infinite lattices with nearest neighbor interaction

Abstract

Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, qi¨=Φi−1′(qi−1−qi)−Φi′(qi−qi+1),i∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\ddot{q_{i}}=\\varPhi _{i-1}'(q_{i-1}-q_{i})- \\varPhi _{i}'(q_{i}-q_{i+1}),\\quad i\\in \\mathbb{Z}, \\end{aligned}$$ \\end{document} where q_{i} denotes the co-ordinate of the ith particle and varPhi _{i} denotes the potential of the interaction between the ith and the (i+1)th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.