A class of semilinear elliptic equations on lattice graphs

Abstract

In this paper, we study the semilinear elliptic equation of the formon lattice graphs Z N , where N ≥ 2 and 2 ≤ p < q < +∞. By the Brézis-Lieb lemma and concentration compactness principle, we prove the existence of positive solutions to the above equation with constant coefficients ā, b and the decomposition of bounded Palais-Smale sequences for the functional with variable coefficients, which tend to some constants ā, b at infinity, respectively.