Fractional logarithmic Schr\"{o}dinger equations on lattice graphs

Abstract

In this paper, we study the fractional logarithmic Schr\"{o}dinger equation $$ (-\Delta)^{s} u+h(x) u=u \log u^{2} $$ on lattice graphs $\mathbb{Z}^d$, where $s\in (0,1)$. If $h(x)$ is a bounded periodic potential, we prove the existence of ground state solution by mountain pass theorem and Lions lemma. If $h(x)$ is a coercive potential, we show the existence of ground state sign-changing solutions by the method of Nehari manifold.