Fractional logarithmic Schrödinger equations on lattice graphs

Abstract

In this paper, we study the fractional logarithmic Schrödinger equation $$(-\hat{I}”{)}^{s}u+h(x)u=u\log u^2$$ 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.

19 pages