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

TL;DR

This paper investigates fractional logarithmic Schrödinger equations on lattice graphs, proving existence of ground state solutions under different potential conditions using variational methods.

Contribution

It introduces new existence results for ground state solutions of fractional logarithmic Schrödinger equations on lattice graphs with periodic and coercive potentials.

Findings

Existence of ground state solutions with periodic potential

Existence of sign-changing solutions with coercive potential

Application of mountain pass theorem and Nehari manifold methods

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.