Ground states for logarithmic Schr\"{o}dinger equations on locally finite graphs

TL;DR

This paper investigates ground state solutions for a logarithmic Schrödinger equation on locally finite graphs, establishing Sobolev embeddings and using variational methods to prove existence results.

Contribution

It introduces new Sobolev embedding theorems on graphs and applies variational techniques to find ground states for the equation with sign-changing potentials.

Findings

Established Sobolev compact embeddings on graphs.

Proved existence of ground state solutions using Nehari manifold and mountain pass theorem.

Handled cases with different assumptions on the potential a(x).

Abstract

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $\Delta$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on $a(x)$. It leads to two kinds of associated energy functionals, one of which is not well-defined under the logarithmic nonlinearity, while the other is $C^1$. The existence of ground state solutions are then obtained by using the Nehari manifold method and the mountain pass theorem respectively.