Global existence and multiplicity of solutions for logarithmic Schr\"{o}dinger equations on graphs

TL;DR

This paper proves the existence of global solutions and infinitely many high-energy solutions for a logarithmic Schrödinger equation on graphs, extending classical Euclidean results to discrete graph settings using variational methods.

Contribution

It introduces a variational approach to establish solutions for the logarithmic Schrödinger equation on graphs, including cases with sign-changing potentials, which is novel compared to classical Euclidean methods.

Findings

Existence of global solutions on graphs.

Infinitely many high-energy solutions with sign-changing potential.

Extension of Euclidean space results to discrete graph structures.

Abstract

We consider the following logarithmic Schr\"{o}dinger equation $$ -\Delta u+h(x)u=u\log u^{2} $$ on a locally finite graph $G=(V,E)$, where $\Delta$ is a discrete Laplacian operator on the graph, $h$ is the potential function. Different from the classical methods in Euclidean space, we obtain the existence of global solutions to the equation by using the variational method from local to global, which is inspired by the works of Lin and Yang in \cite{LinYang}. In addition, when the potential function $h$ is sign-changing, we prove that the equation admits infinitely many solutions with high energy by using the symmetric mountain pass theorem. We extend the classical results in Euclidean space to discrete graphs.