Calculus of variations on locally finite graphs

TL;DR

This paper develops a variational approach on locally finite graphs to find solutions to local equations like Schrödinger, mean field, and Yamabe equations, providing uniform estimates and constructing global solutions.

Contribution

It introduces a variational method from local to global on graphs, combining calculus of variations, mountain-pass theory, and uniform estimates.

Findings

Sequences of solutions to local equations are constructed.

Uniform estimates for solution sequences are established.

Global solutions are obtained through convergence of local solutions.

Abstract

Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schr\"odinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.