  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.
