On finite-energy solutions of Kazan-Warner equations on the lattice graph

Abstract

We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$-\Delta u=\epsilon e^{\kappa u}+\beta {\delta }_0\mathrm{in}{\mathbb{Z}}^2,$$ where $\epsilon =\pm 1$, $\kappa >0$ and $\beta \in \mathbb{R}$. When $\epsilon =1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $\kappa$. This provides a resolution of the open problem on the existence of finite-energy solutions to the Liouville equation. When $\epsilon =-1$ and $\beta >\frac{4\pi }{\kappa }$, we prove that the set of finite-energy solutions exhibits a layer structure. Moreover, we derive the extremal solution in this case.

28 pages