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 $$ - \Delta u= \varepsilon e^{\kappa u} +\beta\delta_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $\kappa>0$ and $\beta\in\mathbb{R}$. When $\varepsilon=1$, 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 $\varepsilon=-1$ 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.