On Gaussian curvature equations in $\mathbb{R}^2$ with prescribed non-positive curvature

TL;DR

This paper investigates solutions to a Gaussian curvature equation in lat space with non-positive curvature, establishing existence, uniqueness, and asymptotic behaviors, including novel solutions with logarithmic growth and non-uniform infinity behavior.

Contribution

It introduces a new framework for analyzing solutions with prescribed non-positive curvature, including conditions for existence, uniqueness, and asymptotic analysis, and provides examples of solutions with unusual growth patterns.

Findings

Existence and uniqueness of solutions with specific asymptotic behavior.

Construction of examples with solutions exhibiting logarithmic growth.

Identification of solutions with non-uniform behavior at infinity.

Abstract

The purpose of this paper is to study the solutions of $$ \Delta u +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$\alpha_p(K)=\sup\left\{\alpha \in \mathbb{R}:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$: $\alpha_p(K)> -\infty$ for some $p>1$ and $\alpha_1(K) > 0$, we show that for any $0 < \alpha < \alpha_1(K)$, there is a unique solution $u_\alpha$ with $u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big)$ at infinity and $\beta\in (0,\,\alpha_1(K)-\alpha)$. Furthermore, we show an example $K_0 \leq 0$ such that $\alpha_p(K_0) = -\infty$ for any $p>1$ and $\alpha_1(K_0) > 0$, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution $u_*$ such that $u_* -\alpha_*\ln|x| = O(1)$ at infinity for some $\alpha_* > 0$, but who does not converge to a constant at infinity. This example exhibits a new phenomenon of solutions with logarithmic growth and non-uniform behavior at infinity.