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

Abstract

The purpose of this paper is to study the solutions of $$\Delta u+K(x)e^{2u}=0\mathrm{in}{\mathbb{R}}^2$$ with $K\le 0$. We introduce the following quantity: $${\alpha }_p(K)=\sup \left\{\alpha \in \mathbb{R}:{\int }_{{\mathbb{R}}^2}|K(x){|}^p(1+|x|{)}^{2\alpha p+2(p-1)}dx<+\infty \right\},\forall p\ge 1.$$ 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 $u_{\alpha }(x)=\alpha \ln |x|+c_{\alpha }+o(|x{|}^{-\frac{2\beta }{1+2\beta }})$ at infinity and $\beta \in (0,{\alpha }_1(K)-\alpha )$. Furthermore, we show an example $K_0\le 0$ 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 $u_{\ast }$ such that $u_{\ast }-{\alpha }_{\ast }\ln |x|=O(1)$ at infinity for some ${\alpha }_{\ast }>0$, 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.

13 pages