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

The purpose of this paper is to study the solutions of $$\mathrm{\Delta }u+K(x)e^{2u}=0\mathrm{i}\mathrm{n}{\mathbb{R}}^2$$ with $K\le 0$. We introduce the following quantity: $${\alpha }_p(K)=\sup \{\alpha \in \mathbb{R}:{\int }_{{\mathbb{R}}^2}\mathrm{\mid }K(x){\mathrm{\mid }}^p(1+\mathrm{\mid }x\mathrm{\mid }{)}^{2\alpha p+2(p-1)}dx<+\mathrm{\infty }\},\mathrm{\forall }p\ge 1.$$ Under the assumption $({\mathbb{H}}_1)$: ${\alpha }_p(K)>-\mathrm{\infty }$ 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 $u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big)$atinfinityand$\beta\in (0,,\alpha_1(K)-\alpha)$. Furthermore, we show an example $K_0\le 0$ such that ${\alpha }_p(K_0)=-\mathrm{\infty }$ 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 $u_{\ast }-{\alpha }_{\ast }\ln \mathrm{\mid }x\mathrm{\mid }=O(1)$ 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.