On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature

Abstract

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