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

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 quantities:

$ \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_{\alpha_*} $ such that $ u_{\alpha_*} -\alpha_*\ln|x| = O(1) $ at infinity for some $ \alpha_* > 0 $, 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.