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