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 $$ Δu +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$α_p(K)=\sup\left\{α\in \mathbb{R}:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2αp+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$: $α_p(K)> -\infty$ for some $p>1$ and $α_1(K) > 0$, we show that for any $0 < α< α_1(K)$, there is a unique solution $u_α$ with $u_α(x) = α\ln |x|+ c_α+o\big(|x|^{-\frac{2β}{1+2β}} \big)$ at infinity and $β\in (0,\,α_1(K)-α)$. Furthermore, we show an example $K_0 \leq 0$ such that $α_p(K_0) = -\infty$ for any $p>1$ and $α_1(K_0) > 0$, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution $u_*$ such that $u_* -α_*\ln|x| = O(1)$ at infinity for some $α_* > 0$, 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.