On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in \(\mathbb R^ 2\) (Q1359498)

We study the asymptotic behavior at infinity of a solution \(u\) of \(\Delta u+ K(x) e^{2u} =0\) in \(\mathbb R^2\). With some mild assumptions on \(K\) and \(u\), we conclude that \(u(x)= \alpha\log |x|+O(1)\) at infinity for some real \(\alpha\). Applying this result, we prove that solutions \(u\) of the equation must be radially symmetric provided that \(K=K(|x|)\) is radial and nonincreasing in \(|x|\). We also prove that the above equation has no solution with a finite total curvature provided that \(K\) is not identically a constant, bounded between two positive constants and monotonic along some direction.