On the conformal Gaussian curvature equation in \(\mathbb{R}^2\) (Q1265126)

The authors study the equation \[ \Delta u+K(x)e^{2u}=0\quad\text{on }\mathbb{R}^2 \] where \(K=K(| x|)\) and \(K\) does not decay for large values of \(| x|\). The best possible range for the total curvature is investigated in the context of radial solutions. Moreover, it is shown by examples that when \(K(x)\) is negative for \(| x|\) large, the equation has a branch of solutions which satisfies some monotonicity property.