On the Conformal Gaussian Curvature Equation inR2

In this paper, we consider the equationΔu+K(x) e2u=0 inR2 (0.1)whereK(x)=K(x) inR2andK(x) does not decay at x large. From the geometric viewpoint, it is quite interesting to ask what the best possible range is of total curvature of all solutions of (0, 1). In this paper, we study this problem for radial solutions. We also construct some particularKto demonstrate the rich phenomenon. In particular, we show by examples that whenK(x) is negative for x large, Eq. (0.1) possess a branch of solutions which satisfies some monotonicity property