On the conformal Gaussian curvature equation in R2

In this paper, we consider the equation Delta u + K(x) e(2u) = 0 in R-2 (0.1) where K(x) = K(\x\) in R-2 and K(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 particular K to demonstrate the rich phenomenon. In particular, we show by examples that when K(x) is negative for \x\ large, Eq. (0.1) possess a branch of solutions which satisfies some monotonicity property. (C) 1998 Academic Press.