On the Conformal Gaussian Curvature Equation in R2

Authors: Cheng K.S.; Lin C.S.

Source: Journal of Differential Equations, Volume 146, Number 1, June 1998 , pp. 226-250(25)

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Abstract:

In this paper, we consider the equationDeltau+K(x) e2u=0 in R2 (0.1)where K(x)=K(|x|) in R2 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. Copyright 1998 Academic Press.

Language: English

Document Type: Research article

Affiliations: Department of Mathematics, National Chung Cheng University, Minghsiung, Chia-Yi, 621, Taiwan

Publication date: 1998-06-01