ON GAUSSIAN CURVATURE EQUATION IN R 2 WITH PRESCRIBED NONPOSITIVE CURVATURE

. The purpose of this paper is to study the solutions of ∆ u + K ( x ) e 2 u = 0 in R 2 with K ≤ 0. We introduce the following quantities: . Under the assumption ( H 1 ): α p ( K ) > −∞ for some p > 1 and α 1 ( K ) > 0, we show that for any 0 < α < α 1 ( K ), there is a unique solution u α with u α ( x ) = α ln | x | + c α + o (cid:0) | x | − 2 β 1+2 β (cid:1) at infinity and β ∈ (0 , α 1 ( K ) − α ). Furthermore, we show an example K 0 ≤ 0 such that α p ( K 0 ) = −∞ for any p > 1 and α 1 ( K 0 ) > 0, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution u α ∗ such that u α ∗ − α ∗ ln | x | = O (1) at infinity for some α ∗ > 0, which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.