On Gaussian curvature equation in R2 with prescribed nonpositive curvature

Abstract

The purpose of this paper is to study the solutions of ∆u + K(x)e2u = 0 in R2 with K ≤ 0. We introduce the following quantities: αp(K) = sup { α ∈ R: ZR2 |K(x)|p(1 + |x|)2αp+2(p−1)dx < +∞ } , ∀ p ≥ 1. Under the assumption (H1): αp(K) > −∞ for some p > 1 and α1(K) > 0, we show that for any 0 < α 1 and α1(K0) > 0, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution uα∗ such that uα∗ − 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.