Conformal metrics with prescribed nonpositive Gaussian curvature on ℝ2
Journal
Calculus of Variations and Partial Differential Equations
Journal Volume
11
Journal Issue
2
Pages
203-231
Date Issued
2000
Author(s)
Cheng, K.-S.
Abstract
In this paper, we consider the equation (0.1) Δu + K(x)e2u = 0 in ℝ2, where K ≢ 0 is a nonpositive function in ℝ2. A solution u is said to be complete if the conformal metric g = e2u|dx|2 is complete in R2. Let α1 = α1(K) = sup{α| (latin small letter esh)ℝ2 K(x)(1 + |x|2)αdx < ∞} . Assuming only that α1 > 0, we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy (0.2) |K(x)| ≤ |x|m for |x| large for some positive constant m, then α1 > 0 is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6].
Type
journal article