An initial value approach to rotationally symmetric harmonic maps

We study the effect of the varying y (0) on the existence and asymptotic behavior of solutions for the initial value problem y (r)+ (n− 1) f (r)y (r) f (r) − (n− 1) g(y(r))g (y(r)) f (r)2 = 0, y(0) = 0, where f and g are some prescribed functions. Global solutions of this ODE on [0,∞) represent rotationally symmetric harmonic maps, with possibly infinite energies, between certain class of Riemannian manifolds. By studying this ODE, we show among other things that (i) all rotationally symmetric harmonic maps from Rn to the hyperbolic space Hn blow up in a finite interval; (ii) all such harmonic maps from Hn to Rn are bounded; and (iii) a trichotomy phenomenon occurs for such harmonic maps from Hn into itself, viz., they blow up in a finite interval, are the identity map, or are bounded according as the initial value y (0) < 1, = 1, or > 1. Finally when n = 2, the above equation can be solved exactly by quadrature method. Our results supplement those of Ratto and Rigoli (J. Differential Equations 101 (1993) 15–27) and Tachikawa (Tokyo J. Math. 11 (1988) 311–316).  2003 Elsevier Inc. All rights reserved.