Arch. Rational Mech. Anal. 144 (1998) 301–311. c  Springer-Verlag 1998 A Bifurcation Analysis for the Ginzburg-Landau Equation Myriam Comte & Petru Mironescu Communicated by H. Brezis I. Introduction We consider the following boundary-value problem for the Ginzburg-Landau equation −u ε = 1 ε 2 u ε (1 −|u ε | 2 ) in B, u ε (z) = z d on ∂B, (1) where B is the unit ball of R 2 , d ∈ N ∗ and ε> 0 is a parameter. Solutions of this equation have been studied in general domains and for arbitrary boundary values by F. Bethuel, H. Brezis & F. H´ elein. In [BBH], these authors consider minimizers u ε for the Ginzburg-Landau energy (2) E ε (u) = E ε (u, G) := 1 2  G |∇u| 2 + 1 4ε 2  G (1 −|u| 2 ) 2 , in the class (3) H 1 g (G, C) := { u ∈ H 1 (G ; C) ; u = g on ∂G }, where G is a smooth domain in R 2 and g is a smooth map from ∂G into the unit circle; see also [S1] and [S2]. The existence of minimizers is obvious for every ε> 0. We refer to these solutions as minimizing solutions. On the other hand, we know from [BBH, FP, G, H], that (1) admits a solution of the form (4) u = u d,ε = u d,ε (re iθ ) = f d,ε (r)e diθ = f(r)e diθ . In fact, there is a unique solution to (1) having the form (4) with f ≧ 0, and f satisfying −f ′′ − f ′ r + d 2 f r 2 = 1 ε 2 f(1 − f 2 ) in [0, 1], f(1) = 1, f(0) = 0,f ≧ 0. (5)