Two results on entire solutions of Ginzburg-Landau system in higher dimensions

In this short article we prove two results on the Ginzburg-Landau system of equations Deltau = u(\u\(2) - 1), where u is an element of C-2 (R-N, R-M) (N, M greater than or equal to 1). First we prove a Liouville-type theorem which asserts that every solution it, satisfying integral(RN)(\u\(2) - 1)(2) < + infinity, is constant (and of unit norm), provided N greater than or equal to 4 (here M greater than or equal to 1). In our second result, we give an answer to a question raised by Brezis (open problem 3 of (Proceedings of the Symposium on Pure Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1999), about the symmetry for the Ginzburg Landau system in the case N = M greater than or equal to 3. We also formulate three open problems concerning the classification of entire solutions of the Ginzburg Landau system in any dimension. (C) 2003 Elsevier Inc. All rights reserved.