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 Δu=u(|u|²−1), where . First we prove a Liouville-type theorem which asserts that every solution u, satisfying , is constant (and of unit norm), provided N?4 (here M?1). In our second result, we give an answer to a question raised by Brézis (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?3. We also formulate three open problems concerning the classification of entire solutions of the Ginzburg-Landau system in any dimension.