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 u∈C2(RN,RM) (N,M⩾1). First we prove a Liouville-type theorem which asserts that every solution u, satisfying ∫RN(|u|2−1)2<+∞, 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.