Multiple solutions of the static Landau-Lifshitz equation from \(B^ 2\) into \(S^ 2\) (Q1909549)

We consider maps \(u: B^2\to S^2\) from the disc in Euclidean space to the Euclidean sphere, which have constant value \(\gamma\) on the boundary of \(B^2\), and we study the question of existence of extremals of the functional \[ E_H(u)= \int_{B^2} (|\nabla u|^2- 2u\cdot H)dx, \] where \(H\) is a constant vector in \(\mathbb{R}^3\) and \(u\cdot H\) is the Euclidean scalar product. The Euler-Lagrange equation is \(\Delta u+ |\nabla u|^2 u- (H\cdot u) u+ H= 0\), an equation introduced in physics by Landau and Lifshitz. When \(H= 0\), it reduces to the equation of harmonic maps, and it is known that the only solution of the problem is the constant map with value \(\gamma\). We show that if \(\gamma\) is not proportional to \(H\), then the problem has at least two solutions as soon as \(H\neq 0\). When \(\gamma= H/|H|\), it has only one solution, and when \(\gamma= - H/|H|\), the number of solutions increases with \(|H|\), and in particular if \(|H|\) is larger than the first eigenvalue of \(\Delta\) on \(B^2\), there are at least three solutions. The proofs use on one hand the methods introduced by [\textit{H. Brezis} and \textit{J. M. Coron}, Commun. Math. Phys. 92, 203-215 (1993; Zbl 0532.58006)] and [\textit{J. Jost}, J. Differ Geom. 19, 393-401 (1984; Zbl 0551.58012)], and on the other a reduction to an ordinary differential equation when \(H\) and \(\gamma\) are proportional.