Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions

We prove that minimizers ${\displaystyle u\in W^{1,n}}$ of the functional ${\displaystyle E_{}\left(u\right)=\frac{1}{n}{\int }_{|\nabla u|}^{n}dx+\frac{1}{4^n}{\int }_{1-{\left|u\right|}^2}^{2}dx}$, ⊂ ${\displaystyle {ℝ}^n}$, n ≥ 3, which satisfy the Dirichlet boundary condition ${\displaystyle u_{}=g}$ on for g: → ${\displaystyle S^{n-1}}$ with zero topological degree, converge in ${\displaystyle W^{1,n}}$ and ${\displaystyle C^{\alpha }\_\left\{loc\right\}}$ for any α<1 - upon passing to a subsequence ${\displaystyle \_\left\{k\right\}\to 0}$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.