Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with $n$-harmonic maps

Abstract

We describe the behavior as $\varepsilon\to 0$ of minimizers for a Ginzburg-Landau functional $$ E_{\varepsilon}(u;\Omega )=\int_{\Omega}\bigl [ \frac {|\nabla u|^n}n +\frac 1{4\varepsilon^n}(1-|u|^2)^2\bigr ]\,dx $$ in the space $H^{1,n}_g(\Omega ;\Bbb R^n)$, where $\Omega\subset\Bbb R^n$ and the boundary data $g:\partial \Omega\to S^{n-1}$ has a nonzero topological degree. Some recent results of Bethuel, Brezis and H\'elein, and of Struwe on the two-dimensional problem, are extended to higher-dimensional cases. New proofs for their results are also presented in this paper.