Asymptotics of the module of minimizers to a Ginzburg-Landau type functional (Q2712162)

The minimizers \(u_{\varepsilon }\) of the Ginzburg-Landau type functional NEWLINE\[NEWLINE E_{\varepsilon }(u,G)=p^{-1}\int_{G}|\nabla u|^{p}+(4\varepsilon ^{p})^{-1}\int_{G}(1-|u|^{2})^{2},\quad p\geq 2, NEWLINE\]NEWLINE are considered, where \(p\geq 2\) and the set \(G\subseteq {\mathbb R}^{n}\) is a bounded simply connected smooth domain. Firstly, it is shown that the module of minimizers denoted by \(|u_{\varepsilon }|\) converges to \(1\) in \(C_{loc}(G,{\mathbb R}^{n})\), where \(p=n\). In the sequel, the rate of the convergence of \(\nabla |u_{\varepsilon }|\) is explored, too. These results are reconstructed for the case of the regularizable minimizers \(\widetilde{u} _{\varepsilon },\) where \(p>n\). For some related results see for instance \textit{Y. Lei} and \textit{Z. Q. Wu} [Electron. J. Differ. Equ. 2000, Paper No. 14, 20 p. (2000; Zbl 0939.35076)] and \textit{Y. Lei} [Electron. J. Differ. Equ. 2001, Paper No. 15, 28 p. (2001; Zbl 0968.35052)].