Singularity analysis of Ginzburg–Landau energy related to p-wave superconductivity

The following Ginzburg–Landau energy in the absence of a magnetic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\varepsilon(\psi) = \int\limits_G\left[\frac{1}{2}|\nabla\psi|^2 + \frac{1}{4\varepsilon^2}(1-|\psi|^2)^2\right]{\rm d}x$$\end{document}was well studied during recent twenty years. Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G \subset \mathbf{R}^2}$$\end{document} is a bounded smooth domain, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\psi}$$\end{document} is an order parameter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon >0 }$$\end{document} . In particular, several global properties including the weighted energy estimation, the concentration compactness properties and the quantization effect of the energy had been established. This paper is concerned with another Ginzburg–Landau type free energy associated with p-wave superconductivity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_\varepsilon (\psi, u; G) = \frac{1}{2} \int\limits_G(|\nabla \psi|^2 + |\nabla u|^2 - |\nabla|\psi||^2){\rm d}x + \frac{1}{4\varepsilon^2} \int\limits_G(1-|\psi|^2)^2{\rm d}x.$$\end{document}Here, u is also an order parameter. We will prove that those global properties still hold for this more complicated energy functional. Such global properties describe the locations of the regular and the singular domains, and also show the convergence relation between the Ginzburg–Landau minimizers and the harmonic maps.