Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

Generalizing previous results of M. Comte and P. Mironescu, it is shown that for degree d large enough (such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ 8d \varepsilon_{d}^{2} -1 > \sqrt{4d-1} $$ \end{document}), there is a bifurcation branch in the set of the solutions of the Ginzburg-Landau equation, emanating from the branch of radial solutions at the critical value εd of the parameter. Moreover, the solutions on the bifurcation branch admit exactly d zeroes, and the energy on the bifurcation branch is strictly smaller than the energy on the radial branch.