A bifurcation analysis for the Ginzburg-Landau equation
The authors consider the following boundary-value problem for the Ginzburg-Landau equation \[ \begin{aligned}-\Delta u={1\over\varepsilon^2} u_\varepsilon(1-|u_\varepsilon|^2)\quad &\text{in }B,\\ u_\varepsilon(z)= z^d\quad &\text{on }\partial B,\end{aligned}\tag{1} \] where \(B\) is the unit ball of \(\mathbb{R}^2\), \(d\in\mathbb{N}^*\) and \(\varepsilon>0\) is a parameter. They establish that for \(d=2\), there exists a bifurcation branch, defined for \(\varepsilon< \varepsilon_2\), \(\varepsilon\) near to \(\varepsilon_2\), emanating from the radial solutions to (1) at \(\varepsilon= \varepsilon_2\). In a neighborhood of \(\varepsilon_2\), the solution on the bifurcation branch admit exactly two zeroes.
scientific article; zbMATH DE number 1239962