A remark on multiplicity of solutions for the Ginzburg-Landau equation
The authors study the critical points of the Ginzburg-Landau functional \[ E_{\varepsilon}(u) = {1 \over 2} \int_{\Omega} \left\{| \nabla u| ^{2} + {1 \over 2\varepsilon^{2}} (1 - | u| ^{2})^{2}\right\} \] defined on \[ \left\{u \in H^{1}(\Omega,{\mathbb C}): u = g \quad\text{on \(\partial\Omega\)}\right\} , \] where \(\varepsilon>0\) and \(g:\partial\Omega \to {\mathbb C}\) is a given smooth map with \(| g(x)| =1\) for any \(x \in \partial\Omega\). Minimizers of this functional have been studied in the book \textit{F. Bethuel}, \textit{H. Brezis} and \textit{F. Hélein} [Ginzburg-Landau Vortices, Birkhäuser (1994; Zbl 0802.35142)]. The main result asserts that, if \(\text{ deg}(g,\partial\Omega) \geq 2\), then for any sufficiently small \(\varepsilon\) the functional \(E_{\varepsilon}\) admits at least \(\text{ deg}(g,\partial\Omega)+1\) distinct critical points. The existence of three critical points was already proved by \textit{L. Almeida} and \textit{F. Bethuel}, [J. Math. Pures Appl., IX. Sér. 77, No. 1, 1-49 (1998; Zbl 0904.35023)].
scientific article; zbMATH DE number 1278762