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Topological methods for the Ginzburg-Landau equations. (English) Zbl 0904.35023
This paper is concerned with the Dirichlet problem for the Ginzburg-Landau equation for complex-valued function \(u:\Omega\to\mathbb{C}\), \[ -\Delta u= {1\over\varepsilon^2} u(1-| u|^2)\quad\text{in }\Omega,\quad u= g\quad\text{on }\partial\Omega, \] where \(\Omega\) is a smooth, bounded and simply connected domain in \(\mathbb{R}^2\), \(g\) is a smooth map from \(\partial\Omega\) to \(S^1= \{z\in \mathbb{C};| z|= 1\}\), and where \(\varepsilon> 0\) is a small parameter. Solutions to this problem are critical points of the functional: \[ E_\varepsilon(u)= {1\over 2} \int_\Omega \Biggl\{|\nabla u|^2+{1\over 2\varepsilon^2} (1-| u|^2)^2\Biggr\}. \] \(E_\varepsilon\) is a \(C^1\)-functional (satisfying the Palais-Smale condition) on the Sobolev space \(H^1_g(\Omega;\mathbb{R}^2)= \{u\in H^1(\Omega; \mathbb{R}^2)\); \(u= g\) on \(\partial\Omega\}\). Minimizers of this functional have been studied extensively in the book of F. Bethuel, H. Brezis and F. Hélein [Ginzburg-Landau Vortices, Birkhäuser (1994; Zbl 0802.35142)], where an asymptotic analysis as \(\varepsilon\to 0\) was carried out. The aim of the present paper is to develop a variational framework in order to find non-minimizing solutions. The main result is:
Theorem 1. Assume that \(\text{deg}(g,\partial\Omega)\geq 2\). Then there is some \(\varepsilon_0> 0\) such that if \(\varepsilon\leq\varepsilon_0\), then at least three distinct solutions exist and one of these is not minimizing.
The proof of Theorem 1 is based on the study of the topology of the level sets of \(E_\varepsilon\), and this study is reduced to a similar problem for the renormalized energy (introduced in the above book) defined on a finite-dimensional space.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
82B26 Phase transitions (general) in equilibrium statistical mechanics
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