Minimax Critical Points in Ginzburg-Landau Problems with Semi-stiff Boundary Conditions: Existence and Bubbling
Résumé
Let $\Omega\subset{\mathbb R}^2$ be smooth bounded simply connected. We consider the simplified Ginzburg-Landau energy $E_\varepsilon (u)$, where $u:\Omega\to{\mathbb C}$. On the boundary, we prescribe $|u|=1$ and deg$\, (u, \partial\Omega)=1$. In this setting, there are no minimizers of $E_\varepsilon$. Using a mountain pass approach, we obtain existence, for large $\varepsilon$, of critical points of $E_\varepsilon$. Our analysis relies on Wente estimates and on the analysis of bubbling phenomena for Palais-Smale sequences.
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