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Radial minimizer of a variant of the \(p\)-Ginzburg-Landau functional. (English) Zbl 1107.35047
Summary: We study the asymptotic behavior of the radial minimizer of a variant of the \(p\)-Ginzburg-Landau functional \[ E_\varepsilon(u, B)=\int_G\left\{{1\over p}|\nabla u|^p+{1\over 4\varepsilon^p}|u|^2(1-|u|^2)^2\right\}\,dx \] in the set \[ W=\{u(x)=f(|x|){x\over|x|}\in W^{1,p}(B,\mathbb R^n); f(1)=1\}, \] where \(B\) is a unit ball in \(\mathbb R^n\), \(n\geq 2\), \(p\geq n\), and \(\varepsilon>0\). The location of the zeros and the uniqueness of the radial minimizer are derived. We also prove the \(W^{1,p}\) convergence of the radial minimizer for this functional.

MSC:
35J20 Variational methods for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
49J10 Existence theories for free problems in two or more independent variables
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