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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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