## Lower bounds for generalized Ginzburg–Landau functionals.(English)Zbl 0928.35045

The author studies properties of Ginzburg-Landau functionals expressed by the integral of $| \nabla u| ^n/n+(1-| u| ^2)^2/(4\varepsilon^2),$ where $$u\in W^{1,n}(U; R^n)$$, $$U\subset R^n$$. Such information can be useful in the analysis of PDE’s associated with those functionals. The lower bounds relating the energy to the Brower degree of $$u$$ are established, and it is proved that the energy concentrates on a small number of small sets. As a consequence, some compactness theorems are derived. Also, the results are applied to the functional which is related to the magnetic potential.

### MSC:

 35J50 Variational methods for elliptic systems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35Q99 Partial differential equations of mathematical physics and other areas of application
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