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In this paper, we study the asymptotic behaviour of minimizing solutions of aGinzburg-Landau type functional with a positive weight and with convexpotential near $0$ and we estimate the energy in this case. We also generalizea lower bound for the energy of unit vector field given initially byBrezis-Merle-Rivi\`ere.

A Ginzburg-Landau type energy with weight and with convex potential near zero