A Liouville type result and quantization effects on the system $-\Delta u = u J'(1-|u|^{2})$ for a potential convex near zero

Abstract

We consider a Ginzburg-Landau type equation in $\R^2$ of the form $-\Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H.Brezis, F.Merle, T.Rivi\`ere from \cite{BMR} who treat the case when $J$ behaves polinomially near 0, as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.