A LIOUVILLE TYPE RESULT AND QUANTIZATION EFFECTS ON THE SYSTEM −∆u = uJ′(1 − |u|^2) FOR A POTENTIAL CONVEX NEAR ZERO

We consider a Ginzburg-Landau type equation in R^2 of the form −∆u = uJ′(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. Rivìere from [10] 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.