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

TL;DR

This paper extends the analysis of a Ginzburg-Landau type equation in two dimensions, establishing Liouville type results and quantization effects for solutions with potentials that may have infinite order zeros.

Contribution

It generalizes previous results by allowing potential functions with zeroes of infinite order, expanding understanding of solution behavior in this broader context.

Findings

Quantization of finite potential solutions established

Finite energy solutions characterized for more general potentials

Extension of previous polynomial-based results to infinite order zeroes

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.