A Liouville type result and quantization effects on the system -Δu = u J'(1-|u|²) for a potential convex near zero

2022/03/16 by U. De Maio, De Maio, U., Rejeb Hadiji +5
Mathematics · Computer Science · Engineering · #Advanced Mathematical Physics Problems #Advanced Mathematical Modeling in Engineering #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.08660

Abstract

We consider a Ginzburg-Landau type equation in \R² of the form -Δu = 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.Rivière from \citeBMR 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.

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