Harnack type inequality: The method of moving planes
type: Item
instance of: scholarly article
title: Harnack type inequality: The method of moving planes
zbMATH Open document ID: 0928.35057
published in: Communications in Mathematical Physics
publication date: 1999-12-16T00:00:00Z
review text: Let \((M,g)\) be a compact Riemann surface without boundary and let \(\Delta_{g}\) be the associated Laplace-Beltrami operator. The main purpose of the paper is to provide a careful analysis of blowup solutions \((\xi_{n})\) in \(C^{2}(M)\) to the problems \[ - \Delta_{g} \xi_{n} = \lambda_{n} (V_{n} e^{\xi_{n}} - W_{n}) \quad\text{on \(M\)} , \qquad \int_{M} V_{n} e^{\xi_{n}} dv_{g} = 1 , \] where \((\lambda_{n})\) is convergent in \({\mathbb R}\) and \(V_{n},W_{n}\) are subjected to suitable uniform estimates. Such a result turns out to be a crucial step in the study of some semilinear elliptic problems, which originated from some Chern-Simons Higgs model. The main result is derived by a Harnack type inequality for a semilinear elliptic equation in dimension two, which is proved by the technique of moving planes.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35B99
zbMATH DE Number: 1271623
zbMATH Keywords: Chern-Simons Higgs model
zbMATH Keywords: blowup solutions
author: Yan Yan Li
reviewed by: Marco Degiovanni
MaRDI profile type: Publication
full work available at URL: s002200050536
OpenAlex ID: W2039470062
DOI: 10.1007/S002200050536
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description: scientific article; zbMATH DE number 1271623