On Liouville systems at critical parameters, Part 1: One bubble

Lin, Chang–Shou; Zhang, Lei

doi:10.1016/j.jfa.2013.02.022

Cited by 28 publications

(46 citation statements)

References 43 publications

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“…So it is natural for us to study (1.3) from analysis viewpoints as well as the perspectives from integrable systems. From the analytic side, the most important issue is to derive a degree counting formula for equation (1.3), a generalization of the previous works of Chen-Lin [10,11] for (1.1) and Lin-Zhang [31,32,33] for general Liouville systems. However this generalization is very challenging because the bubbling phenomena are more complicated and the concentration has not yet been proven even for SU (3) Toda system.…”

Section: Introductionmentioning

confidence: 99%

Energy Concentration and A Priori Estimates forB₂andG₂Types of Toda Systems

Lin¹,

Zhang²

2015

Int Math Res Notices

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For Toda systems with Cartan matrix either B 2 or G 2 , we prove that the local mass of blowup solutions at its blowup points converges to a finite set. Further more this finite set can be completely determined for B 2 Toda systems, while for G 2 systems we need one additional assumption. As an application of the local mass classification we establish a priori estimates for corresponding Toda systems defined on Riemann surfaces.Recently equation (1.2) is found to have deep connection with the classical Lame equation and also the Painleve VI equation. For example for the Painleve VI equation with certain parameters, some non-existence theorem of (1.2) plays a key part in the proof of the smoothness of the solutions with unitary monodromy group. The interested readers may read into [9] and [12] for more in-depth discussions.

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Section: Introductionmentioning

confidence: 99%

Energy Concentration and A Priori Estimates forB₂andG₂Types of Toda Systems

Lin¹,

Zhang²

2015

Int Math Res Notices

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“…where r = |x − x j,ε |. Then, using the same argument of Lin-Zhang(page 2608 of [38]), we have an estimate for the radial part ofη i, j,ε . pro42 Proposition 5.2.…”

Section: Profile Of the Bubbling Solutionsmentioning

confidence: 89%

The Domain Geometry and the Bubbling Phenomenon of Rank Two Gauge Theory

Huang

Zhang

2016

Commun. Math. Phys.

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ABSTRACT. Let Ω be a flat torus and G be the green's function of −∆ on Ω. One intriguing mystery of G is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for the following equation that describes a Chern-Simons model in Gauge theory: e103 e103 (0.1)if fully bubbling solutions of Liouville type exist, G has exactly three critical points. In addition we establish necessary conditions for the existence of fully bubbling solutions with multiple bubbles.

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“…The construction we perform here is inspired by a recent result obtained by Grossi and Pistoia [16], where they consider the sinh-Poisson equation Let us comment on some recent related works. In [29,30,31], Lin and Zhang studied general Liouville type systems with nonnegative coefficients. For Toda systems with singularities, the classification of local masses is given in Lin, Wei and Zhang [27].…”

Section: )mentioning

confidence: 99%