Topological degree for a mean field equation on Riemann surfaces

By Chiun-Chuan Chen, Changshou Lin. 2003

338 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/440a8ef86c83a37a713b984c19baf2d5a7c998df

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Compactness of solutions to some geometric fourth-order equations (2004, background): “Corollary 1.3 has a counterpart in [35] (see also [ 21 ]), where compactness of solutions is proved for a mean field equation on compact surfaces.”
On the existence of blowing-up solutions for a mean field equation ? ? The first and second authors (2005, background): “We are aware of only another successful appro to handle multiple peak blow-up solutions which has been introduced by Chen and Lin in [16] for the an A general degree formula for the corresponding Fredholm map can be found in [15].”
Some existence results for the Toda system on closed surfaces (2007, methodology): “For > 4 , instead of using degree theory, as in [9] and [ 10 ] one can indeed employ directly a minimax scheme based on improvements of (6).”
Blow-up analysis of a Finsler–Liouville equation in two dimensions☆ (2012, background): “(2) have been intensively studied by many mathematicians, for there are many applications in geometric and physical problems, for example, in the problem of prescribing Gaussian curvature [9,12, 8], in the theory of the mean field equation [16,17,10,11,19,32] and in the Chern–Simons theory [38, 41,1”
Supercritical conformal metrics on surfaces with conical singularities (2011, result): “Other results, in the same direction of [10], have been recently announced in [11] (see [12]).”
Weighted barycentric sets and singular Liouville equations on compact surfaces (2011, methodology): “In the related recent literature, two guidelines can be highlighted: on the one hand, topological methods relying on the degree theory by Leray-Schauder (see [13]), on the other purely variational methods based on an improvement of the Moser-Trudinger inequality (5).”
Nondegeneracy of entire solutions of a singular Liouvillle equation (2012, neutral): “See also [3, 4, 7, 10] for related constructions.”
Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions (2005, methodology): “For the Leray-Schauder degree for the Liouville type equation, see [14] and [7].”
Mean field equations, hyperelliptic curves and modular forms: I (2015, background): “Recently, an explicit degree counting formula has been proved in [12, 14], which has the following consequence: Suppose that ρ ∈ (R>0r8πN), αj ∈ N for all j and the genus g(M) of M is at least 1.”
Blowup Solutions of Some Nonlinear Elliptic Equations Involving Exponential Nonlinearities (2006, methodology): “Chen-Lin [6] computed the topological degree of (8) by establishing a refined estimate of Li’s result for (8).”
Classification of blowup limits for SU(3) singular Toda systems (2013, background): “After decades of extensive study, many important questions related to the scalar Liouville equation are answered and the behavior of blowup s olutions is well understood (see [3, 4, 5, 13, 14] for related discussions).”
Singular limits in Liouville-type equations (2005, methodology): “The proof of Theorem 2 relies on the construction of an approximate solution, different from those in [2,15], which turns out to be precise enough, not only with its local maxima near a critical point of φm but everywhere in its domain.”
Classification and nondegeneracy of SU(n+1) Toda system with singular sources (2011, methodology): “For the past thirty years, the Liouville equation has been extensively studied by the method of nonlinear partial differential equations, see [4], [6], [7], [8], [23], [25],[31], [32], [34] and references therein.”
Mean Field Equation of Liouville Type with Singular Data: Topological Degree (2015, methodology): “From [19, 20], we have the following result: LEMMA 2.”