Asymptotic behavior of SU(3) Toda system in a bounded domain

Lin, Chang–Shou; Wei, Juncheng; Zhao, Chunyi

doi:10.1007/s00229-011-0451-z

Cited by 17 publications

(9 citation statements)

References 23 publications

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“…Even though we are not dealing with a closed surface, most of the variational theory for the Liouville equations and the Toda system can be applied in the very same way to Euclidean domains (or surfaces with boundary) with Dirichlet boundary conditions. This was explicitly pointed out in [2,4] for the Liouville equations, but still holds true for the Toda system, since blow-up on the boundary was excluded in [27]. In particular, the general existence result contained in [6] holds on any non-simply connected open domain of the plane, since such domains can be retracted on a bouquet of circles.…”

Section: Improved Moser-trudinger Inequalitiesmentioning

confidence: 88%

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

Battaglia

Malchiodi

2016

Journal of Functional Analysis

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We consider the SU (3) singular Toda system on a compact surface (Σ, g), where hi are smooth positive functions on Σ, ρi ∈ R+, pm ∈ Σ and αim > −1.We give both existence and non-existence results under some conditions on the parameters ρi and αim. Existence results are obtained using variational methods, which involve a geometric inequality of new type; non-existence results are obtained using blow-up analysis and localized Pohožaev-type identities.and it was shown that if a function u has low energy, then the normalized measure he u must distributionally approach the following set of measures (appeared also in [18])Using variational methods, a compactness result in [3] and a monotonicity argument in [37] it was also shown that, endowing Σ ρ,α with the weak topology of distributions, solutions to (6) (up to a discrete set of ρ's, for compactness reasons) exist provided Σ ρ,α is non-contractible. We notice that the problem is not always solvable, as in the classical case of the teardrop: the sphere with only one singular point. Sufficient and necessary conditions for contractibility were given in [11]. The case of positive singularities was treated in [1] on surfaces with positive genus. There are some other existence results ([2, 33]) which also work for the case of the sphere or of the real projective plane. We also refer to [14] for the derivation of a degree-counting formula.

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Section: Improved Moser-trudinger Inequalitiesmentioning

confidence: 88%

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

Battaglia

Malchiodi

2016

Journal of Functional Analysis

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“…In order to localize the previous arguments we give now a definition similar to the selection process in Proposition 2.1, which plays an important role in the following arguments. This kind of approach can be found also in [44,31,52].…”

Section: Exclusion Of Boundary Blow-upmentioning

confidence: 99%

“…To this end, we will follow the argument presented in [52], where the sinh-Gordon case (4) is studied. The argument was originally introduced in [31,44] in treating a fourth order mean field equation and the SU (3) Toda system, respectively. Recently, this strategy was refined in [52] to attack the sinh-Gordon problem (4).…”

Section: Introductionmentioning

confidence: 99%

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

Jevnikar

Yang

2017

Calc. Var.

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We are concerned with the following class of equations with exponential nonlinearities:which is related to the Tzitzéica equation. Here h 1 , h 2 are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue.In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions.In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result.2000 Mathematics Subject Classification. 35J61, 35J20, 35R01, 35B44.

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“…For Toda systems with singularities, the classification of local masses is given in Lin, Wei and Zhang [27]. Sharp estimates for fully blow-up solutions for SU (3) Toda system are given in Lin, Wei and Zhao [24,25]. See also related studies by Malchiodi-Ndiaye [32], Ohtsuka and Suzuki [37].…”

Section: )mentioning

confidence: 99%

“…In the case of a general domain Ω with no symmetry, for SU (3) Toda system with blow-up mass (8π, 4π), (4π, 8π), (8π, 8π), some necessary conditions are needed. For example, in the fully blowingup case, there are six necessary conditions (see Lin, Wei and Zhao [24,25]). For our problem, in the case of a general domain with no symmetry, we expect that there should be at least four necessary conditions.…”

Section: Introductionmentioning

confidence: 99%

New blow-up phenomena for SU(n+ 1) Toda system

Musso

Pistoia

Wei

2016

Journal of Differential Equations

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Asymptotic behavior of SU(3) Toda system in a bounded domain

Abstract: We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.

Cited by 17 publications

References 23 publications

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

Existence and non-existence results for the SU(3) singular Toda system on compact surfaces

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

New blow-up phenomena for SU(n+ 1) Toda system

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