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

Musso, Monica; Pistoia, Angela; Wei, Juncheng

doi:10.1016/j.jde.2015.12.036

Cited by 21 publications

(26 citation statements)

References 39 publications

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“…• In Theorem 1.3 it is essential to assume the matrix A to be positive definite. Otherwise, in [29,33,34,1] the authors build solutions to (1.3) whose masses can be arbitrarily large also on simply connected domains.…”

Section: Introductionmentioning

confidence: 99%

Uniform bounds for solutions to elliptic problems on simply connected planar domains

Battaglia

2019

Proc. Amer. Math. Soc.

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We consider the following elliptic problems on simply connected planar domainsWe show that any solution to each problem must satisfy a uniform bound on the mass, which is given respectively by λˆΩ |x| 2α K(x)e u dx and pˆΩ |x| 2α K(x)u p+1 dx. The same results applies to some systems and more general non-linearities. The proofs are based on the Riemann mapping theorem and a Pohožaev-type identity.

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Section: Introductionmentioning

confidence: 99%

Uniform bounds for solutions to elliptic problems on simply connected planar domains

Battaglia

2019

Proc. Amer. Math. Soc.

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“…α 1 = α 2 = 2, Jost, Lin and Wang in [34] found that the local masses can only take 5 values. Moreover, all these blow-up values can occur as shown by Musso, Pistoia and Wei in [49] (see also Ao and Wang in [1]).…”

Section: Introductionmentioning

confidence: 77%

A unified approach of blow-up phenomena for two-dimensional singular Liouville systems

Battaglia

Pistoia

2018

Rev. Mat. Iberoam.

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We consider generic 2 × 2 singular Liouville systemswhere Ω ∋ 0 is a smooth bounded domain in R 2 possibly having some symmetry with respect to the origin, δ0 is the Dirac mass at 0, λ1, λ2 are small positive parameters and a, b, α1, α2 > 0.We construct a family of solutions to (1) which blow up at the origin as λ1 → 0 and λ2 → 0 and whose local mass at the origin is a given quantity depending on a, b, α1, α2.In particular, if ab < 4 we get finitely many possible blow-up values of the local mass, whereas if ab ≥ 4 we get infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.

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“…It is interesting to see whether any pair of the above is really the local masses of some sequence of blowup solutions of (1.2). For K = A 2 the existence of such a local solution has been obtained (see [30] and [24]). We remark that parts of Corollary 1.5 was already proved by Jost-Lin-Wang [15], and by the first author and the fourth author in [27].…”

Section: Definition 14 a Pair Of Local Massesmentioning

confidence: 93%

On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

et al. 2018

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For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses (σ 1 , σ 2 ) at each blowup point has the expressionthen all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point q is not a vortex point, then u k (x) + 2 log |x − x k | ≤ C, where x k is the local maximum point of u k near q. (iv) If the blow up point q is a vortex point p t and α 1 t , α 2 t and 1 are linearly independent over Q, then u k (x) + 2 log |x − p t | ≤ C.The Harnack type inequalities of (iii) or (iv) are important for studying the behavior of bubbling solutions near each blowup points.

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

Abstract: Abstract. We consider the SU (n + 1) Toda system (S λ )

Cited by 21 publications

References 39 publications

Uniform bounds for solutions to elliptic problems on simply connected planar domains

Uniform bounds for solutions to elliptic problems on simply connected planar domains

A unified approach of blow-up phenomena for two-dimensional singular Liouville systems

On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

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