A general existence result for the Toda system on compact surfaces

Battaglia, Luca; Jevnikar, Aleks; Malchiodi, Andrea; Ruiz, David

doi:10.1016/j.aim.2015.07.036

Cited by 71 publications

(199 citation statements)

References 65 publications

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“…In case of no singularities, we get Λ = 4πN, which means that Theorem 1.1 can be extended, for the regular G 2 Toda system, to any ρ ∈ 4πN × R ∪ R × 4πN under no upper bound on ρ 1 , ρ 2 . The results in [20] also hold true for the singular A 2 Toda system, thus giving an extension of some of the variational existence results from [4,2,5] …”

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confidence: 95%

“…Arguing as in [4,2], we can infer by the following Lemma that if J B2,ρ (u) ≪ 0 (respectively, J G2,ρ (u) ≪ 0) then either f 1,u or f 2,u can accumulate mass only around a fixed number of points, hence it must be close to the corresponding space (Σ) Ki .…”

Section: Improved Moser-trudinger Inequalitymentioning

confidence: 99%

“…The proof of Theorem 4.1 can be deduced by arguing as in [4,2], Section 4. We present in details only the main new ingredient, which is a so-called improved Moser-Trudinger inequality.…”

Section: Improved Moser-trudinger Inequalitymentioning

confidence: 99%

“…In this paper we show for the first time the existence of min-max solutions for the B 2 and G 2 Toda system, by extending the general result from [4].…”

mentioning

confidence: 98%

“…A general existence result have been given in [10,9] for (6), as well as some existence results for (5) under an upper bound on one or both the ρ i (see [25,26,17]). Moreover, in [4] existence of solutions is shown under no assumptions on ρ but rather on the topology of Σ.…”

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confidence: 99%

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B 2 and G2 Toda systems on compact surfaces: A variational approach

Battaglia

2017

Journal of Mathematical Physics

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We consider the B2 and G2 Toda systems on a compact surface (Σ, g)where A = (aij) = 2 −1 −2 2 or 2 −1 −3 2 and hi ∈ C ∞ >0 (Σ), ρi ∈ R>0 are given. We attack the problem using variational techniques, following the previous work [4] concerning the A2 Toda system, namely the case A = 2 −1 −1 2 . We get existence and multiplicity of solutions as long as χ(Σ) ≤ 0 and ρ1, ρ2 ∈ 4πN. We also extend some of the results to the case of general systems. IntroductionLet (Σ, g) be a closed Riemann surface with surface area equal to 1. The B 2 and G 2 Toda systems are respectively the following systems of PDEs on Σ:Here, −∆ = −∆ g is the Laplace-Beltrami operator, ρ 1 , ρ 2 are positive parameters and h 1 , h 2 are positive smooth functions on Σ.Such systems are particularly interesting because their matrices of coefficients−3 2 * Sapienza Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 5, 00185 Romabattaglia@mat.uniroma1.it 1 are the Cartan matrices of the special orthonormal group SO(5) and of the symplectic group Sp(4), respectively. Together with A 2 = 2 −1 −1 2 , corresponding to SU (3), these are the only 2-dimensional Cartan matrices. Such Toda systems have important applications in both algebraic geometry and mathematical physics. In geometry, they appear in the study of complex holomorphic curves (see [8,13,7,11]); in physics, they arise in non-Abelian gauge field theory (see [19,18,13,12,14]).Both (1) and (2) are variational problems. In fact, solutions are respectively critical points of the following energy functionals:logˆΣ h 1 e u1 dV g −ˆΣ u 1 dV g − ρ 2 2 logˆΣ h 2 e u2 dV g −ˆΣ u 2 dV g ; (3) J G2,ρ (u) :=ˆΣ Q G2 (u)dV g − ρ 1 logˆΣ h 1 e u1 dV g −ˆΣ u 1 dV g − ρ 2 3 logˆΣ h 2 e u2 dV g −ˆΣ u 2 dV g . (4)

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mentioning

confidence: 95%

Section: Improved Moser-trudinger Inequalitymentioning

confidence: 99%

“…The proof of Theorem 4.1 can be deduced by arguing as in [4,2], Section 4. We present in details only the main new ingredient, which is a so-called improved Moser-Trudinger inequality.…”

Section: Improved Moser-trudinger Inequalitymentioning

confidence: 99%

“…In this paper we show for the first time the existence of min-max solutions for the B 2 and G 2 Toda system, by extending the general result from [4].…”

mentioning

confidence: 98%