April  2016, 36(4): 1759-1788. doi: 10.3934/dcds.2016.36.1759

Sharp estimates for fully bubbling solutions of $B_2$ Toda system

1. 

Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada

Received  January 2015 Revised  May 2015 Published  September 2015

In this paper, we obtain sharp estimates of fully bubbling solutions of the $B_2$ Toda system in a compact Riemann surface. Our main goal in this paper are (i) to obtain sharp convergence rate, (ii) to completely determine the location of bubbles, (iii) to derive the $\partial_z^2$ condition.
Citation: Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759
References:
[1]

W. W. Ao, C. S. Lin and J. C. Wei, On Non-topological Solutions of the $A_2$ and $B_2$ Chern-Simons System, Memoirs of Amer. Math. Soc., to appear.

[2]

W. W. Ao, C. S. Lin and J. C. Wei, On Non-topological Solutions of the $G_2$ Chern-Simons System, Comm. Analysis and Geometry, to appear.

[3]

D. Bartolucci, C. C. Chen, C. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data. Comm. Partial Diff. Equ., 29 (2004), 1241-1265.

[4]

J. Bolton, G. R. Jensen, M. Rigoli and L. M. Woodward, On conformal minimal immersions of $S^2$ into $CP^n$, Mathematische Annalen, 279 (1988), 599-620. doi: 10.1007/BF01458531.

[5]

L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz, A general existence result for the Toda system on compact surfaces, Advances in Mathematics, 285 (2015), 937-979, arXiv:1306.5404. doi: 10.1016/j.aim.2015.07.036.

[6]

L. Battaglia and A. Malchiodi, A Moser-Trudinger inequality for the singular Toda system, Bull. Inst. Math. Acad. Sin., 9 (2014), 9-23.

[7]

D. Chae and O. Y. Imanuvilov, The existence of non-topological multi-vortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.

[8]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771. doi: 10.1002/cpa.3014.

[9]

C. C. Chen and C. S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727. doi: 10.1002/cpa.10107.

[10]

C. C. Chen and C. S. Lin, Mean field equations of Liouville type with singular data: Sharper estimates, Discrete Contin. Dyn. Syst., 28 (2010), 1237-1272. doi: 10.3934/dcds.2010.28.1237.

[11]

C. C. Chen and C. S. Lin, Mean field equations of Liouville type with the singular data: Topological formula, Comm. Pure Appl. Math., 68 (2015), 887-947. doi: 10.1002/cpa.21532.

[12]

S. S. Chern and J. G. Wolfson, Maps of the two-sphere into a complex Grassmann manifold. II, Annal. Math., 125 (1987), 301-335. doi: 10.2307/1971312.

[13]

G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B,345 (1995), 452-457. doi: 10.1016/0370-2693(94)01649-W.

[14]

G. Dunne, Self-dual Chern-Simons Theories, Lect. Note Phys., 36 (1995), Berlin-New York, Spring-Verlag. doi: 10.1007/978-3-540-44777-1.

[15]

G. Dunne, Vacuum mass spectra for $SU(N)$ self-dual Chern-Simons-Higgs, Nucl. Phys. B, 433 (1995), 333-348. doi: 10.1016/0550-3213(94)00476-U.

[16]

A. Doliwa, Holomorphic curves and Toda system, Lett. Math. Phys., 39 (1997), 21-32. doi: 10.1007/s11005-997-1032-7.

[17]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.

[18]

M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, London Mathematical Society Student Texts, vol. 38. Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9781139174848.

[19]

J. Jost, C. S. Lin and G. F. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., 59 (2006), 526-558. doi: 10.1002/cpa.20099.

[20]

J. Jost and G. F. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., 6 (2002), 277-290. doi: 10.1155/S1073792802105022.

[21]

T. J. Kuo and C. S. Lin, Sharp estimate of solutions to mean field equation with integer singular sources: the first order approximation, preprint, 2013.

[22]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536.

[23]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equations on tori, Annal. Math., 172 (2010), 911-954. doi: 10.4007/annals.2010.172.911.

[24]

C. S. Lin and J. C. Wei, Sharp estimates for bubbling solutions of a fourth order mean field equation, Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 6 (2007), 599-630.

[25]

C. S。 Lin and J. C. Wei, Locating the peaks of solutions via the maximum principle. II. A local version of the method of moving planes, Comm. Pure Appl. Math., 56 (2003), 784-809. doi: 10.1002/cpa.10073.

[26]

C. S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arch. Ration. Mech. Anal., 207 (2013), 353-392. doi: 10.1007/s00205-012-0575-7.

[27]

C. S. Lin and S. Yan, Bubbling solutions for the $SU(3)$ Chern-Simon model on a torus, Comm. Pure Appl. Math, 66 (2013), 991-1027. doi: 10.1002/cpa.21454.

[28]

C. S. Lin and S. Yan, Bubbling solutions for relativistic Abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758. doi: 10.1007/s00220-010-1056-1.

[29]

C. S. Lin, L. P. Wang and J. C. Wei, Topological degree for solutions of a fourth order mean field equation, Math. Zeit., 268 (2011), 675-705. doi: 10.1007/s00209-010-0690-9.

[30]

C. S. Lin, J. C. Wei and C. Zhao, Sharp estimates for fully bubbling solutions of a $SU(3)$ Toda system, Geom. Funct. Anal., 22 (2012), 1591-1635. doi: 10.1007/s00039-012-0193-4.

[31]

C. S. Lin, J. C. Wei and C. Y. Zhao, Asymptotic behavior of $SU(3)$ Toda system in a bounded domain}, Manus. Math., 137 (2012), 1-18. doi: 10.1007/s00229-011-0451-z.

[32]

C. S. Lin, J. C. Wei and D. Ye, Classification and non-degeneracy of $SU(n + 1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207. doi: 10.1007/s00222-012-0378-3.

[33]

C. S. Lin, J. C. Wei and L. Zhang, Classification of blowup limits for $SU(3)$ Toda singular Toda system, Analysis and PDE, 8 (2015), 807-837. doi: 10.2140/apde.2015.8.807.

[34]

C. S. Lin, J. C. Wei and L. Zhang, Local profile of fully bubbling solutions to $SU(n+1)$ Toda system, preprint, arXiv:1308.1579 .

[35]

C. S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Annal. Inst. H. Poincar Anal. Non Lineaire, 27 (2010), 117-143. doi: 10.1016/j.anihpc.2009.09.001.

[36]

C. S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field equations, Comm. Pure Appl. Math., 64 (2011), 556-590. doi: 10.1002/cpa.20355.

[37]

C. S. Lin and L. Zhang, On Liouville systems at critical parameters, part 1: One bubble, J. Funct. Anal., 264 (2013), 2584-2636. doi: 10.1016/j.jfa.2013.02.022.

[38]

A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geom. Funct. Anal., 21 (2011), 1196-1217. doi: 10.1007/s00039-011-0134-7.

[39]

A. Malchiodi and D. Ruiz, A variational analysis of the Toda system on compact surfaces, Comm. Pure Appl. Math., 66 (2013), 332-371. doi: 10.1002/cpa.21433.

[40]

A. Malchiodi and C. B. Ndiaye, Some existence results for the Toda system on closed surfaces, Atti Dell'accademia Pontificia Dei Nuovi Lincei, 18 (2007), 391-412. doi: 10.4171/RLM/504.

[41]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons theory, Cal. Var. Partial Diff. Equ., 9 (1999), 31-94. doi: 10.1007/s005260050132.

[42]

M. Nolasco and G. Tarantello, Vortex condensates for the $SU(3)$ Chern-Simons theory, Comm. Math. Phys., 213 (2000), 599-639. doi: 10.1007/s002200000252.

[43]

F. Robert and J. C. Wei, Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition, Indiana University Math. J., 57 (2008), 2039-2060. doi: 10.1512/iumj.2008.57.3324.

[44]

J. C. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Diff. Equ., 21 (1996), 1451-1467. doi: 10.1080/03605309608821234.

[45]

Y. S. Yang, The relativistic non-abelian Chern-Simons equation, Comm. Math. Phys., 186 (1999), 199-218. doi: 10.1007/BF02885678.

show all references

References:
[1]

W. W. Ao, C. S. Lin and J. C. Wei, On Non-topological Solutions of the $A_2$ and $B_2$ Chern-Simons System, Memoirs of Amer. Math. Soc., to appear.

[2]

W. W. Ao, C. S. Lin and J. C. Wei, On Non-topological Solutions of the $G_2$ Chern-Simons System, Comm. Analysis and Geometry, to appear.

[3]

D. Bartolucci, C. C. Chen, C. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data. Comm. Partial Diff. Equ., 29 (2004), 1241-1265.

[4]

J. Bolton, G. R. Jensen, M. Rigoli and L. M. Woodward, On conformal minimal immersions of $S^2$ into $CP^n$, Mathematische Annalen, 279 (1988), 599-620. doi: 10.1007/BF01458531.

[5]

L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz, A general existence result for the Toda system on compact surfaces, Advances in Mathematics, 285 (2015), 937-979, arXiv:1306.5404. doi: 10.1016/j.aim.2015.07.036.

[6]

L. Battaglia and A. Malchiodi, A Moser-Trudinger inequality for the singular Toda system, Bull. Inst. Math. Acad. Sin., 9 (2014), 9-23.

[7]

D. Chae and O. Y. Imanuvilov, The existence of non-topological multi-vortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.

[8]

C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771. doi: 10.1002/cpa.3014.

[9]

C. C. Chen and C. S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727. doi: 10.1002/cpa.10107.

[10]

C. C. Chen and C. S. Lin, Mean field equations of Liouville type with singular data: Sharper estimates, Discrete Contin. Dyn. Syst., 28 (2010), 1237-1272. doi: 10.3934/dcds.2010.28.1237.

[11]

C. C. Chen and C. S. Lin, Mean field equations of Liouville type with the singular data: Topological formula, Comm. Pure Appl. Math., 68 (2015), 887-947. doi: 10.1002/cpa.21532.

[12]

S. S. Chern and J. G. Wolfson, Maps of the two-sphere into a complex Grassmann manifold. II, Annal. Math., 125 (1987), 301-335. doi: 10.2307/1971312.

[13]

G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B,345 (1995), 452-457. doi: 10.1016/0370-2693(94)01649-W.

[14]

G. Dunne, Self-dual Chern-Simons Theories, Lect. Note Phys., 36 (1995), Berlin-New York, Spring-Verlag. doi: 10.1007/978-3-540-44777-1.

[15]

G. Dunne, Vacuum mass spectra for $SU(N)$ self-dual Chern-Simons-Higgs, Nucl. Phys. B, 433 (1995), 333-348. doi: 10.1016/0550-3213(94)00476-U.

[16]

A. Doliwa, Holomorphic curves and Toda system, Lett. Math. Phys., 39 (1997), 21-32. doi: 10.1007/s11005-997-1032-7.

[17]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.

[18]

M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, London Mathematical Society Student Texts, vol. 38. Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9781139174848.

[19]

J. Jost, C. S. Lin and G. F. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., 59 (2006), 526-558. doi: 10.1002/cpa.20099.

[20]

J. Jost and G. F. Wang, Classification of solutions of a Toda system in $\mathbb{R}^2$, Int. Math. Res. Not., 6 (2002), 277-290. doi: 10.1155/S1073792802105022.

[21]

T. J. Kuo and C. S. Lin, Sharp estimate of solutions to mean field equation with integer singular sources: the first order approximation, preprint, 2013.

[22]

Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444. doi: 10.1007/s002200050536.

[23]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equations on tori, Annal. Math., 172 (2010), 911-954. doi: 10.4007/annals.2010.172.911.

[24]

C. S. Lin and J. C. Wei, Sharp estimates for bubbling solutions of a fourth order mean field equation, Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 6 (2007), 599-630.

[25]

C. S。 Lin and J. C. Wei, Locating the peaks of solutions via the maximum principle. II. A local version of the method of moving planes, Comm. Pure Appl. Math., 56 (2003), 784-809. doi: 10.1002/cpa.10073.

[26]

C. S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arch. Ration. Mech. Anal., 207 (2013), 353-392. doi: 10.1007/s00205-012-0575-7.

[27]

C. S. Lin and S. Yan, Bubbling solutions for the $SU(3)$ Chern-Simon model on a torus, Comm. Pure Appl. Math, 66 (2013), 991-1027. doi: 10.1002/cpa.21454.

[28]

C. S. Lin and S. Yan, Bubbling solutions for relativistic Abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758. doi: 10.1007/s00220-010-1056-1.

[29]

C. S. Lin, L. P. Wang and J. C. Wei, Topological degree for solutions of a fourth order mean field equation, Math. Zeit., 268 (2011), 675-705. doi: 10.1007/s00209-010-0690-9.

[30]

C. S. Lin, J. C. Wei and C. Zhao, Sharp estimates for fully bubbling solutions of a $SU(3)$ Toda system, Geom. Funct. Anal., 22 (2012), 1591-1635. doi: 10.1007/s00039-012-0193-4.

[31]

C. S. Lin, J. C. Wei and C. Y. Zhao, Asymptotic behavior of $SU(3)$ Toda system in a bounded domain}, Manus. Math., 137 (2012), 1-18. doi: 10.1007/s00229-011-0451-z.

[32]

C. S. Lin, J. C. Wei and D. Ye, Classification and non-degeneracy of $SU(n + 1)$ Toda system with singular sources, Invent. Math., 190 (2012), 169-207. doi: 10.1007/s00222-012-0378-3.

[33]

C. S. Lin, J. C. Wei and L. Zhang, Classification of blowup limits for $SU(3)$ Toda singular Toda system, Analysis and PDE, 8 (2015), 807-837. doi: 10.2140/apde.2015.8.807.

[34]

C. S. Lin, J. C. Wei and L. Zhang, Local profile of fully bubbling solutions to $SU(n+1)$ Toda system, preprint, arXiv:1308.1579 .

[35]

C. S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Annal. Inst. H. Poincar Anal. Non Lineaire, 27 (2010), 117-143. doi: 10.1016/j.anihpc.2009.09.001.

[36]

C. S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field equations, Comm. Pure Appl. Math., 64 (2011), 556-590. doi: 10.1002/cpa.20355.

[37]

C. S. Lin and L. Zhang, On Liouville systems at critical parameters, part 1: One bubble, J. Funct. Anal., 264 (2013), 2584-2636. doi: 10.1016/j.jfa.2013.02.022.

[38]

A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geom. Funct. Anal., 21 (2011), 1196-1217. doi: 10.1007/s00039-011-0134-7.

[39]

A. Malchiodi and D. Ruiz, A variational analysis of the Toda system on compact surfaces, Comm. Pure Appl. Math., 66 (2013), 332-371. doi: 10.1002/cpa.21433.

[40]

A. Malchiodi and C. B. Ndiaye, Some existence results for the Toda system on closed surfaces, Atti Dell'accademia Pontificia Dei Nuovi Lincei, 18 (2007), 391-412. doi: 10.4171/RLM/504.

[41]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons theory, Cal. Var. Partial Diff. Equ., 9 (1999), 31-94. doi: 10.1007/s005260050132.

[42]

M. Nolasco and G. Tarantello, Vortex condensates for the $SU(3)$ Chern-Simons theory, Comm. Math. Phys., 213 (2000), 599-639. doi: 10.1007/s002200000252.

[43]

F. Robert and J. C. Wei, Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition, Indiana University Math. J., 57 (2008), 2039-2060. doi: 10.1512/iumj.2008.57.3324.

[44]

J. C. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Diff. Equ., 21 (1996), 1451-1467. doi: 10.1080/03605309608821234.

[45]

Y. S. Yang, The relativistic non-abelian Chern-Simons equation, Comm. Math. Phys., 186 (1999), 199-218. doi: 10.1007/BF02885678.

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