American Institute of Mathematical Sciences

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

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

Weiwei Ao 1,

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.
Keywords: convergence rate, Toda system, sharp estimates, bubbling solutions., classification of solution.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J6.
Citation: Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete & Continuous Dynamical Systems - A, 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., ().   Google Scholar

[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, ().   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01458531.  Google Scholar

[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.  doi: 10.1016/j.aim.2015.07.036.  Google Scholar

[6]

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

[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.  doi: 10.1007/s002200000302.  Google Scholar

[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.  doi: 10.1002/cpa.3014.  Google Scholar

[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.  doi: 10.1002/cpa.10107.  Google Scholar

[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.  doi: 10.3934/dcds.2010.28.1237.  Google Scholar

[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.  doi: 10.1002/cpa.21532.  Google Scholar

[12]

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

[13]

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

[14]

G. Dunne, Self-dual Chern-Simons Theories,, Lect. Note Phys., 36 (1995).  doi: 10.1007/978-3-540-44777-1.  Google Scholar

[15]

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

[16]

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

[17]

P. Griffiths and J. Harris, Principles of Algebraic Geometry,, Wiley-Interscience, (1978).   Google Scholar

[18]

M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems,, London Mathematical Society Student Texts, (1997).  doi: 10.1017/CBO9781139174848.  Google Scholar

[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.  doi: 10.1002/cpa.20099.  Google Scholar

[20]

J. Jost and G. F. Wang, Classification of solutions of a Toda system in $\mathbbR^2$,, Int. Math. Res. Not., 6 (2002), 277.  doi: 10.1155/S1073792802105022.  Google Scholar

[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).   Google Scholar

[22]

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

[23]

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

[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.   Google Scholar

[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.  doi: 10.1002/cpa.10073.  Google Scholar

[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.  doi: 10.1007/s00205-012-0575-7.  Google Scholar

[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.  doi: 10.1002/cpa.21454.  Google Scholar

[28]

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

[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.  doi: 10.1007/s00209-010-0690-9.  Google Scholar

[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.  doi: 10.1007/s00039-012-0193-4.  Google Scholar

[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.  doi: 10.1007/s00229-011-0451-z.  Google Scholar

[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.  doi: 10.1007/s00222-012-0378-3.  Google Scholar

[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.  doi: 10.2140/apde.2015.8.807.  Google Scholar

[34]

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

[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.  doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[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.  doi: 10.1002/cpa.20355.  Google Scholar

[37]

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

[38]

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

[39]

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

[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.  doi: 10.4171/RLM/504.  Google Scholar

[41]

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

[42]

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

[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.  doi: 10.1512/iumj.2008.57.3324.  Google Scholar

[44]

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

[45]

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

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., ().   Google Scholar

[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, ().   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01458531.  Google Scholar

[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.  doi: 10.1016/j.aim.2015.07.036.  Google Scholar

[6]

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

[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.  doi: 10.1007/s002200000302.  Google Scholar

[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.  doi: 10.1002/cpa.3014.  Google Scholar

[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.  doi: 10.1002/cpa.10107.  Google Scholar

[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.  doi: 10.3934/dcds.2010.28.1237.  Google Scholar

[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.  doi: 10.1002/cpa.21532.  Google Scholar

[12]

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

[13]

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

[14]

G. Dunne, Self-dual Chern-Simons Theories,, Lect. Note Phys., 36 (1995).  doi: 10.1007/978-3-540-44777-1.  Google Scholar

[15]

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

[16]

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

[17]

P. Griffiths and J. Harris, Principles of Algebraic Geometry,, Wiley-Interscience, (1978).   Google Scholar

[18]

M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems,, London Mathematical Society Student Texts, (1997).  doi: 10.1017/CBO9781139174848.  Google Scholar

[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.  doi: 10.1002/cpa.20099.  Google Scholar

[20]

J. Jost and G. F. Wang, Classification of solutions of a Toda system in $\mathbbR^2$,, Int. Math. Res. Not., 6 (2002), 277.  doi: 10.1155/S1073792802105022.  Google Scholar

[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).   Google Scholar

[22]

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

[23]

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

[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.   Google Scholar

[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.  doi: 10.1002/cpa.10073.  Google Scholar

[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.  doi: 10.1007/s00205-012-0575-7.  Google Scholar

[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.  doi: 10.1002/cpa.21454.  Google Scholar

[28]

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

[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.  doi: 10.1007/s00209-010-0690-9.  Google Scholar

[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.  doi: 10.1007/s00039-012-0193-4.  Google Scholar

[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.  doi: 10.1007/s00229-011-0451-z.  Google Scholar

[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.  doi: 10.1007/s00222-012-0378-3.  Google Scholar

[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.  doi: 10.2140/apde.2015.8.807.  Google Scholar

[34]

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

[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.  doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[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.  doi: 10.1002/cpa.20355.  Google Scholar

[37]

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

[38]

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

[39]

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

[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.  doi: 10.4171/RLM/504.  Google Scholar

[41]

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

[42]

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

[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.  doi: 10.1512/iumj.2008.57.3324.  Google Scholar

[44]

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

[45]

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

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