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

Weiwei  Ao

doi:10.3934/dcds.2016.36.1759

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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

Abstract
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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, 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), 1241-1265. 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), 599-620. 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), 937-979, arXiv:1306.5404. 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), 9-23. 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), 119-142. 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), 728-771. 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), 1667-1727. 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), 1237-1272. 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), 887-947. 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), 301-335. doi: 10.2307/1971312. Google Scholar
[13]	G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B,345 (1995), 452-457. doi: 10.1016/0370-2693(94)01649-W. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	A. Doliwa, Holomorphic curves and Toda system, Lett. Math. Phys., 39 (1997), 21-32. doi: 10.1007/s11005-997-1032-7. Google Scholar
[17]	P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. Google Scholar
[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. 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), 526-558. 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), 277-290. 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), 421-444. 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), 911-954. 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), 599-630. 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), 784-809. 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), 353-392. 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), 991-1027. 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), 733-758. 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), 675-705. 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), 1591-1635. 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), 1-18. 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), 169-207. 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), 807-837. 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), 117-143. 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), 556-590. 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), 2584-2636. 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), 1196-1217. 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), 332-371. 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), 391-412. 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), 31-94. 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), 599-639. 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), 2039-2060. 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), 1451-1467. doi: 10.1080/03605309608821234. Google Scholar
[45]	Y. S. Yang, The relativistic non-abelian Chern-Simons equation, Comm. Math. Phys., 186 (1999), 199-218. 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), 1241-1265. 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), 599-620. 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), 937-979, arXiv:1306.5404. 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), 9-23. 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), 119-142. 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), 728-771. 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), 1667-1727. 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), 1237-1272. 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), 887-947. 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), 301-335. doi: 10.2307/1971312. Google Scholar
[13]	G. Dunne, Mass degeneracies in self-dual models, Phys. Lett. B,345 (1995), 452-457. doi: 10.1016/0370-2693(94)01649-W. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	A. Doliwa, Holomorphic curves and Toda system, Lett. Math. Phys., 39 (1997), 21-32. doi: 10.1007/s11005-997-1032-7. Google Scholar
[17]	P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. Google Scholar
[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. 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), 526-558. 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), 277-290. 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), 421-444. 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), 911-954. 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), 599-630. 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), 784-809. 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), 353-392. 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), 991-1027. 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), 733-758. 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), 675-705. 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), 1591-1635. 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), 1-18. 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), 169-207. 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), 807-837. 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), 117-143. 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), 556-590. 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), 2584-2636. 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), 1196-1217. 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), 332-371. 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), 391-412. 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), 31-94. 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), 599-639. 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), 2039-2060. 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), 1451-1467. doi: 10.1080/03605309608821234. Google Scholar
[45]	Y. S. Yang, The relativistic non-abelian Chern-Simons equation, Comm. Math. Phys., 186 (1999), 199-218. doi: 10.1007/BF02885678. Google Scholar

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