Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 4 (2015), 807-837.

Classification of blowup limits for $\mathrm{SU}(3)$ singular Toda systems

Chang-Shou Lin, Jun-cheng Wei, and Lei Zhang

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For singular SU(3) Toda systems, we prove that the limit of energy concentration is a finite set. In addition, for fully bubbling solutions we use a Pohozaev identity to prove a uniform estimate. Our results extend previous results of Jost, Lin and Wang on regular SU(3) Toda systems.

Article information

Anal. PDE, Volume 8, Number 4 (2015), 807-837.

Received: 24 April 2014
Revised: 21 January 2015
Accepted: 6 March 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J55

$\operatorname{SU}(n+1)$-Toda system asymptotic analysis a priori estimate classification theorem topological degree blowup solutions


Lin, Chang-Shou; Wei, Jun-cheng; Zhang, Lei. Classification of blowup limits for $\mathrm{SU}(3)$ singular Toda systems. Anal. PDE 8 (2015), no. 4, 807--837. doi:10.2140/apde.2015.8.807.

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  • D. Bartolucci and A. Malchiodi, “An improved geometric inequality via vanishing moments, with applications to singular Liouville equations”, Comm. Math. Phys. 322:2 (2013), 415–452.
  • D. Bartolucci and G. Tarantello, “The Liouville equation with singular data: a concentration-compactness principle via a local representation formula”, J. Differential Equations 185:1 (2002), 161–180.
  • D. Bartolucci and G. Tarantello, “Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory”, Comm. Math. Phys. 229:1 (2002), 3–47.
  • 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 Differential Equations 29:7-8 (2004), 1241–1265.
  • L. Battaglia and A. Malchiodi, “A Moser–Trudinger inequality for the singular Toda system”, Bull. Inst. Math., Acad. Sin. $($N.S.$)$ 9:1 (2014), 1–23.
  • W. H. Bennet, “Magnetically self-focusing streams”, Phys. Rev. 45 (1934), 890–897.
  • J. Bolton and L. M. Woodward, “Some geometrical aspects of the $2$-dimensional Toda equations”, pp. 69–81 in Geometry, topology and physics (Campinas, 1996), edited by B. N. Apanasov et al., de Gruyter, Berlin, 1997.
  • J. Bolton, G. R. Jensen, M. Rigoli, and L. M. Woodward, “On conformal minimal immersions of $S\sp 2$ into $\mathbb{C} \mathrm{P}\sp n$”, Math. Ann. 279:4 (1988), 599–620.
  • E. Calabi, “Isometric imbedding of complex manifolds”, Ann. of Math. $(2)$ 58 (1953), 1–23.
  • S. Chanillo and M. K.-H. Kiessling, “Conformally invariant systems of nonlinear PDE of Liouville type”, Geom. Funct. Anal. 5:6 (1995), 924–947.
  • W. X. Chen and C. Li, “Classification of solutions of some nonlinear elliptic equations”, Duke Math. J. 63:3 (1991), 615–622.
  • W. X. Chen and C. Li, “Qualitative properties of solutions to some nonlinear elliptic equations in $\mathbb{R}\sp 2$”, Duke Math. J. 71:2 (1993), 427–439.
  • C.-C. Chen and C.-S. Lin, “Estimate of the conformal scalar curvature equation via the method of moving planes, II”, J. Differential Geom. 49:1 (1998), 115–178.
  • C.-C. Chen and C.-S. Lin, “Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces”, Comm. Pure Appl. Math. 55:6 (2002), 728–771.
  • C.-C. Chen and C.-S. Lin, “Topological degree for a mean field equation on Riemann surfaces”, Comm. Pure Appl. Math. 56:12 (2003), 1667–1727.
  • S. S. Chern and J. G. Wolfson, “Harmonic maps of the two-sphere into a complex Grassmann manifold, II”, Ann. of Math. $(2)$ 125:2 (1987), 301–335.
  • S. Childress and J. K. Percus, “Nonlinear aspects of chemotaxis”, Math. Biosci. 56:3-4 (1981), 217–237.
  • M. Chipot, I. Shafrir, and G. Wolansky, “On the solutions of Liouville systems”, J. Differential Equations 140:1 (1997), 59–105.
  • P. Debye and E. Huckel, “Zur Theorie der Electrolyte, II: Das Grengzgesetz für die elektrische Leitfähigkeit”, Phys. Zft. 24:10 (1923), 305–325.
  • A. Doliwa, “Holomorphic curves and Toda systems”, Lett. Math. Phys. 39:1 (1997), 21–32.
  • G. Dunne, Self-dual Chern–Simons theories, Lecture Notes in Physics Monographs 36, Springer, Berlin, 1995.
  • G. V. Dunne, R. Jackiw, S.-Y. Pi, and C. A. Trugenberger, “Self-dual Chern–Simons solitons and two-dimensional nonlinear equations”, Phys. Rev. D $(3)$ 43:4 (1991), 1332–1345.
  • N. Ganoulis, P. Goddard, and D. Olive, “Self-dual monopoles and Toda molecules”, Nuclear Phys. B 205:4 (1982), 601–636.
  • M. A. Guest, Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts 38, Cambridge University Press, Cambridge, 1997.
  • J. Jost, C. Lin, and G. Wang, “Analytic aspects of the Toda system, II: Bubbling behavior and existence of solutions”, Comm. Pure Appl. Math. 59:4 (2006), 526–558.
  • E. F. Keller and L. A. Segel, “Traveling bands of Chemotactic Bacteria: A theoretical analysis”, J. Theor. Biol. 30:2 (1971), 235–248.
  • M. K.-H. Kiessling and J. L. Lebowitz, “Dissipative stationary plasmas: kinetic modeling, Bennett's pinch and generalizations”, Phys. Plasmas 1:6 (1994), 1841–1849.
  • A. N. Leznov, “On complete integrability of a nonlinear system of partial differential equations in two-dimensional space”, Teoret. Mat. Fiz. 42:3 (1980), 343–349. In Russian; translated in Theor. Math. Phys. 42 (1980), 224–229.
  • A. N. Leznov and M. V. Saveliev, Group-theoretical methods for integration of nonlinear dynamical systems, Progress in Physics 15, Birkhäuser, Basel, 1992.
  • Y. Y. Li, “Prescribing scalar curvature on $S\sp n$ and related problems, I”, J. Differential Equations 120:2 (1995), 319–410.
  • Y. Y. Li, “Harnack type inequality: the method of moving planes”, Comm. Math. Phys. 200:2 (1999), 421–444.
  • C.-S. Lin and L. Zhang, “Profile of bubbling solutions to a Liouville system”, Ann. Inst. H. Poincaré Anal. Non Linéaire 27:1 (2010), 117–143.
  • C.-S. Lin and L. Zhang, “A topological degree counting for some Liouville systems of mean field type”, Comm. Pure Appl. Math. 64:4 (2011), 556–590.
  • C.-S. Lin, J. Wei, and D. Ye, “Classification and nondegeneracy of $\mathrm{SU}(n+1)$ Toda system with singular sources”, Invent. Math. 190:1 (2012), 169–207.
  • C.-S. Lin, J. Wei, and C. Zhao, “Sharp estimates for fully bubbling solutions of a $ \mathrm{SU}(3)$ Toda system”, Geom. Funct. Anal. 22:6 (2012), 1591–1635.
  • A. Malchiodi and C. B. Ndiaye, “Some existence results for the Toda system on closed surfaces”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei $(9)$ Mat. Appl. 18:4 (2007), 391–412.
  • A. Malchiodi and D. Ruiz, “A variational analysis of the Toda system on compact surfaces”, Comm. Pure Appl. Math. 66:3 (2013), 332–371.
  • P. Mansfield, “Solution of Toda systems”, Nuclear Phys. B 208:2 (1982), 277–300.
  • M. S. Mock, “Asymptotic behavior of solutions of transport equations for semiconductor devices”, J. Math. Anal. Appl. 49 (1975), 215–225.
  • M. Musso, A. Pistoia, and J. Wei, “New concentration phenomena for $\mathrm{SU}(3)$ Toda system”, preprint, 2015.
  • M. Nolasco and G. Tarantello, “Double vortex condensates in the Chern–Simons–Higgs theory”, Calc. Var. Partial Differential Equations 9:1 (1999), 31–94.
  • M. Nolasco and G. Tarantello, “Vortex condensates for the $\mathrm{SU}(3)$ Chern–Simons theory”, Comm. Math. Phys. 213:3 (2000), 599–639.
  • J. Prajapat and G. Tarantello, “On a class of elliptic problems in $\mathbb{R}\sp 2$: symmetry and uniqueness results”, Proc. Roy. Soc. Edinburgh Sect. A 131:4 (2001), 967–985.
  • R. Schoen, “Topics in differential geometry”, lecture notes, Stanford University, 1988.
  • Y. Yang, “The relativistic non-abelian Chern–Simons equations”, Comm. Math. Phys. 186:1 (1997), 199–218.
  • Y. Yang, Solitons in field theory and nonlinear analysis, Springer, New York, 2001. 0982.35003