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Asymptotic behavior of SU(3) Toda system in a bounded domain

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Abstract

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Baraket S., Pacard F.: Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Diff. Equ. 6, 1–38 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezis H., Li Y.Y., Shafrir I.: A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities. J. Funct. Anal. 115(2), 344–358 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezis H., Merle F.: Uniform estimates and blow-up behavior for solutions of −Δu = V(x)e u in two dimensions. Comm. Partial Diff. Equ. 16(8–9), 1223–1253 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Busca J., Sirakov B.: Symmetry results for semilinear elliptic systems in the whole space. J. Diff. Equ. 163(1), 41–56 (1991)

    Article  MathSciNet  Google Scholar 

  6. Chen C.-C., Lin C.-S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surface. Comm. Pure Appl. Math. 55, 728–771 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen C.-C., Lin C.-S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56, 1667–1727 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Del Pino M., Kowalczyk M., Musso M.: Singular limits in Liouville-type equations. Calc. Var. Partial Diff. Equ. 24, 47–81 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dunne, G.: Self-Dual Chern-Simons Theories, Lecture Notes in Physics: Monographs, vol. 36. Springer-Verlag, Berlin Heidelberg (1995)

  10. Jost J., Lin C.-S., Wang G.F.: Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions. Comm. Pure Appl. Math. 59(4), 526–558 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jost J., Wang G.F.: Classification of solutions of a Toda system in \({{\mathbb R}^2}\). Int. Math. Res. Not. 6, 277–290 (2002)

    Article  MathSciNet  Google Scholar 

  12. Jost J., Wang G.F.: Analytic aspects of the Toda system: I Moser-Trudinger inequality. Comm. Pure Appl. Math. 54(11), 1289–1319 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Li Y.Y.: Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200(2), 421–444 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Li Y.Y., Shafrir I.: Blow-up analysis for solutions of −Δu = V e u in dimension two. Indiana Univ. Math. J. 43(4), 1255–1270 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin C.-S., Wei J.C.: Sharp estimates for bubbling solutions of a fourth order mean field equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5–6(4), 599–630 (2007)

    MathSciNet  Google Scholar 

  16. Lin C.-S., Zhang L.: Profile of bubbling solutions to a Liouville system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(1), 117–143 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin, C.-S., Wei, J.C., Zhao, C.Y.: Refined asymptotic behavior of SU(3) Toda system in a bounded domain (preprint)

  18. Ma L., Wei J.C.: Convergence for a Liouville equation. Comment. Math. Helv. 76(3), 506–514 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nagasaki K., Suzuki T.: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Anal. 3(2), 173–188 (1990)

    MATH  MathSciNet  Google Scholar 

  20. Ohtsuka H., Suzuki T.: Blow-up analysis for SU(3) Toda system. J. Diff. Equ. 232(2), 419–440 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Robert F., Wei J.C.: Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition. Indiana Univ. Math. J. 57(5), 2039–2060 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Troy W.: Symmetry properties in systems of semilinear elliptic equations, J. Diff. Equ. 42, 400–413 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tarantello, G.: Selfdual Gauge Field Vortices. An analytical approach. Progress in Nonlinear Differential Equations and their Applications, vol. 72. Birkhäuser Boston, Inc., Boston (2008)

  24. Wei J.C., Zhao C.Y., Zhou F.: On nondegeneracy of solutions to SU(3) Toda system. C.R.A.S. 349, 185–190 (2011)

    MATH  MathSciNet  Google Scholar 

  25. Yang,Y.S.: Solitons in Field theory and Nonlinear Analysis. Springer Monographs in Mathematics. Springer, New York (2001)

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Correspondence to Juncheng Wei.

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Lin, CS., Wei, J. & Zhao, C. Asymptotic behavior of SU(3) Toda system in a bounded domain. manuscripta math. 137, 1–18 (2012). https://doi.org/10.1007/s00229-011-0451-z

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  • DOI: https://doi.org/10.1007/s00229-011-0451-z

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