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.
Similar content being viewed by others
References
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)
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)
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)
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)
Busca J., Sirakov B.: Symmetry results for semilinear elliptic systems in the whole space. J. Diff. Equ. 163(1), 41–56 (1991)
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)
Chen C.-C., Lin C.-S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56, 1667–1727 (2003)
Del Pino M., Kowalczyk M., Musso M.: Singular limits in Liouville-type equations. Calc. Var. Partial Diff. Equ. 24, 47–81 (2005)
Dunne, G.: Self-Dual Chern-Simons Theories, Lecture Notes in Physics: Monographs, vol. 36. Springer-Verlag, Berlin Heidelberg (1995)
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)
Jost J., Wang G.F.: Classification of solutions of a Toda system in \({{\mathbb R}^2}\). Int. Math. Res. Not. 6, 277–290 (2002)
Jost J., Wang G.F.: Analytic aspects of the Toda system: I Moser-Trudinger inequality. Comm. Pure Appl. Math. 54(11), 1289–1319 (2001)
Li Y.Y.: Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200(2), 421–444 (1999)
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)
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)
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)
Lin, C.-S., Wei, J.C., Zhao, C.Y.: Refined asymptotic behavior of SU(3) Toda system in a bounded domain (preprint)
Ma L., Wei J.C.: Convergence for a Liouville equation. Comment. Math. Helv. 76(3), 506–514 (2001)
Nagasaki K., Suzuki T.: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Anal. 3(2), 173–188 (1990)
Ohtsuka H., Suzuki T.: Blow-up analysis for SU(3) Toda system. J. Diff. Equ. 232(2), 419–440 (2007)
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)
Troy W.: Symmetry properties in systems of semilinear elliptic equations, J. Diff. Equ. 42, 400–413 (1981)
Tarantello, G.: Selfdual Gauge Field Vortices. An analytical approach. Progress in Nonlinear Differential Equations and their Applications, vol. 72. Birkhäuser Boston, Inc., Boston (2008)
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)
Yang,Y.S.: Solitons in Field theory and Nonlinear Analysis. Springer Monographs in Mathematics. Springer, New York (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-011-0451-z