Abstract
In this paper, using a geometric method we show that the blow-up values of the elliptic sinh-Gordon equation are multiples of 8π.
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Abresch U. (1987). Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374: 169–192
Aubin, T.: Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, xviii+395 pp. Springer, Berlin (1998)
Bobenko A.I. (1991). All constant mean curvature tori in \({\mathbb{R}}^3, S^3, {\mathbb{H}}^3\) Math. Ann. 290: 209–245
Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. Harmonic maps and integrable systems, pp. 83–127, Aspects Math. E23, Vieweg, Braunschweig (1994)
Brezis H. and Coron J.-M. (1984). Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37: 149–187
Brezis H. and Coron J.-M. (1983). Large solutions of harmonic maps in two dimensions. Commun. Math. Phys. 92: 203–215
Brezis H. and Merle F. (1991). Uniform estimates and blow-up behavior for solutions of −Δ u = V(x)e u in two dimensions. Commun. Partial Diff. Equ. 16: 1223–1253
Chen X.X. (1999). Remarks on the existence of branch bubbles on the blowup analysis of equation −Δu = e 2u in dimension two. Commun. Anal. Geom. 7: 295–302
Chorin A.J. (1994). Vorticity and Turbulence. Springer, New York
Ding W., Jost J., Li J. and Wang G. (1999). Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 16: 653–666
Ding W.Y. and Tian G. (1995). Energy identity for a class of approximate harmonic maps from surfaces. Commun. Anal. Geom. 3: 543–554
Hélein F. (1991). Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I 312: 591–596
Hopf, H.: Lectures on differential geometry in the large. Stanford Lecture Notes 1955; reprinted in Lecture Notes in Math. 1000, Springer, Heidelberg (1984)
Jost J. (1984). The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with non-constant boundary values. J. Diff. Geom. 19: 393–401
Jost, J.: Two-dimensional geometric variational problems. Pure and Applied Mathematics (New York), Wiley, Chichester (1991)
Jost J. and Wang G. (2001). Analytic aspects of the Toda system. I. A Moser-Trudinger inequality I. A Moser- Trudinger inequality. Commun. Pure Appl. Math. 54: 1289–1319
Jost J., Lin C.S. and Wang G. (2006). Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions. Commun. Pure Appl. Math. 59: 526–558
Joyce G. and Montgomery D. (1973). Negative temperature states for the two-dimensional guiding-centre plasma. J. Plasma Phys. 10: 107–121
Li Y.Y. and Shafrir I. (1994). Blow-up analysis for solutions of −Δu = Ve u in dimension two. Indiana Univ. Math. J. 43: 1255–1270
Lin F.H. and Wang C.Y. (1998). Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. Partial Differ. Equ. 6: 369–380
Lions P.-L. (1997). On Euler Equations and Statistical Physics. Scuola Normale Superiore, Pisa
Lucia M. and Nolasco M. (2002). SU(3) Chern-Simons vortex theory and Toda Systems. J. Differ. Equ. 184: 443–474
Marchioro C. and Pulvirenti M. (1994). Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York
Newton P.K. (2001). The N-Vortex Problem: Analytical Techniques. Springer, New York
Ohtsuka H. and Suzuki T. (2006). Mean field equation for the equilibrium turbulence and a related functional inequality. Adv. Differ. Equ. 11: 281–304
Parker T.H. (1996). Bubble tree convergence for harmonic maps. J. Differ. Geom. 44: 595–633
Pinkall U. and Sterling I. (1989). On the classification of constant mean curvature tori. Ann. Math. 130: 407–451
Pointin Y.B. and Lundgren T.S. (1976). Statistical mechanics of two dimensional vortices in a bounded container. Phys. Fluids 19: 1459–1470
Ricciardi, T.: Mountain pass solutions for a mean field equation from two-dimensional turbulence. ArXiv: math. AP/0612123 (2006)
Rivière, T.: Conservation laws for conformal invariant variational problems. ArXiv math. AP/0603380 (2006)
Sacks J. and Uhlenbeck K. (1981). The existence of minimal immersions of 2-spheres. Ann. Math. 113: 1–24
Sawada K., Suzuki T. and Takahashi F. (2007). Mean field equation for equilibrium vortices with neutral orientation. Nonlinear Anal. 66(2): 509–526
Shafrir I. and Wolansky G. (2005). Moser-Trudinger and logarithmic HLS inequalities for systems. J. Eur. Math. Soc. 7: 413–448
Spruck, J.: The elliptic sinh Gordon equation and the construction of toroidal soap bubbles. Calculus of variations and partial differential equations (Trento, 1986), pp. 275–301, Lecture Notes in Math. vol. 1340, Springer, Berlin (1988)
Steffen K. (1986). On the nonuniqueness of surfaces with prescribed constant mean curvature spanning a given contour. Arch. Rat. Mech. Anal. 94: 101–122
Struwe M. (1986). Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Rat. Mech. Anal. 93: 135–157
Struwe M. (1988). Plateau’s problem and the calculus of variations. Mathematical Notes 35. Princeton University Press, Princeton
Struwe M. and Tarantello G. (1998). On multivortex solutions in Chern-Simons gauge theory. Boll. Unione M at. Ital. Sez. B Artic. Ric. Mat. 1(8): 109–121
Wente H. (1969). An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26: 318–344
Wente H. (1980). Large solutions of the volume constrained Plateau problem. Arch. Rat. Mech. Anal. 75: 59–77
Wente H. (1986). Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121: 193–243
Zhou, C.Q.: Existence of solution for mean field equation for the equilibrium turbulence. Preprint (1986)
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The fourth named author supported partially by NSFC of China (No. 10301020).
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Jost, J., Wang, G., Ye, D. et al. The blow up analysis of solutions of the elliptic sinh-Gordon equation. Calc. Var. 31, 263–276 (2008). https://doi.org/10.1007/s00526-007-0116-7
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DOI: https://doi.org/10.1007/s00526-007-0116-7