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Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

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Abstract

We are concerned with the following class of equations with exponential nonlinearities:

$$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$

which is related to the Tzitzéica equation. Here \(h_1, h_2\) are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser–Trudinger inequality related to this problem. Finally, we give a general existence result.

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References

  1. Ao, W., Jevnikar, A., Yang, W.: On the boundary behavior for the blow up solutions of the sinh-Gordon equation and \({\bf {B}}_2,{\bf {G}}_2\) toda systems in bounded domain (preprint)

  2. Bahri, A., Coron, J.M.: The scalar curvature problem on the standard three dimensional sphere. J. Funct. Anal. 95(1), 106–172 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Battaglia, L., Jevnikar, A., Malchiodi, A., Ruiz, D.: A general existence result for the Toda system on compact surfaces. Adv. Math. 285, 937–979 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Battaglia, L., Malchiodi, A.: A Moser–Trudinger inequality for the singular Toda system. Bull. Inst. Math. Acad. Sin. (N.S.) 9(1), 1–23 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Battaglia, L., Pistoia, A.: A unified approach of blow-up phenomena for two-dimensional singular Liouville systems. Rev. Mat. Iberoam (to appear)

  6. Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u = V(x)\, e^u\) in two dimensions. Commun. Partial Differ. Equ. 16, 1223–1254 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang, S.Y.A., Gursky, M.J., Yang, P.C.: The scalar curvature equation on \(2\)- and \(3\)- spheres. Calc. Var. Partial Differ. Equ. 1(2), 205–229 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, S.Y.A., Yang, P.C.: Prescribing Gaussian curvature on \(S^2\). Acta Math. 159(3–4), 215–259 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, W., Li, C.: Prescribing scalar curvature on \(S^n\). Pac. J. Math. 199(1), 61–78 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C.C., Lin, C.S.: Estimate of the conformal scalar curvature equation via the method of moving planes II. J. Differ. Geom. 49(1), 115–178 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conte, R., Musette, M., Grundland, A.M.: Bäcklund transformation of partial differential equations from the Painlevé–Gambier classification. II. Tzitzéica equation. J. Math. Phys. 40, 2092–2106 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. D’Aprile, T., Pistoia, A., Ruiz, D.: Asymmetric blow-up for the \(SU(3)\) Toda System. J. Funct. Anal. 271(3), 495–531 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, W.: On the best constant in a Sobolev inequality on compact \(2\)-manifolds and application. (Unpublished manuscript) (1984)

  15. Dunajski, M., Plansangkate, P.: Strominger–Yau–Zaslow geometry, affine spheres and Painlevé III. Commun. Math. Phys. 290, 997–1024 (2009)

    Article  MATH  Google Scholar 

  16. Esposito, P., Wei, J.: Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation. Calc. Var. PDEs 34(3), 341–375 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gaffet, B.: \(SU(3)\) symmetry of the equations of unidimensional gas flow, with arbitrary entropy distribution. J. Math. Phys. 25, 245–255 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gaffet, B.: A class of \(1\)-d gas flows soluble by the inverse scattering transform. Phys. D 26, 123–139 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grossi, M., Pistoia, A.: Multiple blow-up phenomena for the sinh-Poisson equation. Arch. Ration. Mech. Anal. 209(1), 287–320 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hirota, R., Ramani, A.: The Miura transformations of Kaup’s equation and of Mikhailov’s equation. Phys. Lett. 76A, 95–96 (1980)

    Article  MathSciNet  Google Scholar 

  21. Hirota, R., Satsuma, J.: \(N\)-soliton solutions of model equations for shallow water waves. J. Phys. Soc. Jpn Lett. 40, 611–612 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jevnikar, A.: An existence result for the mean field equation on compact surfaces in a doubly supercritical regime. Proc. R. Soc. Edinb. Sect. A 143(5), 1021–1045 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jevnikar, A.: New existence results for the mean field equation on compact surfaces via degree theory. Rend. Semin. Mat. Univ. Padova 136, 11–17 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jevnikar, A.: A note on a multiplicity result for the mean field equation on compact surfaces. Adv. Nonlinear Stud. 16(2), 221–229 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jevnikar, A., Kallel, S., Malchiodi, A.: A topological join construction and the Toda system on compact surfaces of arbitrary genus. Anal. PDE 8(8), 1963–2027 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jevnikar, A., Wei, J., Yang, W.: Classification of blow-up limits for the sinh-Gordon equation. Differential and Integral Equations (to appear)

  27. Jevnikar, A., Wei, J., Yang, W.: On the Topological degree of the Mean field equation with two parameters. Indiana Univ. Math. J (to appear)

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Jost, J., Wang, G., Ye, D., Zhou, C.: The blow-up analysis of solutions of the elliptic sinh- Gordon equation. Calc. Var. PDEs 31, 263–276 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kiessling, M.K.H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46(1), 27–56 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related problems I. J. Differ. Equ. 120(2), 319–410 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin, C.S., Wei, J., Zhao, C.Y.: Asymptotic behavior of \(SU(3)\) Toda system in a bounded domain. Manuscr. Math. 137, 1–18 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Lin, C.S., Wei, J.C., Zhang, L.: Classifcation of blowup limits for \(SU(3)\) singular Toda systems. Anal. PDE 8(4), 807–837 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lin, C.S., Wei, J.C., Yang, W., Zhang, L.: On Rank Two Toda System with Arbitrary Singularities: Local Mass and New Estimates. arXiv:1609.02771

  37. Lucia, M.: A deformation lemma with an application to a mean field equation. Topol. Methods Nonlinear Anal. 30(1), 113–138 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Malchiodi, A.: Topological methods for an elliptic equation with exponential nonlinearities. Discrete Contin. Dyn. Syst. 21(1), 277–294 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ohtsuka, H., Suzuki, T.: Mean field equation for the equilibrium turbulence and a related functional inequality. Adv. Differ. Equ. 11, 281–304 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Ohtsuka, H., Suzuki, T.: A blowup analysis of the mean field equation for arbitrarily signed vortices. Self-Similar Sol Nonlinear PDEs 74, 185–197 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pistoia, A., Ricciardi, T.: Concentrating solutions for a Liouville type equation with variable intensities in \(2\text{ D }\)-turbulence. Nonlinearity 29(2), 271–297 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ricciardi, T., Suzuki, T.: Duality and best constant for a Trudinger–Moser inequality involving probability measures. J. Eur. Math. Soc. 16, 1327–1348 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ricciardi, T., Takahashi, R.: Blow-up behavior for a degenerate elliptic sinh-Poisson equation with variable intensities. Calc. Var. Partial Differ. Equ. 55(6), 55–152 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ricciardi, T., Takahashi, R., Zecca, G., Zhang, X.: On the existence and blow-up of solutions for a mean field equation with variable intensities. arXiv:1509.05204

  45. Ricciardi, T., Zecca, G.: Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics. Differ. Integral Equ. 25(3–4), 201–222 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. Schief, W., Rogers, C.: The affinsphären equation. Moutard Bäcklund Transform., Inverse Probl. 10, 711–731 (1994)

    Article  MATH  Google Scholar 

  48. Schoen, R. Stanford Classnotes

  49. Schoen, R., Zhang, D.: Prescribed scalar curvature on the \(n\)-sphere. Calc. Var. 4(1), 1–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tarantello, G.: Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete Contin. Dyn. Syst. 28(3), 931–973 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tzitzéca, G.: Sur une nouvelle classe de surfaces. Rend. Circolo Mat. Palermo 25, 180–187 (1908)

    Article  Google Scholar 

  52. Tzitzéica, G.: Sur une nouvelle classe de surfaces. C. R. Acad. Sci. Paris 150, 955–956 (1910)

    MATH  Google Scholar 

  53. Tzitzéica, G.: Géométrie différentielle projective des réseaux. Gauthier-Villars, Paris (1924)

    MATH  Google Scholar 

  54. Zhou, C.: Existence result for mean field equation of the equilibrium turbulence in the super critical case. Commun. Contemp. Math. 13(4), 659–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Aleks Jevnikar

  2. Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, 10617, Taiwan

    Wen Yang

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  1. Aleks Jevnikar
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  2. Wen Yang
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Correspondence to Aleks Jevnikar.

Additional information

Communicated by A. Malchiodi.

We thank Professor Juncheng Wei for suggesting us the problem and Francesca De Marchis for pointing out Remark 1.5.

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Jevnikar, A., Yang, W. Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results. Calc. Var. 56, 43 (2017). https://doi.org/10.1007/s00526-017-1136-6

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  • DOI: https://doi.org/10.1007/s00526-017-1136-6

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