Advertisement

Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1190–1204 | Cite as

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

  • Zhi Jun ZhangEmail author
Article
  • 16 Downloads

Abstract

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD2u(x) = b(x)f (u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in ℝN with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and bC (Ω) is positive in Ω, but may be appropriate singular on the boundary.

Keywords

The Monge-Ampère equations strictly convex large solutions a borderline case boundary behavior 

MR(2010) Subject Classification

35J60 35B40 35J67 35B65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Alarcón, S., Díaz, G., Rey, J. M.: Large solutions of elliptic semilinear equations in the borderline case. An exhaustive and intrinsic point of view. J. Math. Anal. Appl., 431, 365–405 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Anedda, C., Porru, G.: Boundary behaviour for solutions of boundary blow-up problems in a borderline case. J. Math. Anal. Appl., 352, 35–47 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bandle, C., Marcus, M.: Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior. J. Anal. Math., 58, 9–24 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987Google Scholar
  5. [5]
    Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations, I. Monge-Ampère equation. Comm. Pure Appl. Math., 37, 369–402 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Cheng, S. Y., Yau, S. T.: On the existence of a complete Kahler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure Appl. Math., 33, 507–544 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Cheng, S. Y., Yau, S. T.: The real Monge-Ampère equation and affine flat structures, in: Chern, S.S., Wu, W. (eds.), Proceedings of 1980 Beijing Symposium on Differential Geometry and Differential Equations, vol. 1, pp. 339–370, Beijing. Science Press, New York, 1982Google Scholar
  8. [8]
    Cîrstea, F. C., Du, Y.: Large solutions of elliptic equations with a weakly superlinear nonlinearity. J. Anal. Math., 103, 261–277 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Cîrstea, F. C., Trombetti, C.: On the Monge-Ampère equation with boundary blow-up: existence, uniqueness and asymptotics. Cal. Var. Part. Diff. Equattions, 31, 167–186 (2008)CrossRefzbMATHGoogle Scholar
  10. [10]
    Colesanti, A., Salani, P., Francini, E.: Convexity and asymptotic estimates for large solutions of Hessian equations. Differ. Integral Equations, 13, 1459–1472 (2000)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 3nd edn. Springer, Berlin, 1998zbMATHGoogle Scholar
  12. [12]
    Guan, B., Jian, H.: The Monge-Ampère equation with infinite boundary value. Pacific J. Math., 216, 77–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Huang, Y.: Boundary asymptotical behavior of large solutions to Hessian equations. Pacific J. Math., 244, 85–98 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Amer. Math. Soc., 138, 175–188 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Jian, H.: Hessian equations with infinite Dirichlet boundary. Indiana Univ. Math. J., 55, 1045–1062 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Keller, J. B.: On solutions of Δu = f (u). Comm. Pure Appl. Math., 10, 503–510 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Lazer, A. C., McKenna, P. J.: Asymptotic behavior of solutions of boundary blowup problems. Differential Integral Equations, 7, 1001–1019 (1994)MathSciNetzbMATHGoogle Scholar
  18. [18]
    Lazer, A. C., McKenna, P. J.: On singular boundary value problems for the Monge-Ampère operator. J. Math. Anal. Appl., 197, 341–362 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Lions, P. L.: Two remarks on Monge-Ampère equations. Ann. Mat. Pura Appl., 142, 263–275 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations, in: Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), 245–272, Academic Press, New York, 1974CrossRefGoogle Scholar
  21. [21]
    López-Gómez, J.: Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, 2015CrossRefzbMATHGoogle Scholar
  22. [22]
    Maric, V.: Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, Springer-Verlag, Berlin, 2000CrossRefzbMATHGoogle Scholar
  23. [23]
    Matero, J.: The Bieberbach-Rademacher problem for the Monge-Ampère operator. Manuscripta Math., 91, 379–391 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Mohammed, A.: On the existence of solutions to the Monge-Ampère equation with infinite boundary values. Proc. Amer. Math. Soc., 135, 141–149 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Mohammed, A.: Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary value. J. Math. Anal. Appl., 325, 480–489 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Osserman, R.: On the inequality Δuf (u). Pacific J. Math., 7, 1641–1647 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Resnick, S. I.: Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987CrossRefzbMATHGoogle Scholar
  28. [28]
    Salani, P.: Boundary blow-up problems for Hessian equations. Manuscripta Math., 96, 281–294 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Seneta, R.: Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976Google Scholar
  30. [30]
    Takimoto, K.: Solution to the boundary blowup problem for k-curvature equation. Calc. Var. Partial Differ. Equations, 26, 357–377 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Yang, H., Chang, Y.: On the blow-up boundary solutions of the Monge-Ampère equation with singular weights. Comm. Pure Appl. Anal., 11, 697–708 (2012)CrossRefzbMATHGoogle Scholar
  32. [32]
    Zhang, X., Du, Y.: Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation. Cal. Var. Part. Diff. Equations, 57(30), 1–24 (2018)zbMATHGoogle Scholar
  33. [33]
    Zhang, X., Feng, M.: Boundary blow-up solutions to the k-Hessian equation with singular weights. Nonlinear Anal., 167, 51–66 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Zhang, Z.: Boundary behavior of large solutions for semilinear elliptic equations with weights. Asymptotic Analysis, 96, 309–329 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Zhang, Z.: Boundary behavior of large solutions for semilinear elliptic equations in borderline cases. Electronic J. Diff. Equations, 2012(136), 1–11 (2012)MathSciNetGoogle Scholar
  36. [36]
    Zhang, Z.: Boundary behavior of large solutions to the Monge-Ampère equations with weights. J. Diff. Equations, 259, 2080–2100 (2015)CrossRefzbMATHGoogle Scholar
  37. [37]
    Zhang, Z.: A unified boundary behavior of large solutions to Hessian equations. Chin. Ann. Math. B, in pressGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceYantai UniversityYantaiP. R. China

Personalised recommendations