Skip to main content
SpringerLink
Log in

A Class of Singular Coupled Systems of Superlinear Monge-Ampère Equations

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations

$$\left\{ {\begin{array}{*{20}{c}} {\det {D^2}{u_1} = \lambda {h_1}(\left| x \right|){f_1}( - {u_2}),}&{\text{in}\;\Omega ,} \\ {\det {D^2}{u_2} = \lambda {h_2}(\left| x \right|){f_2}( - {u_1}),}&{\text{in}\;\Omega ,} \\ {{u_1} = {u_2} = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{\text{on}\;\partial \Omega } \end{array}} \right.$$

for a certain range of λ > 0, hi are weight functions, fi are continuous functions with possible singularity at 0 and satisfy a combined N-superlinear growth at ∞, where i ∈ {1, 2}, Ω is the unit ball in ℝN. We establish the existence of a nontrivial radial convex solution for small λ, multiplicity results of nontrivial radial convex solutions for certain ranges of λ, and nonexistence results of nontrivial radial solutions for the case λ ≫ 1. The asymptotic behavior of nontrivial radial convex solutions is also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amann, H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev., 18: 620–709 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bao, J., Li, H., Zhang, L. Monge-Ampère equation on exterior domains. Calc. Var. Partial Differential Equations 52: 39–63 (2013)

    Article  MATH  Google Scholar 

  3. Caffarelli, L. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. Math., 131: 129–134 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Caffarelli, L. Interior W2,p estimates for solutions of the Monge-Ampère equation. Ann. Math., 131: 135–150 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli, L., Kohn, J.J., Nirenberg, L., Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations II. Complex Monge-Ampère equations, and uniformly elliptic equations. Comm. Pure Appl. Math., 38: 209–252 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Nirenberg, L., Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equations. Comm. Pure Appl. Math., 37: 369–402 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L., Nirenberg, L., Spruck, J. The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian. Acta Math., 155: 261–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng, S.Y., Yau, S.T. On the regularity of the Monge-Ampère equation det((∂2u/∂xi∂xj)) = F(x, u). Comm. Pure Appl. Math., 30: 41–68 (1977)

    Article  MathSciNet  Google Scholar 

  9. Chhetri, M., Girg, P. Existence of positive solutions for a class of superlinear semipositone systems. J. Math. Anal. Appl., 408: 781–788 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cîrstea, F.-C., Rădulescu, V. Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl. 81: (2002) 827–846 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cîrstea, F.-C., Trombetti, C. On the Monge-Ampère equation with boundary blow-up: existence, uniqueness and asymptotics. Calc. Var. Partial Differential Equations 31: 167–186 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colesanti, A., Salani, P., Francini, E. Convexity and asymptotic estimates for large solutions of Hessian equations. Differential Integral Equations 13: 1459–1472 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Dalmasso, R. Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal., 39: 559–568 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. D’Ambrosio, L., Mitidieri, E. Entire solutions of quasilinear elliptic systems on Carnot groups. Proc. Steklov Inst. Math., 283: 3–19 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. D’Ambrosio, L., Mitidieri, E. Quasilinear elliptic systems in divergence form associated to general non-linearities. Adv. Nonlinear Anal., 7: 425–447 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dolcetta, I.C., Vitolo, A. Glaeser’s type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete Cont. Dyn. S., 28: 539–557 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng, M. New results of coupled system of k-Hessian equations. Appl. Math. Lett., 94: 196–203 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feng, M. Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior. Adv. Nonlinear Anal., 10:371–399 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feng, M., Zhang, X. On a k-Hessian equation with a weakly superlinear nonlinearity and singular weights. Nonlinear Anal., 190: 111601 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Galise, G., Koike, S., Ley, O., Vitolo, A. Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term. J. Math. Anal. Appl., 441: 194–210 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ghergu, M., Rădulescu, V. Explosive solutions of semilinear elliptic systems with gradient term. Rev. R. Acad. Cienc. Ser. A Mat., 97: 437–445 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Gilbarg, D., Trudinger, N.S. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, Heidelberg, 2001.

    Book  MATH  Google Scholar 

  23. Guan, B., Jian, H. The Monge-Ampère equation with infinite boundary value. Pacific J. Math., 216: 77–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hai, D.D. On a class of semilinear systems. J. Math. Anal. Appl., 285: 477–486 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hai, D.D., Shivaji, R. Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball. J. Differential Equations 266: 2232–2243 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hai, D.D., Shivaji, R. Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions. J. Math. Anal. Appl., 456: 872–881 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jian, H., Li, Y. Optimal boundary regularity for a singular Monge-Ampère equation. J. Differential Equations 264: 6873–6890 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jian, H., Wang, X.-J., Zhao, Y. Global smoothness for a singular Monge-Ampère equation. J. Differential Equations 263 7250–7262 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jiang, F., Trudinger, N.S., Yang, X.-P. On the Dirichlet problem for Monge-Ampère type equations. Calc. Var. Partial Differential Equations 49: 1223–1236 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lair, A.V., Wood, A.W. Existence of entire large positive solutions of semilinear elliptic systems. J. Differential Equations 164: 380–394 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lazer, A.C., McKenna, P.J. On singular boundary value problems for the Monge-Ampère operator. J. Math. Anal. Appl., 197: 341–362 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, E.K., Shivaji, R., Sim, I., Son, B. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Commun. Pure Appl. Anal., 18: 1139–1154 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, Y., Lu, S. Existence and nonexistence to exterior Dirichlet problem for Monge-Ampère equation. Calc. Var. Partial Differential Equations 57: 161 (2018)

    Article  MATH  Google Scholar 

  34. Lou, Z., Weth, T., Zhang. Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type. Z. Angew. Math. Phys., 70: 35 (2019)

    Article  MATH  Google Scholar 

  35. Mavinga, N., Pardo, R. A priori bounds and existence of positive solutions for semilinear elliptic systems. J. Math. Anal. Appl., 449: 1172–1188 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mohammed, A. On the existence of solutions to the Monge-Ampère equation with infinite boundary values. Proc. Amer. Math. Soc., 135: 141–149 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mohammed, A., Rădulescu, V.D., Vitolo, A. Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness. Adv. Nonlinear Anal., 9: 39–64 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pliś, S. On boundary blow-up problems for the complex Monge-Ampère equation. Proc. Amer. Math. Soc., 136: 4355–4364 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Quittner, P., Souplet, P.H. A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces. Arch. Ration. Mech. Anal., 174: 49–81 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Savin, O. Pointwise C2,α estimates at the boundary for the Monge-Ampère equation. J. Amer. Math. Soc., 26: 63–99 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Salani, P. Boundary blow-up problems for Hessian equations. Manuscripta Math., 96: 281–294 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sánchez, J., Vergara, V. Bounded solutions of a k-Hessian equation involving a weighted nonlinear source. J. Differential Equations 263: 687–708 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Son, B., Wang, P. Analysis of positive radial solutions for singular superlinear p-Laplacian systems on the exterior of a ball. Nonlinear Anal., 192: 111657 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Trudinger, N.S., Wang, X.-J. The Monge-Ampère equation and its geometric applications. Handbook of geometric analysis 1: 467–524 (2008)

    MATH  Google Scholar 

  45. Trudinger, N.S., Wang, X.-J. Boundary regularity for the Monge-Ampère and affine maximal surface equations. Ann. Math., 167: 993–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang, F., An, Y. Triple nontrivial radial convex solutions of systems of Monge-Ampère equations. Appl. Math. Lett., 25: 88–92 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, H. Convex solutions of systems arising from Monge-Ampère equations. Electron. J. Qual. Theory Differ. Equ. Special Edition I., 26: 1–8 (2009)

    Google Scholar 

  48. Wang, H. Radial convex solutions of boundary value problems for systems of Monge-Ampère equations. arXiv:1008.4614vl.

  49. Wei, W. Existence and multiplicity for negative solutions of k-Hessian equations. J. Differential Equations 263: 615–640 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, X.-J. Existence of multiple solutions to the equations of Monge-Ampère type. J. Differential Equations 100: 95–118 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Yang, H., Chang, Y. On the blow-up boundary solutions of the Monge-Ampère equation with singular weights. Commun. Pure Appl. Anal., 11: 697–708 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang, X. On a singular k-Hessian equation. Appl. Math. Lett. 97: 60–66 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang, X., Du, Y. Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation. Calc. Var. Partial Differential Equations 57: 30 (2018)

    Article  MATH  Google Scholar 

  54. Zhang, X., Feng, M. Blow-up solutions to the Monge-Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates. Calc. Var. Partial Differential Equations 61: 208 (2022)

    Article  MATH  Google Scholar 

  55. Zhang, X., Feng, M. The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation. J. Differential Equations 267: 4626–4672 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, Z. Large solutions to the Monge-Ampère equations with nonlinear gradient terms: Existence and boundary behavior. J. Differential Equations 264: 263–296 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang, Z. Boundary behavior of large solutions for semilinear elliptic equations with weights. Asymptot. Anal., 96: 309–329 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, Z., Wang, K. Existence and non-existence of solutions for a class of Monge-Ampère equations. J. Differential Equations 246: 2849–2875 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang, Z., Qi, Z. On a power-type coupled system of Monge-Ampère equations. Topol. Method. Nonl. An., 46: 717–729 (2015)

    MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.

Funding

This work is supported by Beijing Natural Science Foundation under Grant No. 1212003 and the Promoting the Classified Development of Colleges and Universities-application and Cultivation of Scientific Research Awards of BISTU under Grant No. 2021JLPY408.

Author information

Authors and Affiliations

  1. School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, China

    Mei-qiang Feng

Authors
  1. Mei-qiang Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mei-qiang Feng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, Mq. A Class of Singular Coupled Systems of Superlinear Monge-Ampère Equations. Acta Math. Appl. Sin. Engl. Ser. 38, 925–942 (2022). https://doi.org/10.1007/s10255-022-1024-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-022-1024-5

Keywords

2000 MR Subject Classification

Search

Navigation