Skip to main content
SpringerLink
Log in

The Keller–Osserman Problem for the k-Hessian Operator

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

A delicate problem is to obtain existence of solutions to the boundary blow-up elliptic equation

$$\begin{aligned} \sigma _{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) =g\left( u\right) \text { in }\Omega , \,\underset{x\rightarrow x_{0}}{\lim } u\left( x\right) =+\infty \,\, \forall x_{0}\in \partial \Omega , \end{aligned}$$

where \(\sigma _{k}^{1/k}\left( \lambda \left( D^{2}u\right) \right) \) is the k-Hessian operator and \(\Omega \subset \mathbb {R}^{N}\) is a smooth bounded domain. Our goal is to provide a necessary and sufficient condition on g to ensure existence of at least one positive blow-up solution. The main tools for proving existence are the comparison principle and the method of sub and supersolutions.

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. Ivochkina, N.M.: The integral method of barrier functions and the Dirichlet problem for equations with operators of Monge–Ampère type. Mat. Sb. 112(154), 193–206 (1980). English transl. in Math. USSR Sb. 40 (1981)

  2. Ivochkina, N.M.: Second order equations with d-elliptic operators. Trudy Mat. Inst. Steclov. 147, 45–56 (1980). English transl. in Proc. Steclov Inst Math., no. 2 (1981)

  3. Ivochkina, N.M.: A description of the stability cones generated by differential operators of Monge–Ampère type. Math. USSR Sb. 122(164), 265–275 (1983). English transl. in Math. USSR Sb. 50 (1985)

  4. Ivochkina, N.M.: Description of cones of stability generated by differential operators of Monge–Ampère type. Math. USSR Sb. 50, 259–268 (1985). English transl

  5. Ivochkina, N.M.: On the possibility of integral formulae in R\(^{\mathit{n}}\). Zap. Nauchn. Semin. LOMI 52, 35–51 (1975)

    Google Scholar 

  6. Ivochkina, N.M.: Solution of the Dirichlet problem for some equations of Monge–Ampère type. Math. USSR Sb. 56, 2 (1987)

    Article  Google Scholar 

  7. Ivochkina, N.M.: Solution of the Dirichlet problem for curvature equations of order \(m\). Math. USSR Sb. 67, 2 (1989)

    Google Scholar 

  8. Ivochkina, N.M., Trudinger, N.S., Wang, X.-J.: The Dirichlet problem for degenerate Hessian equations. Commun. Partial Differ. Equ. 29(1–2), 219–235 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Ivochkina, N.M.: On some properties of the positive \(m-\)Hessian operators in \(C^{2}( \Omega )\). J. Fixed Point Theory Appl. 14, 79–90 (2013)

    Article  MathSciNet  Google Scholar 

  10. Ivochkina, N.M., Filimonenkova, N.V.: On algebraic and geometric conditions in the theory of Hessian equations. J. Fixed Point Theory Appl. 16, 11–25 (2014)

    Article  MathSciNet  Google Scholar 

  11. Escudero, C.: On polyharmonic regularizations of k-Hessian equations: variational methods. Nonlinear Anal. 125, 732–758 (2015)

    Article  MathSciNet  Google Scholar 

  12. Reilly, R.C.: On the Hessian of a function and the curvatures of its graph. Michigan Math. J.20, 373–383 (1973–1974)

  13. Rund, H.: Integral formulae associated with Euler-Lagrange operators of multiple integral problems in the calculus of variations. Aequ. Math. 11(2/3), 212–229 (1974)

    Article  MathSciNet  Google Scholar 

  14. Wang, X.-J.: The k-Hessian Equation. Lectures Notes in Mathematics, pp. 177–252. Springer, Berlin (2009)

  15. 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(1), 261–301 (1985)

    Article  MathSciNet  Google Scholar 

  16. Osserman, R.: On the inequality \(\Delta u\ge f(u)\). Pac. J. Math. 7, 1641–1647 (1957)

    Article  Google Scholar 

  17. Keller, J.B.: On solution of \(\Delta u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)

    Article  Google Scholar 

  18. Keller, J.B.: Electrohydrodynamics I. The equilibrium of a charged gas in a container. J. Ration. Mech. Anal. 5, 4 (1956)

  19. Diaz, J.I.: Nonlinear Partial Differential Equations and Free Boundaries. Pitman Research Notes in Mathematics, vol. 106. Pitman, London (1985)

  20. Matero, J.: Quasilinear elliptic equations with boundary blow-up. J. Anal. Math. 69, 229–247 (1996)

    Article  MathSciNet  Google Scholar 

  21. Matero, J.: The Bieberbach-Rademacher problem for the Monge-Ampère operator. Manuscr. Math. 91, 379–391 (1996)

    Article  MathSciNet  Google Scholar 

  22. Pohozaev, S.I.: The Dirichlet problem for the equation \(\Delta u=u^{2}\). Dokl. Acad Sci. USSR 136(3), 769–772 (1960). English translation: Soviet. Mathematics Doklady 1(1961), 1143–1146

  23. Lychagin, V.V.: Contact geometry and non linear second order differential equations. Russ. Math. Surv. 34(1), 149–180 (1979)

    Article  Google Scholar 

  24. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk. SSSR Ser. Mat. 47, 75–108 (1983). English transl. Math. USSR Izv. 22(1984), 67–97

  25. Bao, J., Ji, X.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138(1), 175–188 (2010)

    Article  MathSciNet  Google Scholar 

  26. Bao, J., Ji, X., Li, H.: Existence and nonexistence theorem for entire subsolutions of \(k\)-Yamabe type equations. J. Differ. Equ. 253, 2140–2160 (2012)

    Article  MathSciNet  Google Scholar 

  27. Covei, D.-P.: Existence of solutions to quasilinear elliptic problems with boundary blow-up. Ann. Univ. Oradea Fasc. Mat. XVI I(1), 77–84 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Ball, J.M.: Convexity conditions and existence theorems in nonlonear elastisity. Arch. Ration. Mech. Anal. 63(4), 339–403 (1977)

    Google Scholar 

  30. Bedford, E., Taylor, B.A.: Variational properties of the complex Monge-Ampère equation I: Dirichlet principle. Duke Math. J. 45, 375–405 (1978)

    Article  MathSciNet  Google Scholar 

  31. Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s. Commun. Pure Appl. Math. XLII 15–45, (1989)

  32. Wang, B., Bao, J.: Mirror symmetry for a Hessian over-determined problem and its generalization. Commun. Pure Appl. Anal. 13(6), 2305–2316 (2014)

    Article  MathSciNet  Google Scholar 

  33. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

  34. Diaz, G., Diaz, J.I.: Partially at surfaces solving \(k\)-Hessian perturbed equations. In: A mathematical tribute to Professor Jose Mara Montesinos Amilibia / coord. por Marco Castrillon Lpez, Elena Martin Peinador, Jose Manuel Rodriguez Sanjurjo, Jesus Maria Ruiz Sancho, pp. 265–293 (2016)

  35. Faraco, D., Zhong, X.: Quasiconvex functions and Hessian equations. Arch. Ration. Mech. Anal. 168, 245–252 (2003)

    Article  MathSciNet  Google Scholar 

  36. Jian, H.: Hessian equations with infinite Dirichlet boundary value. Indiana Univ. Math. J. 55(3), 1045–1062 (2006)

    Article  MathSciNet  Google Scholar 

  37. Evans, L.C.: Classical solutions of fully nonlinear convex second order elliptic equations. Commun. Pure Appl. Math. 25, 333–363 (1982)

    Article  MathSciNet  Google Scholar 

  38. Guan, B.: The Dirichlet problem for a class of fully nonlinear elliptic equations. Commun. Partial Differ. Equ. 19(3–4), 399–416 (1994)

    Article  MathSciNet  Google Scholar 

  39. Trudinger, N.S., Wang, X.-J.: Hessian measures I. Topol. Methods Nonlinear Anal. 10, 225–239 (1997)

    Article  MathSciNet  Google Scholar 

  40. Trudinger, N.S., Wang, X.-J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)

    Article  MathSciNet  Google Scholar 

  41. Colesanti, A., Salani, P., Francini, E.: Convexity and asymptotic estimates for large solutions of Hessian equations. Differ. Integral Equ. 13(10–12), 1459–1472 (2000)

    MathSciNet  MATH  Google Scholar 

  42. Garding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)

    MathSciNet  MATH  Google Scholar 

  43. Usami, H.: Nonexistence results of entire solutions for superlinear elliptic inequalities. Math. Anal. Appl. 164, 59–82 (1992)

    Article  MathSciNet  Google Scholar 

  44. Covei, D.-P.: Boundedness and blow-up of solutions for a nonlinear elliptic system. Int. J. Math. 25(9), 1–12 (2014)

    Article  MathSciNet  Google Scholar 

  45. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1998)

    MATH  Google Scholar 

  46. Dynkin, E.B.: A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Relat. Fields 89, 89–115 (1991)

    Article  MathSciNet  Google Scholar 

  47. Dynkin, E.B., Yushkevich, A.A., Seitz, G.M., Onishchik, A.L. (eds.): Selected Papers of E. B. Dynkin with Commentary. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  48. Dynkin, E.B.: An Introduction to Branching Measure-Valued Processes. AMS, Providence, RI (1994)

    Book  Google Scholar 

  49. Xiang, N., Yang, X.-P.: The complex Hessian equation with infinite Dirichlet boundary value. Proc. Am. Math. Soc. 136(6), 2103–2111 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank to the editors and reviewers for their advice and guidance in the review process.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st District, Postal Office 22, 010374, Bucharest, Romania

    Dragos-Patru Covei

Authors
  1. Dragos-Patru Covei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dragos-Patru Covei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Covei, DP. The Keller–Osserman Problem for the k-Hessian Operator. Results Math 75, 48 (2020). https://doi.org/10.1007/s00025-020-1174-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-1174-9

Keywords

Mathematics Subject Classification

Search

Navigation