Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 423–433 | Cite as

Multiple solutions to Hessian equations

  • Limei DaiEmail author


This paper concerns with the multiple solutions of Hessian equations σ k (λ(D 2 u)) = f (x, u) in a (k − 1)-convex domain \({\Omega\subset \mathbb{R}^{n}}\). Using the methods of degree theory and a priori estimates we prove the existence of two or more solutions to the Hessian equations.

Mathematics Subject Classification (2010)



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  1. 1.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chou K., Wang X.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. 54, 1029–1064 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Guan B.: The Dirichlet problem for a class of fully nonlinear elliptic equations. Commun. Partial Differ. Equ. 19, 399–416 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Jacobsen J.: Global bifurcation problems associated with K-Hessian operators. Topol. Methods Nonlinear Anal. 14, 81–130 (1999)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Jacobsen J.: A Liouville-Gelfand equation for k-Hessian operators. Rocky Mt. J. Math. 34, 665–683 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lions P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24, 441–467 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Trudinger N.S., Wang X.: Hessian measures. I. Topol. Methods Nonlinear Anal. 10, 225–239 (1997)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Trudinger N.S., Wang X.: Hessian measures. II. Ann. Math. 150, 579–604 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tso K.: On a real Monge-Ampère functional. Invent. Math. 101, 425–448 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Wang X.: Existence of multiple solutions to the equations of Monge-Ampère type. J. Differ. Equ. 100, 95–118 (1992)zbMATHCrossRefGoogle Scholar
  11. 11.
    Wang X.: A class of fully nonlinear elliptic equations and related functionals. Indiana Univ. Math. J. 43, 25–54 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceWeifang UniversityWeifangPeople’s Republic of China

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