Multiple solutions for semilinear cone elliptic equations without Ambrosetti–Rabinowitz condition

  • Nhu Thang Nguyen
  • Thi Thu Huong Nguyen


In this paper we establish the existence of multiple solutions for a class of semilinear cone degenerate elliptic Dirichlet boundary value problems involving subcritical nonlinearity (cone Sobolev exponent) without the Ambrosetti–Rabinowitz condition. The paper uses singular analysis to control the linear part to provide appropriate functional setting that a variation of the Moutain Pass argument can be applied.


Cone-degenerate operators Cone Laplace–Beltrami Cerami sequences Fountain theorem Subcritical growth 

Mathematics Subject Classification

35J20 35J10 35J70 35A15 35D30 



The authors would like to thank the anonymous referees for their valuable comments and suggestions to improve the quality of the paper. This work is supported by the Hanoi University of Science and Technology under Project No. T2016–PC–204.


  1. 1.
    Anh, C.T., My, B.K.: Existence of solutions to \(\varDelta _{\lambda } \)-Laplace equations without the Ambrosetti–Rabinowitz condition. Complex Var Elliptic Equ 61(7), 1002–1013 (2016)CrossRefGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical theory and applications. J. Funct. Anal. 14, 349–381 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bony, J.-M.: Second microlocalization and propagation of singularities for semilinear hyperbolic equations. In: Mizohata, S. (ed) Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), pp. 11–49. Academic Press, Boston (1986)Google Scholar
  4. 4.
    Chen, H., Liu, X., Wei, Y.: Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities. Calc. Var. Partial Differ. Equ. 43(3–4), 463–484 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, H., Liu, X., Wei, Y.: Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents. J. Differ. Equ. 252(7), 4200–4228 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Egorov, Y.V., Schulze, B.W.: Pseudo-differential Operators, Singularities, Applications. Birkhäuser Verlag, Basel (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Edmund, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Basel (2008)Google Scholar
  8. 8.
    Gil, J.B., Mendoza, G.A.: Adjoints of elliptic cone operators. Am. J. Math. 125, 357–408 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Guillemin, V., Uhlmann, G.: Oscillatory integrals with singular symbols. Duke Math. J. 48(1), 251–267 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kogoj, A.E., Lanconelli, E.: On semilinear \(\varDelta \)-Laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lam, N., Lu, G.: Elliptic equations and systems with subscritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. J. Geom. Anal. 24, 118–143 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lesch, M.: Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart (1997)zbMATHGoogle Scholar
  13. 13.
    Liu, X., Zhang, S.: Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities. J. Pseudo Differ. Oper. Appl. 7(4), 451–471 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Luyen, D.T., Tri, N.M.: Existence of solutions to boundary-value problems for semilinear \(\varDelta _{\lambda }\) differential equations. Math. Notes 97, 73–84 (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Melrose, R., Mendoza, G.: Elliptic Operators of Totally Characteristic Type. MSRI, Berkeley (1983). (preprint)Google Scholar
  16. 16.
    Melrose, R.B., Uhlmann, G.A.: Lagrangian intersection and the Cauchy problem. Commun. Pure Appl. Math. 32(4), 483–519 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Miyagaki, O.H., Souto, M.A.S.: Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245, 3628–3638 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Thuy, N.T.C., Tri, N.M.: Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators. Russ. J. Math. Phys. 9, 365–370 (2002)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Roidos, N., Schrohe, E.: The Cahn-Hilliard equation and the Allen-Cahn equation on manifolds with conical singularities. Commun. Partial Differ. Equ. 38(5), 925–943 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Roidos, N., Schrohe, E.: Existence and maximal \(L^p\)-regularity of solutions for the porous medium equation on manifolds with conical singularities. Commun. Partial Differ. Equ. 41(9), 1441–1471 (2016)CrossRefzbMATHGoogle Scholar
  21. 21.
    Schechter, M., Zou, W.: Superlinear problems. Pac. J. Math. 214, 145–160 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Schrohe, E., Seiler, J.: The resolvent of closed extensions of cone differential operators. Can. J. Math. 57, 771–811 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Willem, M.: Minimax Theorem. Birkhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationCau GiayViet Nam
  2. 2.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHai Ba TrungViet Nam

Personalised recommendations