Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities

Article
First Online:
Received:
Accepted:

Abstract

In this paper, we use the Nehari manifold method and Ljusternik-Schnirelmann category to prove the multiplicity result of positive solutions for the semi-linear elliptic systems with critical cone Sobolev exponent on manifolds with conical singularities.

Keywords

Conical singularity Critical cone Sobolev exponent Nehari manifold method Ljusternik-Schnirelmann category 

Mathematics Subject Classification

35J20 58J05 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The first author would like to thank the support from the Collaborative Innovation Center of Mathematics in Wuhan University.

References

  1. 1.
    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, 463–484 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, H., Liu, X., Wei, Y.: Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents. Ann. Glob. Anal. Geom. 39, 27–43 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, H., Liu, X., Wei, Y.: Multiple solutions for semilinear totally characteristic elliptic equations with subscritical or critical cone Sobolev exponents. J. Differ. Equ. 252, 4200–4228 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, H., Liu, X., Wei, Y.: Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term. J. Differ. Equ. 252, 4289–4314 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fan, H., Liu, X.: Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev exponents on singular manifolds. Electron. J. Differ. Equ. 181, 1–22 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Alves, C.O., de Morais Filho, D.C., Souto, M.A.S.: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal. 42, 771–787 (2000)Google Scholar
  7. 7.
    Cingolani, S., Lazzo, M.: Multiple positive solutionsto ninlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160, 118–138 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Williem, M.: Minimax theorems, pp. 71–90. Birkhäuser, Boston, Basel, Berlin (1996)Google Scholar
  9. 9.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1, 145–201 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fan, H., Liu, X.: The maximum principle of an elliptic operator with totally characteristic degeneracy on manifolds with conical singularities. Wuhan University (2012, Preprint)Google Scholar
  11. 11.
    Fan, H., Liu, X.: Multiple positive solutions for semilinear elliptic systems involving sign-changing weight. Math. Methods Appl. Sci. 38, 1342–1351 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lin, H.: Multiple positive solutions for semilinear elliptic systems. J. Math. Anal. Appl. 391, 107–118 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Li, Q., Yang, Z.: Mutiple positive solutions for quasilinear elliptic systems. Electron. J. Differ. Equ. 2013, 1–14 (2013)CrossRefGoogle Scholar
  14. 14.
    Adriouch, K., EL Hamidi, A.: The Nehari manifold for systems of nonlinear elliptic equations. Nonlinear Anal. 64, 2149–2167 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Struwe, M.: Variational methods, pp. 36–57. Springer-Verlag, Berlin, Heidelberg, New York (2000)Google Scholar
  16. 16.
    Tian, S.Y.: Some researches for elliptic boundary vaule problems on singular manifolds with multiple totally characteristic degenerate directions and evolution equations, PHD thesis of Wuhan University (2015)Google Scholar
  17. 17.
    Liu, N.: Research on properties of solutions for a class of degenerate evolution equations, PHD thesis of Wuhan University (2015)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations