Multiple positive solutions for degenerate elliptic equations with singularity and critical cone Sobolev exponents

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Abstract

In this paper, we study the existence of multiple positive solutions for degenerate elliptic systems with singular and critical cone Sobolev exponents on singular manifolds. With the help of the variational method, we obtain a multiplicity result.

Keywords

Multiple positive solutions Variational method Critical cone Sobolev exponent Singular term 

Mathematics Subject Classification

35R01 58J05 58J32 

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.
    Kondrat’ev, V.A.: Bundary value problems for elliptic equations in domains with conical points. Tr. Mosk. Mat. Obs. 16, 209–292 (1967)Google Scholar
  3. 3.
    Ju, V., Egorov, B.-W.: Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory, Advances and Applications 93. Birkhäuser, Basel (1997)Google Scholar
  4. 4.
    Mazzeo, R.: Elliptic theory of differential edge operators I. Commun. Partial Differ. Equ. 16, 1615–1664 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Melrose, R.B., Mendoza, G.A.: Elliptic operators of totally characteristic type, Preprint, Math. Sci. Res. Institute, MSRI, pp. 047–83 (1983)Google Scholar
  6. 6.
    Melrose, R.B., Piazza, P.: Analytic K-theory on manifolds with cornners. Adv. Math. 92, 1–26 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Schulze, B.-W.: Boundary value problems and singular pseudo-differential operators. Wiley, Chichester (1998)MATHGoogle Scholar
  8. 8.
    Schrohe, E., Seiler, J.: Ellipticity and invertiblity in the cone algebra on \(L_p\)-Sobolev spaces. Integral Equ. Oper. Theory 41, 93–114 (2001)CrossRefMATHGoogle Scholar
  9. 9.
    Chen, H., Liu, X., Wei, Y.: Existence theorem for a class of semilinear elliptic equations with critical cone Sobolev exponent. Ann. Glob. Anal. Geom. 39, 27–43 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, H., Liu, X., Wei, Y.: Multiple solutions for semilinear totally characteristic equations with subcritical or critical cone sobolev exponents. J. Differ. Equ. 252, 4200–4228 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Liu, X., Yuan, M.: Existence of nodal solution for semi-linear elliptic equations with critical Sobolev exponent on singular manifold. Acta Mathematica Scientia 33, 543–555 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Fan, H.: Existence theorems for a class of edge-degenerate elliptic equations on singular manifolds. Proc. Edinb. Math. Soc. 2015, 1–23 (2015)MathSciNetGoogle Scholar
  14. 14.
    Fan, H., Liu, X.: Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev Exponents on singular manifolds. Electron. J. Differ. Equ. 2013, 1–22 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fan, H.: Existence results for degenerate elliptic equations with critical cone Sobolev exponents. Acta Mathematica Scientia 34B(6), 1–15 (2014)MathSciNetGoogle Scholar
  16. 16.
    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, 451–471 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chen, H., Wei, Y., Zhou, B.: Existence of solutions for degenerate elliptic equations with singular potential on singular manifolds. Mathematische Nachrichten 285, 1370–1384 (2012)MathSciNetMATHGoogle Scholar
  18. 18.
    Struwe, M.: Variational Methods, 2nd edn. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  19. 19.
    Fan, H., Liu, X.: Positive and negative solutions for a class of Kirchhoff type problems on unbounded domain. Nonlinear Anal. 114, 186–196 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lei, C., Liao, J., Tang, C.: Multiple positive solutions for Kirchhoff type problems with singularity and critical exponents. J. Math. Anal. Appl. 421, 521–538 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Servadei, R., Enrico, V.: Mountain Pass solutions for nonlocal elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Liu, J., Wang, L., Zhao, P.: Positive solutions for a nonlocal problem with a convection term and small perturbations. Math. Methods Appl. Sci. 40(3), 720–728 (2017)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fan, H., Liu, X.: The Maximum Principle of an Elliptic Operator with Totally Characteristic Degeneracy on Manifolds with Conical Singularities. Wuhan University, Preprint (2012)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouChina

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