Positive Solutions for Elliptic Problems Involving Hardy–Sobolev–Maz’ya Terms

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Abstract

In the present paper, we study the semilinear elliptic problem \(\displaystyle -\Delta u -\mu \frac{u}{|y|^{2}}=\frac{|u|^{2^{*}(s)-2}u}{|y|^{s}}+ f(x,u)\) in bounded domain. Replacing the Ambrosetti–Rabinowitz condition by general superquadratic assumptions and the nonquadratic assumption, we establish the existence results of positive solutions.

Keywords

Positive solutions Hardy–Sobolev–Maz’ya terms Hardy–Sobolev critical exponents Mountain Pass Lemma Local Palais–Smale condition 

References

  1. 1.
    Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponents: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253–294 (1988)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bhakta, M., Sandeep, K.: Hardy–Sobolev–Maz’ya type equation in bounded domain. J. Differ. Equ. 247, 119–139 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Castorina, D., Fabbri, I., Mancini, G., Sandeep, K.: Hardy–Sobolev inequalities and hyperbolic symmetry. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19, 189–197 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Castorina, D., Fabbri, I., Mancini, G., Sandeep, K.: Hardy–Sobolev extremals, hyperbolic symmetry and scalar curvature equations. J. Differ. Equ. 246, 1187–1206 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cao, D., Han, P.: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J. Differ. Equ. 205, 521–537 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, D., Peng, S.: A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J. Differ. Equ. 193, 424–434 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cao, D., Yan, S.: Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc. Var. Partial Differ. Equ. 38, 471–501 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, C., Kuo, Y., Wu, T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Costa, D.G., Magalhāes, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, J.: Multiple positive solutions of a class of nonlinear elliptic equations. J. Math. Anal. Appl. 295, 341–354 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ding, L., Tang, C.: Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy–Sobolev critical exponents. Appl. Math. Lett. 20, 1175–1183 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ganguly, D.: Sign changing solutions of the Hardy–Sobolev–Maz’ya equation. Adv. Nonlinear Anal. 3, 187–196 (2014)MathSciNetMATHGoogle Scholar
  14. 14.
    Ganguly, D., Sandeep, K.: Sign changing solutions of the Brezis–Nirenberg problem in the hyperbolic space. Calc. Var. Partial Differ. Equ. 50, 69–91 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ghoussoub, N., Yuan, C.: Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 352, 5703–5743 (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on \(\mathbb{R}^{N}\). Proc. R. Soc. Edinburgh A 129(4), 787–809 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kang, D.: On the quasilinear elliptic equations with critical Sobolev–Hardy exponents and Hardy-terms. Nonlinear Anal. 68, 1973–1985 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kang, D., Peng, S.: Positive solutions for singular critical elliptic problems. Appl. Math. Lett. 17, 411–416 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kang, D., Peng, S.: Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential. Appl. Math. Lett. 18, 1094–1100 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu, H., Tang, C.: Positive solutions for semilinear elliptic equations with critical weighted Hardy–Sobolev exponents. Bull. Belg. Math. Soc. Simon Stevin 22, 1–21 (2015)MathSciNetGoogle Scholar
  21. 21.
    Maz’ya, V.G.: Sobolev Space. Springer, Berlin (1985)MATHGoogle Scholar
  22. 22.
    Mancini, G., Sandeep, K.: On a semilinear elliptic equation in \(\mathbb{H}^{n}\). Annali della Scuola Normale Superiore di Pisa Classe di Scienze 7, 635–671 (2007)MATHGoogle Scholar
  23. 23.
    Nguyen, L., Lu, G.: Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. J. Geom. Anal. 24, 118–143 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS. 65. American Mathematical Society, Providence (1985)Google Scholar
  25. 25.
    Schechter, M.: A variation of the mountain pass lemma and applications. J. Lond. Math. Soc. 244(3), 491–502 (1991)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schecher, M., Zou, W.: On the Brezis–Nirenberg problem. Arch. Ration. Mech. Anal. 197, 337–356 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang, C., Wang, J.: Infinitely many solutions for Hardy–Sobolev–Maz’ya equation involving critical growth. Commun. Contemp. Math. 14(6), 1250044 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Yang, J.: Positive solutions for the Hardy–Sobolev–Maz’ya equation with Neumann boundary condition. J. Math. Anal. Appl. 421, 1889–1916 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China

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