Acta Applicandae Mathematicae

, Volume 115, Issue 3, pp 255–264 | Cite as

Multiplicity of Solutions for Elliptic System Involving Supercritical Sobolev Exponent



The multiplicity of positive solutions are established for a class of elliptic systems involving nonlinear Schrödinger equations with critical or supercritical growth. The solutions are obtained by using Moser iteration technique.


Supercritical Sobolev exponent Ljusternik-Schnirelmann theory Moser iteration 

Mathematics Subject Classification (2000)

35Q55 35A15 35B40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alves, C.O., Figueiredo, G.M., Furtado, M.F.: Multiplicity of solutions for elliptic systems via local mountain pass method. Commun. Pure Appl. Anal. 8, 1745–1758 (2009) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alves, C.O.: Local mountain pass for a class of elliptic system. J. Math. Anal. Appl. 335, 135–150 (2007) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alves, C.O., Soares, S.H.M.: On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations. J. Math. Anal. Appl. 296, 563–577 (2004) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alves, C.O., Figueiredo, G.M.: Multiplicity of positive solutions for a quasilinear problem in R N via penalization method. Adv. Nonlinear Stud. 5(4), 551–572 (2005) MathSciNetMATHGoogle Scholar
  5. 5.
    Brézis, H.: Analyse Fonctionnelle-Théorie et applications. Masson, Paris (1983) MATHGoogle Scholar
  6. 6.
    Chabrowski, J., Yang, J.: Existence theorems for elliptic equations involving supercritical Sobolev exponent. Adv. Differ. Equ. 2, 231–256 (1997) MathSciNetMATHGoogle Scholar
  7. 7.
    Chabrowski, J., Yang, J.F.: Multiple semiclassical solutions of the Schrödinger equation involving a critical Sobolev exponent. Port. Math. 57, 3 (2000) 273-284 MathSciNetGoogle Scholar
  8. 8.
    Cingolani, S., Lazzo, M.: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160, 118–138 (2000) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cingolani, S., Lazzo, M.: Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods. Nonlinear Anal. 10, 1–13 (1997) MathSciNetMATHGoogle Scholar
  10. 10.
    Figueiredo, G.M., Furtado, M.F.: Multiple positive solutions for a quasilinear system of Schrödinger equations. Nonlinear Differ. Equ. Appl. 15, 309–333 (2008) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Figueiredo, G.M.: Multiplicity of solutions for a quasilinear problem with supercritical growth. Electron. J. Differ. Equ. 31, 1–7 (2006) Google Scholar
  12. 12.
    Figueiredo, G.M.: Multiplicidade de soluçes positivas para uma class de prblemas quasilineares. Doct. dissertation, Unicamp (2004) Google Scholar
  13. 13.
    Moser, J.: A new proof de Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Stat., Simul. Comput. 13, 457–468 (1960) MATHGoogle Scholar
  14. 14.
    Rabelo, P.: On a class of elliptic systems in R N involving supercritical Sobolev exponent. J. Math. Anal. Appl. 354, 46–59 (2009) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of Mathematics, School of Mathematics SciencesNanjing Normal UniversityNanjingPeople’s Republic of China

Personalised recommendations