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Periodica Mathematica Hungarica

, Volume 70, Issue 2, pp 145–152 | Cite as

Positive solutions of singular elliptic systems with multiple parameters and Caffarelli–Kohn–Nirenberg exponents

  • G. A. Afrouzi
  • Vicenţiu D. RădulescuEmail author
  • S. Shakeri
Article
  • 174 Downloads

Abstract

This paper is concerned with the existence of positive solutions for a class of quasilinear singular elliptic systems with Dirichlet boundary condition. By studying the competition between the Caffarelli–Kohn–Nirenberg exponents, the sign-changing potentials and the nonlinear terms, we establish an interval on the range of multiple parameters over which solutions exist in an appropriate weighted Sobolev space. The arguments rely on the method of weak sub- and super-solutions.

Keywords

Caffarelli–Kohn–Nirenberg exponents Elliptic system  Multiple parameters Sub-supersolution method 

Mathematics Subject Classification

35J55 35J65 

Notes

Acknowledgments

The authors are grateful to the anonymous referee for the careful reading of the paper and numerous useful suggestions. V. Rădulescu acknowledges the support through Grant CNCS PCE-47/2011.

References

  1. 1.
    G.A. Afrouzi, S.H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system. Nonlinear Anal. 71, 445–455 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    N. Akhmediev, A. Ankiewicz, Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999)CrossRefGoogle Scholar
  3. 3.
    J. Ali, R. Shivaji, Positive solutions for a class of \( p \)-Laplacian systems with multiple parameters. J. Math. Anal. Appl. 335, 1013–1019 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    J. Ali, R. Shivaji, M. Ramaswamy, Multiple positive solutions for classes of elliptic systems with combined nonlinear effects. Differ. Integral Equ. 19, 669–680 (2006)zbMATHMathSciNetGoogle Scholar
  5. 5.
    C.O. Alves, D.G. de Figueiredo, Nonvariational elliptic systems. Discrete Contin. Dyn. Syst. 8, 289–302 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    A. Ambrosetti, J.G. Azorero, I. Peral, Existence and multiplicity results for some nonlinear elliptic equations. Rend. Mat. Appl. 7, 167–198 (2000)Google Scholar
  7. 7.
    C. Atkinson, K. El Kalli, Some boundary value problems for the Bingham model. J. Non Newton. Fluid Mech. 41, 339–363 (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    H. Bueno, G. Ercole, W. Ferreira, A. Zumpano, Existence and multiplicity of positive solutions for the \(p\)-Laplacian with nonlocal coefficient. J. Math. Anal. Appl. 343, 151–158 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)zbMATHMathSciNetGoogle Scholar
  10. 10.
    A. Canada, P. Drábek, J.L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion. Trans. Am. Math. Soc. 349, 4231–4249 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    F. Catrina, Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Commun. Pure Appl. Math. 54, 229–258 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear Anal. 39, 559–568 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    E.N. Dancer, Competing species systems with diffusion and large interaction. Rend. Sem. Mat. Fis. Milano 65, 23–33 (1995)CrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1: Physical Origins and Classical Methods (Springer, Berlin, Heidelberg, New York, 1985)Google Scholar
  15. 15.
    P. Drabek, J. Hernandez, Existence and uniqueness of positive solutions for some quasilinear elliptic problem. Nonlinear Anal. 44, 189–204 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    J.F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate. Commun. Pure Appl. Math. 43, 857–883 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    F. Fang, S. Liu, Nontrivial solutions of superlinear \(p\)-Laplacian equations. J. Math. Anal. Appl. 351, 3601–3619 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Filippucci, P. Pucci, V. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions. Commun. Partial Differ. Equ. 33, 706–717 (2008)CrossRefzbMATHGoogle Scholar
  19. 19.
    D.D. Hai, R. Shivaji, An existence result on positive solutions for a class of semi-linear elliptic systems. Proc. R. Soc. Edinb. Sect. A 134, 137–141 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    J.R. Graef, S. Heidarkhani, L. Kong, Multiple solutions for systems of multi-point boundary value problems. Opusc. Math. 33, 293–306 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    D.D. Hai, R. Shivaji, An existence result on positive solutions for a class of \(p\)-Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    O.H. Miyagaki, R.S. Rodrigues, On positive solutions for a class of singular quasilinear elliptic systems. J. Math. Anal. Appl. 334, 818–833 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    M.K.V. Murthy, G. Stampacchia, Boundary value problems for some degenerate elliptic operators. Ann. Mat. Pura Appl. 80, 1–122 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    M. Nagumo, Über die Differentialgleichung \(y^{\prime \prime } = f (x, y, y^{\prime })\). Proc. Phys. Math. Soc. Jpn. 19, 861–866 (1937)Google Scholar
  25. 25.
    H. Poincaré, Les fonctions fuchsiennes et l’équation \(\Delta u = e^{u}\). J. Math. Pures Appl. 4, 137–230 (1898)zbMATHGoogle Scholar
  26. 26.
    S.H. Rasouli, On the existence of positive solution for class of nonlinear elliptic systems with multiple parameters and singular weights. Commun. Korean Math. Soc. 27, 557–564 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    V. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, vol. 6 (Hindawi Publishing Corporation, New York, 2008)CrossRefGoogle Scholar
  28. 28.
    P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    B. Xuan, The eigenvalue problem for a singular quasilinear elliptic equation. Electron. J. Differ. Equ. 16, 1–11 (2004)MathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  • G. A. Afrouzi
    • 1
  • Vicenţiu D. Rădulescu
    • 2
    • 3
    Email author
  • S. Shakeri
    • 1
  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Department of Mathematics, Faculty of SciencesKing Abdulaziz UniversityJeddahSaudi Arabia

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