Science China Mathematics

, Volume 54, Issue 2, pp 243–256 | Cite as

Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents

Articles

Abstract

This paper is concerned with a singular elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. By analytic technics and variational methods, the extremals of the corresponding bet Hardy-Sobolev constant are found, the existence of positive solutions to the system is established and the asymptotic properties of solutions at the singular point are proved.

Keywords

elliptic system nontrivial solution critical exponent variational method 

MSC(2000)

35B33 35J60 

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References

  1. 1.
    Abdellaoui B, Felli V, Peral I. Some remarks on systems of elliptic equations doubly critical in the whole ℝ{srN}. Calc Var Partial Differential Equations, 2009, 34: 97–137MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alves C, Filho D, Souto M. On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal, 2000, 42: 771–787MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ambrosetti A, Rabinowitz H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349–381MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bartsch T, Peng S, Zhang Z. Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities. Calc Var Partial Differential Equations, 2007, 30: 113–136MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bouchekif M, Nasri Y. On a singular elliptic system at resonance. Ann Mat Pura Appl, 2010, 189: 227–240MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequality with weights. Compos Math, 1984, 53: 259–275MATHMathSciNetGoogle Scholar
  8. 8.
    Cao D, Han P. Solutions to critical elliptic equations with multi-singular inverse square potentials. J Differential Equations, 2006, 224: 332–372MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Catrina F, Wang Z. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extermal functions. Comm Pure Appl Math, 2001, 54: 229–257MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dautray R, Lions P. Mathematical Analysis and Numerical Methods for Science and Technology. Physical Origins and Classical Methods, Vol. 1. Berlin: Springer, 1990Google Scholar
  11. 11.
    Felli V, Schneider M. Perturbations results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type. J Differential Equations, 2003, 191: 121–142MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Figueiredo D, Peral I, Rossi J. The critical hyperbola for a Hamiltonian elliptic system with weights. Ann Mat Pura Appl, 2008, 187: 531–545MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ghoussoub N, Yuan C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352: 5703–5743MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Han P. The effect of the domain topology on the number of positive solutions of some elliptic systems involving critical Sobolev exponents. Houston J Math, 2006, 32: 332–372Google Scholar
  15. 15.
    Hardy G, Littlewood J, Polya G. Inequalities, reprint of the 1952 edition, Cambridge Math Lib. Cambridge: Cambridge University Press, 1988Google Scholar
  16. 16.
    Jannelli E, The role played by space dimension in elliptic critcal problems. J Differential Equations, 1999, 156: 407–426MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jin L, Deng Y. A global compact result for a semilinear elliptic problem with Hardy potential and critical nonlinearities on ℝN. Sci China Math, 2010, 53: 385–400MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kang D. On elliptic problems with critical weighted Sobolev-Hardy exponents. Nonlinear Anal, 2007, 66: 1037–1050MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kang D, Huang Y, Liu S. Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J South Central Univ Natl, Nat Sci Ed, 2008, 27: 91–95Google Scholar
  20. 20.
    Li S, Peng S. Asymptotic behavior on the Hénon equation with supercritical exponent. Sci China Ser A, 2009, 52: 2185–2194MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Liu Z, Han P. Existence of solutions for singular elliptic systems with critical exponents. Nonlinear Anal, 2008, 69: 2968–2983MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Peng S. Remarks on singular critical growth elliptic equations. Discrete Contin Dyn Syst, 2006, 14: 707–719MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Xuan B, Wang J. Extremal functions and best constants to an inequality involving Hardy potential and critical Sobolev exponent. Nonlinear Anal, 2009, 71: 845–859MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouth-Central University For NationalitiesWuhanChina
  2. 2.School of Mathematics and StatisticsCentral China Normal UniversityWuhanChina

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