Advertisement

Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus

  • Francesca Gladiali
  • Massimo Grossi
  • Filomena Pacella
  • P. N. Srikanth
Article

Abstract

In this paper we consider the problem
$$\left\{ \begin{array}{ll} -\Delta u=u^p+\lambda u & \quad\hbox{ in }A,\\ u > 0&\quad \hbox{ in }A,\\ u=0 &\quad \hbox{ on }\partial A, \end{array}\right. $$
where A is an annulus of \({\mathbb{R}^N,N\ge2}\) and p > 1. We prove bifurcation of nonradial solutions from the radial solution in correspondence of a sequence of exponents {p k } and for expanding annuli.

Mathematics Subject Classification (2000)

35J61 35B32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bartsch, T., Clapp, M., Grossi, M., Pacella, F.: Asymptotically radial solutions in expanding annular domains. Math. Ann. (to appear)Google Scholar
  2. 2.
    Byeon J.: Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Differ. Equ. 136, 136–165 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Catrina F., Wang Z.Q.: Nonlinear elliptic equations on expanding symmetric domains. J. Differ. Equ. 156, 153–181 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dancer E.N.: Real analyticity and nondegeneracy. Math. Ann. 325(2), 369–392 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Felmer P., Martinez S., Tanaka K.: Uniqueness of radially symmetric positive solutions for −Δu + u = u p in an annulus. J. Differ. Equ. 245, 1198–1209 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Grossi M.: Asymptotic behavior of the Kazdan–Warner solution in the annulus. J. Differ. Equ. 223, 96–111 (2006)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grossi M., Pacella F., Yadava S.L.: Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. 21, 211–226 (2003)MathSciNetGoogle Scholar
  8. 8.
    Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)Google Scholar
  9. 9.
    Li Y.Y.: Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differ. Equ. 83, 348–367 (1990)MATHCrossRefGoogle Scholar
  10. 10.
    Lin S.S.: Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains. Trans. Am. Math. Soc. 332, 775–791 (1992)MATHCrossRefGoogle Scholar
  11. 11.
    Lin S.S.: Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli. J. Differ. Equ. 120, 255–288 (1995)MATHCrossRefGoogle Scholar
  12. 12.
    Ni W.M., Nussbaum R.: Uniqueness and nonuniqueness for positive radial solutions of Δu + f (u, r) = 0. Commun. Pure Appl. Math. 38, 67–108 (1985)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nussbaum R.D.: The fixed point index for local condensing maps. Ann. Mat. Pura Appl. 89, 217–258 (1971)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Smoller J., Wasserman A.: Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions. Commun. Math. Phys. 105, 415–441 (1986)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Smoller J., Wasserman A.: Bifurcation and symmetry-breaking. Invent. Math. 100, 63–95 (1990)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Srikanth P.N.: Symmetry breaking for a class of semilinear elliptic problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 107–112 (1990)MATHMathSciNetGoogle Scholar
  17. 17.
    Tang M.: Uniqueness of positive radial solutions for Δuu + u p = 0 on an annulus. J. Differ. Equ. 189, 148–160 (2003)MATHCrossRefGoogle Scholar
  18. 18.
    Yadava S.L.: Uniqueness of positive radial solutions of the Dirichlet problems −Δu = u p ± u q in an annulus. J. Differ. Equ. 139, 194–217 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Francesca Gladiali
    • 1
  • Massimo Grossi
    • 2
  • Filomena Pacella
    • 2
  • P. N. Srikanth
    • 3
  1. 1.Struttura dipartimentale di Matematica e FisicaUniversità di SassariSassariItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomeItaly
  3. 3.TIFR Centre for Applicable MathematicsBangaloreIndia

Personalised recommendations