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Afrika Matematika

, Volume 28, Issue 3–4, pp 407–415 | Cite as

On the multiplicity of non radial solutions for singular elliptic equations

  • Noria Bekkouche
  • Naima Daoudi-Merzagui
  • Meriem Hellal
Article
  • 73 Downloads

Abstract

The main goal of this work is to analyze the existence and the multiplicity of non radial solutions for a Dirichlet problem associated to an elliptic singular partial differential equation. Our approach is based on a variational method.

Keywords

Singular quasilinear equations Minimization with constraints Periodic solution 

Mathematics Subject Classification

35B10 35B32 35B38 58E30 

References

  1. 1.
    Aduen, H., Castro, A.: Infinitely many non radial solutions to superlinear Dirichlet problem. Proc. Am. Math. Soc. 131, 835–843 (2002)MATHCrossRefGoogle Scholar
  2. 2.
    Alessio, F., Dambrosio, W.: Multiple solutions to a Dirichlet problem on bounded symmetric domains. J. Math. Anal. Appl. 235, 217–226 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bartsch, T., Willem, M.: Infinitely many nonradial solutions of a Euclidean scalar field equation. J. Funct. Anal. 117, 447–460 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Batkam, C.J.: Radial and nonradial solutions of a strongly indefinite elliptic system on \({\mathbb{R}}^{N}\). Afr. Mat. 26, 65–75 (2015)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ben Mabrouk, A., Ben Mohamed, M.: Nonradial solutions of a mixed concave-convex elliptic problem. J. Partial Differ. Equ. 24, 313–323 (2011)MathSciNetMATHGoogle Scholar
  6. 6.
    Bonanno, G.: Some remarks on a three critical points theorem. Nonlinear Anal. 54, 651–665 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Brézis, H.: Analyse Fonctionelle. Masson, Paris (1983)Google Scholar
  8. 8.
    Builter, G.J.: Rapid oscillation non extendability on the existence of periodic solution to second order nonlinear ordinary differential equations. J. Differ. Equ. 22, 467–477 (1976)CrossRefGoogle Scholar
  9. 9.
    Butler, G.J.: Periodic solutions of sublinear second order differential equations. J. Math. Anal. Appl. 62, 676–690 (1978)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Castro, A., Kurepa, A.: Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball. Proc. Am. Math. Soc. 101, 57–64 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Castro, A., Finan, M.B.: Existence of many positive nonradial solutions for a superlinear Dirichlet problem on thin annuli. Nonlinear Differ. Equ. Electron. J. Differ. Equ. Conf. 5, 21–31 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Chen, J.: Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term. Proc. Am. Math. Soc. 132, 3225–3229 (2004)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chen, J.: Multiple positive solutions for a class of nonlinear elliptic equations. J Math. Anal. Appl. 295, 341–354 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dancer, E.N.: On the Dirichlet problem for weakly nonlinear elliptic partial differential equations. Proc. Roy. Soc. 76, 283–300 (1977)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Drabek, P., Kufner, A., Nicolosi, F.: Quasilinear elliptic equations with degenerations and singularities. Gruyter Ser. Nonlinear Anal. 5, (1997)Google Scholar
  16. 16.
    Fabes, E., Jerison, D., Kenig, C.: The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble) 32, 151–182 (1982)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. P.D.E. 7, 77–116 (1982)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Faraci, F., Livrea, R.: Bifurcation theorems for nonlinear problems with lack of compactness. Ann. Polon. Math. 82, 77–85 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ferrero, A., Gazzola, F.: Existence of solutions for singular critical growth semilinear elliptic equations. J. Differ. Equ. 177, 494–522 (2001)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Garcia Azorero, J.P., Peral Alonso, I.: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144, 441–476 (1998)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Guo, Y., Li, B., Wei, J.: Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in \({\mathbb{R}}^{3}\). J. Differ. Equ. 256, 3463–3495 (2014)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Guo, J., Peng, Y., Guo, S.: Study on the nonradial solutions for a semilinear elliptic equation with Hardy term. Acta. Math. Appl. Sin. 36, 666–679 (2013)MathSciNetMATHGoogle Scholar
  23. 23.
    Hai, D.D.: On a class of sublinear quasilinear elliptic problems. Proc. Am. Math. Soc. 131, 2409–2414 (2003)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford. Mathematical Monographs. The Cleredon Press, Oxford University Press, New York (1993)MATHGoogle Scholar
  25. 25.
    Krasnosel’skii, M.A., Perov, A.I., Povolockii, A.I., Zabreiko, P.P.: Plane Vector Fields. Academic Press, New York (1966)Google Scholar
  26. 26.
    Kristaly, A., Varga, C.: Multiple solutions for elliptic problems with singular and sublinear potentials. Proc. Am. Math. Soc. 135, 2121–2126 (2007)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Kyrits, S.T.: Positive solutions for p-Laplacian equations with concave terms. Discrete Contin. Dyn. Syst. Suppl. 31, 922–930 (2011)Google Scholar
  28. 28.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case, part 1. Rev. Mat. Iberoamericana 1, 145–201 (1985)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case, part 2. Rev. Mat. Iberoamericana 1, 45–121 (1985)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Lv, D., Yang, X.: Nonradial solutions for semilinear Schrö dinger equations with sign-changing potential. Electron. J. Qual. 16, 35–91 (2015)Google Scholar
  31. 31.
    Montefusco, E.: Lower semicontinuity of functionals the concentration-compactness principle. J. Math. Anal. Appl. 263, 264–276 (2001)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Norimichi, H., Naoki, S.: Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms. Proc. Am. Math. Soc. 134, 2585–2592 (2006)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Rabinowitz, P.:H. Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, Vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (1986)Google Scholar
  34. 34.
    Ruiz, D., Willem, M.: Elliptic problems with critical exponents and Hardy potential. J. Differ. Equ. 190, 524–538 (2003)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Raymond, J.S.: On the multiplicity of solutions of the equations \(-\Delta u=\lambda f\) (u). J. Differ. Equ. 180, 65–88 (2002)MATHCrossRefGoogle Scholar
  36. 36.
    Shekhter, B.L.: On existence and zeros of solutions of a nonlinear two point boundary value problem. J. Math. Anal. Appl. 97, 1–20 (1983)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Tyagi, J.: Existence of nontrivial solutions for singular quasilinear equations with sign changing non linearity. Electron. J. Differ. Equ. 117, 1–9 (2010)Google Scholar
  38. 38.
    Van Groesen, E.W.C.: Application of natural constraints in critical point theory to boundary value problems on domains with rotation symmetry. Arch. Mat. 44, 171–179 (1985)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Van Groesen, E.W.C.: Existence of multiple normal mode trajectories on convex energy surfaces of even classical hamiltonian systems. J. Differ. Equ. 57, 70–89 (1983)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Wang, Z. Q.: Nonradial solutions of nonlinear Neumann problems in radially symmetric domains. Topol. Nonlinear Anal. 85, 96 (1996). [Banach Center Publ., 35, Polish Acad. Sci., Warsaw]Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Noria Bekkouche
    • 1
  • Naima Daoudi-Merzagui
    • 1
  • Meriem Hellal
    • 1
  1. 1.Department of MathematicsUniversity of TlemcenTlemcenAlgeria

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