Advertisement

Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

  • Leszek Gasiński
  • Dumitru Motreanu
  • Nikolaos S. Papageorgiou
Regular Articles

Abstract

We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro-Lazer-Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.

Keywords

Double resonance reduction method eigenvalue hemivariational inequality locally Lipschitz function Clarke subdifferential critical point local linking nonsmooth Cerami condition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bartolo P, Benci V and Fortunato D, Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity,Nonlin. Anal. 7 (1983) 981–1012MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Berestycki H and de Figueiredo D, Double resonance in semilinear elliptic problems,Comm. Partial Diff. Eqns 6 (1981) 91–120MATHCrossRefGoogle Scholar
  3. [3]
    Brezis H and Nirenberg L, Remarks on finding critical points,Comm. Pure Appl. Math. 44 (1991) 939–963CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Cac N P, On an elliptic boundary value problem at double resonance,J. Math. Anal. Appl. 132 (1988) 473–483MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Castro A and Cossio J, Multiple solutions for a nonlinear Dirichlet problem,SIAM J. Math. Anal. 25 (1994) 1554–1561MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Castro A, Cossio J and Neuberger J, On multiple solutions of a nonlinear Dirichlet problem,Nonlin. Anal. 30 (1997) 3657–3662MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Castro A and Lazer A, Critical point theory and the number of solutions of a nonlinear Dirichlet problem,Annali di Mat. Pura ed Appl. 70 (1979) 113–137CrossRefMathSciNetGoogle Scholar
  8. [8]
    Cerami G, Un criterio di esistenza per i punti critici su varieta illimitate,Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978) 332–336MathSciNetGoogle Scholar
  9. [9]
    Chang K C, Variational methods for nondifferentiable functionals and their applications to partial differential equations,J. Math. Anal. Appl. 80 (1981) 102–129MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Clarke F H, Optimization and nonsmooth analysis (New York: Wiley) (1983)MATHGoogle Scholar
  11. [11]
    Costa D and Silva E A, On a class of resonant problems at higher eigenvalues,Diff. Int. Eqns 8 (1995) 663–671MATHMathSciNetGoogle Scholar
  12. [12]
    Gasinski L and Papageorgiou N S, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance,Proc. R. Soc. Edinburgh 131A (2001) 1091–1111CrossRefMathSciNetGoogle Scholar
  13. [13]
    Gasinski L and Papageorgiou N S, An existence theorem for nonlinear hemivariational inequalities at resonance,Bull. Australian Math. Soc. 63 (2001) 1–14MATHMathSciNetGoogle Scholar
  14. [14]
    Gasinski L and Papageorgiou N S, Multiple solutions for semilinear hemivariational inequalities at resonance,Publ. Math. Debrecen 59 (2001) 121–146MATHMathSciNetGoogle Scholar
  15. [15]
    Goeleven D, Motreanu D and Panagiotopoulos P D, Eigenvalue problems for variational-hemivariational inequalities at resonance,Nonlin. Anal. 33 (1998) 161–180MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Hirano N and Nishimura T, Multiplicity results for semilinear elliptic problems at resonance with jumping nonlinearities,J. Math. Anal. Appl. 180 (1993) 566–586MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Hu S and Papageorgiou N S, Handbook of multivalued analysis, volumeI: Theory (The Netherlands: Kluwer, Dordrecht) (1997)MATHGoogle Scholar
  18. [18]
    Hu S and Papageorgiou N S, Handbook of multivalued analysis, volumeII: Applications (The Netherlands: Kluwer, Dordrecht) (2000)MATHGoogle Scholar
  19. [19]
    Iannacci R and Nkashama M N, Nonlinear elliptic partial differential equations at resonance: Higher eigenvalues,Nonlin. Anal. 25 (1995) 455–471MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Jost J, Post-modern analysis (Berlin: Springer-Verlag) (1998)Google Scholar
  21. [21]
    Kandilakis D, Kourogenis N and Papageorgiou N S, Two nontrivial critical points for nonsmooth functionals via local linking and applications,J. Global Optimization 34 (2006) 219–244MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Kourogenis N and Papageorgiou N S, Nonsmooth critical point theory and nonlinear elliptic equations at resonance,J. Austr. Math. Soc. 69A (2000) 245–271MathSciNetCrossRefGoogle Scholar
  23. [23]
    Landesman E, Robinson S and Rumbos A, Multiple solutions of semilinear elliptic problems at resonance,Nonlin. Anal. 24 (1995) 1049–1059MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Lebourg G, Valeur mayenne pour gradient génèralisé,CRAS Paris 281 (1975) 795–797MATHMathSciNetGoogle Scholar
  25. [25]
    Motreanu D and Panagiotopoulos P D, A minimax approach to the eigenvalue problem of hemivariational inequalities and applications,Applicable Anal. 58 (1995) 53–76MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Motreanu D and Panagiotopoulos P D, On the eigenvalue problem for hemivariational inequalities: Existence and multiplicity of solutions,J. Math. Anal. Appl. 197 (1996) 75–89MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Naniewicz Z and Panagiotopoulos P D, Mathematical theory of hemivariational inequalities and applications (New York: Marcel-Dekker) (1994)MATHGoogle Scholar
  28. [28]
    Niculescu C and Radulescu V, A saddle point theorem for nonsmooth functionals and problems at resonance,Annales Acad. Sci. Fennicae 21 (1996) 117–131MATHMathSciNetGoogle Scholar
  29. [29]
    Radulescu V, Mountain pass theorems for nondifferentiable functions and applications,Proc. Japan Acad. Sci. A69 (1993) 193–198MathSciNetGoogle Scholar
  30. [30]
    Radulescu V and Panagiotopoulos P, Perturbations of hemivariational inequalities with constraints and applications,J. Global Optim. 12 (1998) 285–297MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    Robinson S, Double resonance in semilinear elliptic boundary value problem over bounded and unbounded domains,Nonlin. Anal. 21 (1993) 407–424MATHCrossRefGoogle Scholar
  32. [32]
    Su J, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,Nonlin. Anal. 48 (2002) 881–895MATHCrossRefGoogle Scholar
  33. [33]
    Su J and Tang C, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues,Nonlin. Anal. 44 (2001) 311–321MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    Tang C and Wu X P, Existence and multiplicity of semilinear elliptic equations,J. Math. Anal. Appl. 256 (2001) 1–12MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    Tang C and Wu X P, Periodic solutions for second order systems with not uniformly coercive potential,J. Math. Anal. Appl. 259 (2001) 386–397MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Thews K, A reduction method for some nonlinear Dirichlet problems,Nonlin. Anal. 3 (1979) 795–813MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  • Leszek Gasiński
    • 1
  • Dumitru Motreanu
    • 2
  • Nikolaos S. Papageorgiou
    • 3
  1. 1.Institute of Computer ScienceJagiellonian UniversityCracowPoland
  2. 2.Departement de MathematiquesUniversité de PerpignanPerpignanFrance
  3. 3.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations