Annali di Matematica Pura ed Applicata

, Volume 163, Issue 1, pp 181–198 | Cite as

Resonance at two consecutive eigenvalues for semilinear elliptic equations

  • Pierpaolo Omari
  • Fabio Zanolin


The solvability of the Dirichlet problem for a semilinear elliptic equation is studied in some situations where the classical resonance conditions of Landesman and Lazer may fail.


Elliptic Equation Dirichlet Problem Resonance Condition Classical Resonance Consecutive Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-O-Z]
    Afuwape A. U. -Omari P. -Zanolin F.,Nonlinear perturbations of differential operators with nontrivial kernel and applications to third order periodic boundary value problems, J. Math. Anal. Appl.,143 (1989), pp. 35–56.Google Scholar
  2. [A]
    Agmon S.,The L p approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa,13 (1959), pp. 405–448.Google Scholar
  3. [A-M]
    Amann H. -Mancini G.,Some applications of monotone operator theory to resonance problems, Nonlinear Analysis T.M.A.,3 (1979), pp. 815–830.Google Scholar
  4. [A-L-P]
    Ahmad S. -Lazer A. C. -Paul J. L.,Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J.,25 (1976), pp. 933–944.Google Scholar
  5. [B-B-F]
    Bartolo P. -Benci V. -Fortunato D.,Abstract critical point theorems and applications to some nonlinear problems with «strong» resonance at infinity, Nonlinear Analysis T. M. A.,7 (1983), pp. 981–1012.Google Scholar
  6. [B-DF]
    Berestycki H. -De Figueiredo D.G.,Double resonance in semilinear elliptic problems, Comm. Partial Diff. Equations,9 (1981), pp. 91–120.Google Scholar
  7. [B-N]
    Brezis H. -Nirenberg L.,Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Sc. Norm. Sup. Pisa,5 (1978), pp. 225–326.Google Scholar
  8. [Ca]
    Cac N. P.,On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl.,132 (1988), pp. 473–483.Google Scholar
  9. [Co]
    Costa D. G.,A note on unbounded perturbations of linear resonant problems, preprint (1989).Google Scholar
  10. [C-O]
    Costa D. G. -Oliveira A. S.,Existence of solutions for a class of semilinear elliptic problems at double resonance, Bol. Soc. Bras. Mat.,19 (1988), pp. 21–37.Google Scholar
  11. [Da]
    Dancer E. N.,Non-uniqueness for nonlinear boundary value problems, Rocky Mountain J. Math.,13 (1983), pp. 401–412.Google Scholar
  12. [DF]
    De Figueiredo D. G.,The Dirichlet problem for nonlinear elliptic equations: A Hilbert space approach, Lecture Notes in Mathematics,446, Springer-Verlag, Berlin (1975), pp. 144–165.Google Scholar
  13. [DF-G]
    De Figueiredo D. G. -Gossez J. P.,Conditions de non-résonance pour certain problèmes elliptiques semi-linéaires, C. R. Acad. Sci. Paris,302 (1986), pp. 543–545.Google Scholar
  14. [DF-N]
    De Figueiredo D. G. -Ni W. M.,Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Analysis T.M.A.,5 (1979), pp. 629–634.Google Scholar
  15. [Di]
    Ding T.,Nonlinear oscillations at a point of resonance, Scientia Sinica (Serie A),25 (1982), pp. 918–931.Google Scholar
  16. [Do]
    Dolph C. L.,Nonlinear integral equations of the Hammerstein type, Trans. Amer. Mat. Soc.,66 (1949), pp. 289–307.Google Scholar
  17. [Dr]
    Drabek P.,Solvability of nonlinear problems at resonance, Comment. Math. Univ. Carolinae,23 (1982), pp. 359–368.Google Scholar
  18. [D-T]
    Drabek P. -Tomiczek P.,Remark on the structure of the range of second order nonlinear elliptic operator, Comment. Math. Univ. Carolinae,30 (1989), pp. 455–464.Google Scholar
  19. [F-F]
    Fabry C. -Fonda A.,Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl.,157 (1990), pp. 99–116Google Scholar
  20. [F]
    Friedman A.,Partial Differential Equations, Holt, Rinehart & Winston, New York (1969).Google Scholar
  21. [F-H]
    Fucik S. -Hess P.,Nonlinear perturbations of linear operators having nullspace with strong unique continuation property, Nonlinear Analysis T.M.A.,3 (1979), pp. 271–277.Google Scholar
  22. [F-K]
    Fucik S. -Krbec M.,Boundary value problems with bounded nonlinearity and general nullspace of the linear part, Math. Z.,155 (1977), pp. 129–138.Google Scholar
  23. [Gon]
    Gonçalves J. V. A.,Existence of saddle points for functional on Hilbert spaces: Applications to Hammerstein equations, J. Integral Equations,8 (1985), pp. 229–238.Google Scholar
  24. [Gos]
    Gossez J. P.,Nonresonance near the first eigenvalue of a second order elliptic problem, Lecture Notes in Mathematics,1324, Springer-Verlag, Berlin (1986), pp. 97–104.Google Scholar
  25. [Gu]
    Gupta C. P.,Solvability of a boundary value problem with the nonlinearity satisfying a sign condition, J. Math. Anal. Appl,129 (1988), pp. 482–492.Google Scholar
  26. [He]
    Hess P.,A remark on the preceding paper of Fucik and Krbec, Math. Z.,155 (1977), pp. 138–141.Google Scholar
  27. [Hi]
    Hirano N.,Unbounded nonlinear perturbations of linear elliptic problems at reso- nance, J. Math. Anal. Appl.,132 (1988), pp. 434–446.Google Scholar
  28. [I-N]
    Iannacci R. -Nkashama M. N.,Nonlinear boundary value problems at resonance, Nonlinear Analysis T.M.A.,11 (1987), pp. 455–473.Google Scholar
  29. [I-N-W]
    Iannacci R. -Nkashama M. N. -Ward J. R. jr.,Nonlinear second order elliptic partial differential equations at resonance, Trans. Amer. Math. Soc,311(1989), pp. 711–726.Google Scholar
  30. [L-L]
    Landesman E. M. -Lazer A. C.,Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech.,19 (1970), pp. 609–623.Google Scholar
  31. [M1]
    Mawhin J.,Contractive mappings and periodically perturbed conservative systems, Arch. Math. Scripta Fac. Sci. Mat. UJEP Brunensis,12 (1976), pp. 67–74.Google Scholar
  32. [M2]
    Mawhin J.,Topological Degree Methods in Nonlinear Boundary Value Problems, C.B.M.S. Regional Conference Series in Mathematics,40, A.M.S., Providence (1979).Google Scholar
  33. [M-Wi]
    Mawhin J. -Willem M.,Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. Henri Poincaré,6 (1986), pp. 431–453.Google Scholar
  34. [M-Wa]
    Mawhin J. -Ward J. R. jr.,Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Analysis T.M.A.,6 (1981), pp. 677–684.Google Scholar
  35. [O]
    Omari P.,Resonance at two consecutive eigenvalues for nonlinear differential equations, unpublished internal report (1988).Google Scholar
  36. [O-Z1]
    Omari P. -Zanolin F.,Existence results for forced nonlinear periodic BVPs at resonance, Ann. Mat. Pura Appl.,141 (1985), pp. 127–157.Google Scholar
  37. [O-Z2]
    Omari P. -Zanolin F.,Boundary value problems for forced nonlinear equations at resonance, Lecture Notes in Mathematics,1151, Springer-Verlag, Berlin (1985), pp. 285–294.Google Scholar
  38. [O-Z3]
    Omari P. -Zanolin F.,A note on nonlinear oscillations at resonance, Acta Math. Sinica (New Series),3 (1987), pp. 351–361.Google Scholar
  39. [T]
    Thews K.,A reduction method for some nonlinear Dirichlet problems, Nonlinear Analysis T.M.A.,3 (1979), pp. 795–813.Google Scholar
  40. [W]
    Ward J. R. jr.,Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math.,10 (1984), pp. 291–305.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1993

Authors and Affiliations

  • Pierpaolo Omari
    • 1
  • Fabio Zanolin
    • 2
  1. 1.Dipartimento di Scienze MatematicheUniversitàe di TriesteTriesteItalia
  2. 2.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItalia

Personalised recommendations