Annali di Matematica Pura ed Applicata

, Volume 191, Issue 3, pp 395–430 | Cite as

Neumann problems resonant at zero and infinity

  • Leszek Gasiński
  • Nikolaos S. Papageorgiou
Open Access


We consider a semilinear Neumann problem with a reaction which is resonant at both zero and ±∞. Using a combination of methods from critical point theory, together with truncation techniques, the use of upper–lower solutions and of the Morse theory (critical groups), we show that the problem has at least five nontrivial smooth solutions, four of which have constant sign (two positive and two negative).


Resonance at zero and infinity Critical point theory Morse theory Truncation techniques Regularity theory Multiple solutions Solutions of constant sign 

Mathematics Subject Classification (2000)

35J20 35J60 58E05 



The authors would like to thank the referee for their corrections and remarks.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915) (2008)Google Scholar
  2. 2.
    Aizicovici S., Papageorgiou N.S., Staicu V.: Existence of multiple solutions with precise sign information for superlinear Neumann problems. Ann. Mat. Pura Appl. 188(4), 679–719 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bartolo P., Benci V., Fortunato D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bartsch T.: Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117–152 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bartsch T., Li S.-J.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419–441 (1997)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brézis H., Nirenberg L.: H 1 versus C 1 local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465–472 (1993)MATHGoogle Scholar
  7. 7.
    Chang K.-C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems, vol. 6 of Progress in Nonlinear Differential Equations and Their Applications. Birkhüuser Verlag, Boston (1993)Google Scholar
  8. 9.
    Costa D.G., Silva E.A.B.: On a class of resonant problems at higher eigenvalues. Diff. Integral Equ. 8, 663–671 (1995)MathSciNetMATHGoogle Scholar
  9. 9.
    Dunford N., Schwartz J.T.: Linear Operators, I General Theory, vol. 7 of Pure and Applied Mathematics. Wiley, New York (1958)Google Scholar
  10. 10.
    Filippakis M., Papageorgiou N.S.: Multiple nontrivial solutions for resonant Neumann problems. Math. Nachr. 283, 1000–1014 (2010)MathSciNetMATHGoogle Scholar
  11. 11.
    Garcĺa Azorero J., Manfredi J., Peral Alonso I.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385–404 (2000)MathSciNetGoogle Scholar
  12. 12.
    Gasiński L., Papageorgiou N.S.: Nonlinear Analysis. Chapman and Hall/CRC Press, Boca Raton (2006)MATHGoogle Scholar
  13. 13.
    Guo Z., Zhang Z.: W 1,p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286, 32–50 (2003)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hirano N., Nishimura T.: Multiplicity results for semilinear elliptic problems at resonance with jumping nonlinearities. J. Math. Anal. Appl. 180, 566–586 (1993)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Iannacci R., Nkashama M.N.: Nonlinear two point boundary value problems at resonance without Landesman–Lazer condition. Proc. Am. Math. Soc. 106, 943–952 (1989)MathSciNetMATHGoogle Scholar
  16. 16.
    Iannacci R., Nkashama M.N.: Nonlinear elliptic partial differential equations at resonance: higher eigenvalues. Nonlinear Anal. 25, 455–471 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Iannizzotto A., Papageorgiou N.S.: Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities. Nonlinear Anal. 70, 3285–3297 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kristaly A., Papageorgiou N.S.: Multiple nontrivial solutions for neumann problems involving the p-Laplacian: a Morse theoretic approach. Adv. Nonlinear Stud. 124, 83–87 (1996)MathSciNetGoogle Scholar
  19. 19.
    Kuo C.-C.: On the solvability of a nonlinear second order elliptic equations at resonance. Proc. Am. Math. Soc. 124, 83–87 (1996)MATHCrossRefGoogle Scholar
  20. 20.
    Landesman E.M., Robinson S.B., Rumbos A.: Multiple solutions of semilinear elliptic problems at resonance. Nonlinear Anal. 24, 1049–1059 (1995)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Li C.: The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems. Nonlinear Anal. 54, 431–443 (2003)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Liang Z., Su J.: Multiple solutions for semilinear elliptic boundary value problems with double resonance. J. Math. Anal. Appl. 354, 147–158 (2009)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Li C., Li S.: Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition. J. Math. Anal. Appl. 298, 14–32 (2004)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Liu S.-B.: Nontrivial solutions for elliptic problems. Nonlinear Anal. 70, 1965–1974 (2009)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Li S.-J., Su J.: Existence of multiple solutions of a two-point boundary value problems at resonance. Topol. Methods Nonlinear Anal. 10, 123–135 (1997)MathSciNetMATHGoogle Scholar
  26. 26.
    Li S.-J., Zou W.: The computations of the critical groups with an applications to elliptic resonant problems at a higher eigenvalues. J. Math. Anal. Appl. 235, 237–259 (1999)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Mawhin J.: Semicoercive monotone variational problems. Acad. Roy. Belg. Bull. Cl. Sci. 73(5), 118–130 (1987)MathSciNetMATHGoogle Scholar
  28. 28.
    Mawhin J., Ward J., Willem M.: Variational methods and semi-linear elliptic equations. Arch. Ration. Mech. Anal. 95, 269–277 (1986)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Mawhin J., Willem M.: Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences. Springer, New York (1989)Google Scholar
  30. 30.
    Motreanu D., Motreanu V.V., Papageorgiou N.S.: Nonlinear Neumann problems near resonance. Indiana Univ. Math. J. 58, 1257–1279 (2009)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Motreanu, D., Papageorgiou, N.S.: Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators. Proc. Am. Math. Soc. (to appear)Google Scholar
  32. 32.
    Perera K., Schechter M.: Solutions of nonlinear equations having asymptotic limits at zero and infinity. Calc. Var. Partial Diff. Equ. 12, 359–369 (2001)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Qian A.: Existence of infinitely many solutions for a superlinear Neumann boundary value problem. Boundary Value Probl. 2005, 329–335 (2005)MATHGoogle Scholar
  34. 34.
    Su J., Tang C.-L.: Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues. Nonlinear Anal. 44, 311–321 (2001)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Tang C.-L., Wu X.-P.: Existence and multiplicity for solutions of Neumann problems for elliptic equations. J. Math. Anal. Appl. 288, 660–670 (2003)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Vázquez J.L.: A strong maximum principle for some quasilinear elliptic equation. Appl. Math. Optim. 12, 191–202 (1984)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Zou W.: Multiple solutions results for two-point boundary value problem with resonance. Discrete Contin. Dyn. Syst. 4, 485–496 (1998)MATHCrossRefGoogle Scholar
  38. 38.
    Zou W., Liu J.-Q.: Multiple solutions for resonant elliptic equations via local linking theory and Morse theory. J. Diff. Equ. 170, 68–95 (2001)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute of Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations