Quasilinear problems under local Landesman–Lazer condition

  • D. Arcoya
  • M. C. M. Rezende
  • E. A. B. SilvaEmail author


This paper presents results on the existence and multiplicity of solutions for quasilinear problems in bounded domains involving the p-Laplacian operator under local versions of the Landesman–Lazer condition. The main results do not require any growth restriction at infinity on the nonlinear term which may change sign. The existence of solutions is established by combining variational methods, truncation arguments and approximation techniques based on a compactness result for the inverse of the p-Laplacian operator. These results also establish the intervals of the projection of the solution on the direction of the first eigenfunction of the p-Laplacian operator. This fact is used to provide the existence of multiple solutions when the local Landesman–Lazer condition is satisfied on disjoint intervals.

Mathematics Subject Classification

35J20 35J92 47J30 



This work was done while the second and third authors were visiting the Departamento de Análisis Matemático, Universidad de Granada. They would like to present their gratitude for the warm hospitaltiy of the whole members of that department.

First author is supported by FEDER-MEC (Spain) PGC2018-096422-B-I00 and Junta de Andalucía FQM-116. Third author is supported by CNPq (Brazil) 311808/2014-0 and 312060/2018-1.


  1. 1.
    Alama, S., Del Pino, M.: Solutions of elliptic equations with indefinite nonlinearities via morse theory and linkings. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 95–115 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alama, S., Tarantello, G.: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 1, 439–475 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alama, S., Tarantello, G.: Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141, 159–215 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosetti, A., Arcoya, D.: On a quasilinear problem at strong resonance. Topol. Methods Nonlinear Anal. 6, 255–264 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Anane, A.: Simplicité et isolation de la première valeur du p-Laplacien avec: poids. C. R. Acad. Sci. Paris 305, 725–728 (1987)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Anane, A., Gossez, J.P.: Strongly nonlinear elliptic problems near resonance, a variational approach Comm. Partial Differ. Equ. 15, 1141–1159 (1990)CrossRefGoogle Scholar
  8. 8.
    Arcoya, D., Carmona, J., Leonori, T., Martínez-Aparicio, P.J., Orsina, L., Pettita, F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. 246, 4006–4042 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Arcoya, D., Gámez, J.L.: Bifurcation theory and related problems: anti-maximum principle and resonance. Comm. Partial Differ. Equ. 26, 1879–19011 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Arcoya, D., Orsina, L.: Landesman–Lazer conditions and quasilinear elliptic problems. Nonlinear Anal. 28, 1623–1632 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bartolo, P., Benci, V., Fortunato, D.: Abstract criticl point theorems and application to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4, 59–78 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Boccardo, L., Drábek, P., Kučera, M.: Landesman–Lazer conitions for strongly nonlinear boundary value problems. Comment. Math. Univ. Carol. 30, 411–427 (1989)zbMATHGoogle Scholar
  14. 14.
    Castro, A.: Reduction Methods via Minimax. First Latin American School of Differential Equations (São Paulo, Brazil, 1981), Lecture Notes in Mathematics, Vol. 957, pp. 1–20. Springer, Berlin (1982)Google Scholar
  15. 15.
    Castro, A., Lazer, A.C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120, 113–137 (1979)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chang, K.C., Jiang, M.Y.: Dirichlet problem with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 20, 257–282 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Costa, D.G., Tehrani, H.: Existence of positive solutions for a class of indefinite elliptic problems in \(\mathbb{R}^N\). Calc. Var. Partial Differ. Equ. 13, 159–189 (2001)CrossRefGoogle Scholar
  18. 18.
    De Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199, 452–467 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    De Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity. J. Eur. Math. Soc. 8, 269–286 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Drábek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinb. Sect. A 127, 703–726 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ilyasov, Y., Silva, K.: On branches of positive solutions for p-Laplacian problems at the extreme value of the Nehari manifold method. Proc. Am. Math. Soc. 146, 2925–2935 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ladyzenskaya, O., Uralt’seva, N.: Linear and Quasilinear Elliptic Equations. Translated by Scripta Technica. Academic Press, New York (1968)Google Scholar
  23. 23.
    Landesman, E.M., Lazer, A.C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1969/1970)Google Scholar
  24. 24.
    Landesman, E.M., Lazer, A.C., Meyers, David R.: On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence. J. Math. Anal. Appl. 52, 594–614 (1975)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Medeiros, E.S., Severo, U.B., Silva, E.A.B.: On a class of elliptic problems with indefinite nonlinearities. Calc. Var. Partial Differ. Equ. 50, 751–777 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ouyang, T.: On the positive solutions of semilinear equations \(\Delta u+\lambda u+hu^p=0\) on compact manifolds II. Indiana Univ. Math. J. 40, 1083–1141 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Peral, I.: Multiplicity of Solutions for the p-Laplacian. Second School on Nonlinear Functional Analysis and Applications to Differential Equations (1997)Google Scholar
  28. 28.
    Rabinowitz, P.H.: Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations. Nonlinear Analysis (a Collection of Papers in Honor of Erich Röthe), pp. 161–177. Academic Press, New York (1978)Google Scholar
  29. 29.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (1986)Google Scholar
  30. 30.
    Rezende, M.C.M., Sánchez-Aguilar, P.M., Silva, E.A.B.: A Landesman–Lazer local condition for semilinear elliptic problems. Bull. Braz. Math. Soc. MathSciNetCrossRefGoogle Scholar
  31. 31.
    Silva, E.A.B.: Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 16, 455–477 (1991)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Silva, K., Macedo, A.: Local minimizers over the Nehari manifold for a class of concave–convex problems with sign changing nonlinearity. J. Differ. Equ. 265, 1894–1921 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Stampacchia, G.Le: problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • D. Arcoya
    • 1
  • M. C. M. Rezende
    • 2
  • E. A. B. Silva
    • 2
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de GranadaGranadaSpain
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

Personalised recommendations