Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On nonlinear parametric problems for p-Laplacian-like operators

Sobre los problemas paramétricos no lineales con operadores de tipo p-Laplaciano

  • 62 Accesses

  • 6 Citations


We study a nonlinear parametric problem driven by a p-Laplacian-like operator (which need not be homogeneous) and with a ( p - 1)-superlinear nonlinearity which satisfy weaker conditions than the Ambrosetti-Rabinowitz condition. Using critical point theory, we show that for every λ > 0, the nonlinear parametric problem has a nontrivial solution. Then, by strengthening the conditions on the operator and the nonlinearity, and using variational methods together with suitable truncation techniques and tools from Morse theory, we show that, for every λ > 0, the nonlinear parametric problem has three nontrivial smooth solutions.


En este artículo estudiamos un problema paramétrico no lineal que involucra al operador de tipo p-Laplaciano (que en general no és homogéneo) y donde la derivada del potencial es una función ( p - 1)-superlinear que verifica una condición más débil que la conocida condición de Ambrosetti-Rabinowitz. Utilizando métodos variacionales, mostramos que, para todo λ > 0, el problema paramétrico no lineal tiene una solución no trivial. Entonces, fortaleciendo las condiciones y usando herramientas de la teoría de Morse junto con adecuadas técnicas de truncación, mostramos que, para cada λ > 0, el problema tiene tres soluciones suaves.

This is a preview of subscription content, log in to check access.


  1. [1]

    Ambrosetti, A., Brezis, H. and Cerami, G., (1994). Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519–543.

  2. [2]

    Ambrosetti, A. and Rabinowitz, P., (1973). Dual variational methods in the critical point theory and applications, J. Funct. Anal., 14, 349–381.

  3. [3]

    Bartolo, P., Berti, V. and Fortunato, D., (1983). Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7, 981–1012.

  4. [4]

    Bartsch, T. and Li, S., (1997). Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28, 419–441.

  5. [5]

    Browder, F., (1970). Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc, 76, 999–1005.

  6. [6]

    Chang, K. C., (1993). Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston.

  7. [7]

    Corvellec, J.-N., (2001). On the second deformation lemma, Topol. Meth. Nonlin. Anal.17, 55–66.

  8. [8]

    Costa, D. and Magalhaes, C., (1995). Existence results for perturbations of the p-Laplacian, Nonlin. Anal., 24, 409–418.

  9. [9]

    De Napoli, P. and Mariani, M. C., (2003). Mountain pass solutions to equations of p-Laplacian, Nonlin. Anal., 54, 1205–1219.

  10. [10]

    Dugundji, J., (1966). Topology, Allyn and Bacon Inc., Boston.

  11. [11]

    Gasinski, L. and Papageorgiou, N. S., (2006). Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton.

  12. [12]

    Granas, A. and Dugunsji, J., (2003). Fixed Point Theory, Springer-Verlag, New York.

  13. [13]

    Guo, Z., (1996). On the number of positive solutions for quasilinear elliptic problems, Nonlin. Anal., 27, 229–247.

  14. [14]

    Guo, Z. and Webb, J. R. L., (2002). Large and small solutions of a class of quasilinear elliptic eigenvalue problems, J. Differential Equations, 180, 1–50.

  15. [15]

    Hai, D. D., (2003). On a class of sublinear quasilinear elliptic problems, Proceedings AMS, 131, 2409–2414.

  16. [16]

    Ladyzhenskaya, O. and Uraltseva, N., (1968). Linear and Quasilinear Elliptic Equations, Academic Press, New York.

  17. [17]

    Lieberman, G., (1988). Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12, 1203–1219.

  18. [18]

    Mawhin, J. and Willem, M., (1989). Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York.

  19. [19]

    Miyagaki, O. and Souto, M. A. S., Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, — in press.

  20. [20]

    Montenegro, M., (1999). Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlin. Anal., 37, 431–448.

  21. [21]

    Motreanu, D., Motreanu, V. and Papageorgiou, N. S., (2007). Multiple nontrivial solutions for nonlinear eigenvalue problems, Proceedings AMS, 135, 3649–3658.

  22. [22]

    Motreanu, D., Motreanu, V. and Papageorgiou, N. S., (2007). A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math., 124, 507–531.

  23. [23]

    Palais, R., (1966). Homotopy theory of finite dimensional manifolds, Topology, 5, 1–16.

  24. [24]

    Papageorgiou, N. S., Rocha, E. M. and Staicu, V., (2008). A multiplicity theorem for hemivariational inequalities with a p-Laplacian-like differential operator, Nonlin. Anal., 69, 1150–1163.

  25. [25]

    Papageorgiou, N. S., Rocha, E. M. and Staicu, V., (2008). Multiplicity theorems for superlinear elliptic problems, Calc. Variations PDEs, 33, 199–230.

  26. [26]

    Perera, K., (1997). Critical groups of pairs of critical points produced by linking subsets, J. Differential Equations, 140:1, 142–160.

  27. [27]

    Perera, K., (2003). Multiple positive solutions for a class of quasilinear elliptic boundary value problems, Electronic J. Differential Equations, 7, 1–5.

  28. [28]

    Schechter, M. and Zou, W., (2004). Superlinear problems, Pacific J. Math., 214, 145–160.

  29. [29]

    Ubilla, P., (1999). Existence of nonnegative solutions for a quasilinear Dirichlet problem, Comm. Appl. Nonlinear Anal., 6:2, 89–99.

  30. [30]

    Vazquez, J., (1984). A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12, 191–202.

  31. [31]

    Zhang, Q., (2005). A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl., 312, 24–32.

  32. [32]

    Zhou, H. S., (2002). An application of a mountain pass theorem, Acta Math. Sinica, 18, 27–36.

Download references

Author information

Correspondence to Nikolaos S. Papageorgiou.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Papageorgiou, N.S., Rocha, E.M. On nonlinear parametric problems for p-Laplacian-like operators. Rev. R. Acad. Cien. Serie A. Mat. 103, 177–200 (2009). https://doi.org/10.1007/BF03191850

Download citation


  • p-Laplacian-like operator
  • superlinear nonlinearity
  • strong deformation retract
  • critical groups and Morse theory

Mathematics Subject Classifications

  • 35J25
  • 35J80
  • 58E05