Skip to main content
SpringerLink
Log in

Multiple solutions for semilinear discontinuous variational problems with lack of compactness

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

This article establishes the existence of multiple solutions for a class of elliptic problems with discontinuous nonlinearity with critical growth by using the Lusternik—Schnirelmann category.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. O. Alves, A. M. Bertone and J. V. Gonçalves, A variational approach to discontinuous problems with critical Sobolev exponents, Journal of Mathematical Analysis and Applications 265 (2002), 103–127.

    Article  MathSciNet  Google Scholar 

  2. C. O. Alves and A. M. Bertone, A discontinuous problem involving the p-Laplacian operator and critical exponent in ℝN, Electronic Journal of Differential Equations (2003), Article no. 42.

  3. C. O. Alves and Y. H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity, Journal of Mathematical Analysis and Applications 279 (2003), 508–521

    Article  MathSciNet  Google Scholar 

  4. C. O. Alves, J. V. Gonçalves and J. A. dos Santos, Strongly nonlinear multivalued elliptic equations on a bounded domain, Journal of Global Optimization 58 (2014), 565–593.

    Article  MathSciNet  Google Scholar 

  5. C. O. Alves, J. A. dos Santos and R. C. M. Nemer, Multiple solutions for a problem with discontinuous nonlinearity, Annali di Matematica Pura ed Applicata 197 (2018), 883–903.

    Article  MathSciNet  Google Scholar 

  6. A. Ambrosetti and M. Badiale, The dual variational principle and elliptic problems with discontinuous nonlinearities, Journal of Mathematical Analysis and Applications 140 (1989), 363–373.

    Article  MathSciNet  Google Scholar 

  7. A. Ambrosetti, M. Calahorrano and F. Dobarro, Global branching for discontinuous problems, Commentationes Mathematicae Universitatis Carolinae 31 (1990), 213–222.

    MathSciNet  MATH  Google Scholar 

  8. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, Journal of Functional Analysis 44 (1973), 349–381.

    Article  MathSciNet  Google Scholar 

  9. A. Ambrosetti and M. Struwe, A note on the problem −Δu = λu+u∣u∣2*−2, Manuscripta Mathematica 54 (1986), 373–379.

    Article  MathSciNet  Google Scholar 

  10. A. Ambrosetti and R. E. L. Turner, Some discontinuous variational problems, Differential and Integral Equations 1 (1988), 341–349.

    MathSciNet  MATH  Google Scholar 

  11. M. Badiale and G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Analysis 29 (1997), 639–677.

    Article  MathSciNet  Google Scholar 

  12. A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Communications on Pure and Applied Mathematics 41 (1988), 253–294.

    Article  MathSciNet  Google Scholar 

  13. V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Archive for Rational Mechanics and Analysis 114 (1991), 79–93.

    Article  MathSciNet  Google Scholar 

  14. V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calculus of Variations and Partial Differential Equations 2 (1994), 29–48.

    Article  MathSciNet  Google Scholar 

  15. S. Carl and H. Dietrich, The weak upper and lower solution method for elliptic equations with generalized subdifferentiable perturbations, Applicable Analysis 56 (1995), 263–278.

    Article  MathSciNet  Google Scholar 

  16. S. Carl and S. Heikkila, Quasilinear elliptic equations with discontinuous nonlinearities in ℝN, Nonlinear Analysis 30 (1997), 1743–1751.

    Article  MathSciNet  Google Scholar 

  17. S. Carl and S. Heikkila, Elliptic equations with discontinuous nonlinearities in ℝN, Nonlinear Analysis 31 (1998), 217–227.

    Article  MathSciNet  Google Scholar 

  18. S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities, Springer Monographs in Mathematics, Springer, New York, 2007.

    Book  Google Scholar 

  19. G. Cerami, Metodi variazionalli nello studio di problemi al contorno con parte nonlineare discontinua, Rendiconti del Circolo Matematico di Palermo 32 (1983), 336–357.

    Article  MathSciNet  Google Scholar 

  20. K. C. Chang, On the multiple solutions of the elliptic differential equations with discontinuous nonlinear terms, Scientia Sinica 21 (1978), 139–158.

    MathSciNet  MATH  Google Scholar 

  21. K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Communications on Pure and Applied Mathematics 33 (1980), 117–146.

    Article  MathSciNet  Google Scholar 

  22. K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, Journal of Mathematical Analysis and Applications 80 (1981), 102–129.

    Article  MathSciNet  Google Scholar 

  23. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1983.

    MATH  Google Scholar 

  24. S. Hu, N. Kourogenis and N. S. Papageorgiou, Nonlinear elliptic eigenvalue problems with discontinuities, Journal of Mathematical Analysis and Applications 233 (1999), 406–424.

    Article  MathSciNet  Google Scholar 

  25. O. Kavian, Introduction á la théorie des points critiques et applications aux problémes elliptíques, Mathématiques & Applications, Vol. 13, Springer, Paris, 1993.

    MATH  Google Scholar 

  26. M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec léxposant critique de Sobolev, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 314 (1992), 161–164.

    MathSciNet  MATH  Google Scholar 

  27. D. Motreanu and C. Varga, Some critical point results for locally Lipschitz functionals, Communications on Applied Nonlinear Analysis 4 (1997), 17–33.

    MathSciNet  MATH  Google Scholar 

  28. V. Radulescu, Mountain pass theorems for non-differentiable functions and applications, Japan Academy. Proceedings. Series A. Mathematical Sciences 69 (1993), 193–198.

    Article  MathSciNet  Google Scholar 

  29. O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Analysis 13 (1989), 1241–1249.

    Article  MathSciNet  Google Scholar 

  30. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Université de Grenoble. Annales de l’Institut Fourier 15 (1965), 189–257.

    Article  MathSciNet  Google Scholar 

  31. M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Mathematische Zeitschrift 187 (1984), 511–517.

    Article  MathSciNet  Google Scholar 

  32. A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, International Press, Somerville, MA, 2010, pp. 597–632.

    MATH  Google Scholar 

  33. W. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser, Boston, MA, 1996.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-900, Campina Grande, PB, Brazil

    Claudianor O. Alves & Jefferson A. dos Santos

Authors
  1. Claudianor O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jefferson A. dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jefferson A. dos Santos.

Additional information

J. A. dos Santos was partially supported by CNPq/Brazil 303479/2019-1.

C.O. Alves was partially supported by CNPq/Brazil 304804/2017-7.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alves, C.O., dos Santos, J.A. Multiple solutions for semilinear discontinuous variational problems with lack of compactness. Isr. J. Math. 247, 149–176 (2022). https://doi.org/10.1007/s11856-021-2282-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2282-2

Search

Navigation