Skip to main content
SpringerLink
Log in

Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

  • Chapter
  • First Online:
Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 95))

  • 1584 Accesses

Abstract

This chapter is concerned with parametric Dirichlet boundary value problems involving the p-Laplacian operator. Specifically, this chapter gives an account of recent results that establish the existence and multiplicity of solutions according to different types of nonlinearities in the problem. More precisely, we focus on problems exhibiting nonlinearities of concave–convex type and nonlinearities that are asymptotically \((p-1)\)-linear. In each situation, we point out significant qualitative properties of the solutions, especially, we establish the existence of sign-changing (that is, nodal) solutions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Bartsch, T., Liu, Z., Weth, T.: Nodal solutions of a p-Laplacian equation. Proc. Lond. Math. Soc. 91(3), 129–152 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benyamini, Y., Sternfeld, Y.: Spheres in infinite-dimensional normed spaces are Lipschitz contractible. Proc. Am. Math. Soc. 88, 439–445 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  6. Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the p-Laplacian. Differ. Integral Equ. 20, 309–324 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlinear Anal. 68, 2668–2676 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carl, S., Perera, K.: Sign-changing and multiple solutions for the p-Laplacian. Abstr. Appl. Anal. 7, 613–625 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer, New York (2007)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  11. Cuesta, M., de Figueiredo, D., Gossez, J.-P.: The beginning of the Fu\v cik spectrum for the p-Laplacian. J. Differ. Equ. 159, 212–238 (1999)

    Article  MATH  Google Scholar 

  12. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. García Azorero, J.P., Manfredi, J.J., Peral Alonso, I.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385–404 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Godoy, T., Gossez, J.-P., Paczka, S.: On the antimaximum principle for the p-Laplacian with indefinite weight. Nonlinear Anal. 51, 449–467 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Book  MATH  Google Scholar 

  16. Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo, Z., Zhang, Z.: \(W1,p\) versus C 1local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286, 32–50 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: A unified approach for multiple constant sign and nodal solutions. Adv. Differ. Equ. 12, 1363–1392 (2007)

    MATH  MathSciNet  Google Scholar 

  20. Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: On p-Laplace equations with concave terms and asymmetric perturbations. Proc. R. Soc. Edinb. Sect. A 141, 171–192 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. &00D4#;tani, M.: Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76, 140–159 (1988)

    Google Scholar 

  22. Rabinowitz, P.H.: Some minimax theorems and applications to nonlinear partial differential equations. In: Cesari, L., Rothe, E.H., Kannan, R., Weinberger, H.F. (eds.) Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe, pp. 161–177. Academic Press, New York (1978)

    Google Scholar 

  23. Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author is supported by a Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement No. PIEF-GA-2010-274519).

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Perpignan, Avenue Paul Alduy 52, 66860, Perpignan, France

    D. Motreanu

  2. Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, 84105, Be’er Sheva, Israel

    V. V. Motreanu

Authors
  1. D. Motreanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Motreanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Motreanu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Motreanu, D., Motreanu, V. (2014). Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_15

Download citation

Publish with us

Policies and ethics

Search

Navigation