Multiple solutions for nonlinear discontinuous elliptic problems near resonance

Nikolaos Kourogenis; Nikolaos Papageorgiou

Colloquium Mathematicae (1999)

  • Volume: 81, Issue: 1, page 89-99
  • ISSN: 0010-1354

Abstract

top
We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when λ → λ 1 from the left, λ 1 being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

How to cite

top

Kourogenis, Nikolaos, and Papageorgiou, Nikolaos. "Multiple solutions for nonlinear discontinuous elliptic problems near resonance." Colloquium Mathematicae 81.1 (1999): 89-99. <http://eudml.org/doc/210732>.

@article{Kourogenis1999,
abstract = {We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $λ → λ_1$ from the left, $λ_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.},
author = {Kourogenis, Nikolaos, Papageorgiou, Nikolaos},
journal = {Colloquium Mathematicae},
keywords = {discontinuous function; generalized directional derivative; critical point; coercive functional; multiple solutions; Clarke subdifferential; Rayleigh quotient; first eigenvalue; p-Laplacian; elliptic inclusion; nonsmooth Palais-Smale condition; discontinuous data; existence theory},
language = {eng},
number = {1},
pages = {89-99},
title = {Multiple solutions for nonlinear discontinuous elliptic problems near resonance},
url = {http://eudml.org/doc/210732},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Kourogenis, Nikolaos
AU - Papageorgiou, Nikolaos
TI - Multiple solutions for nonlinear discontinuous elliptic problems near resonance
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 1
SP - 89
EP - 99
AB - We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $λ → λ_1$ from the left, $λ_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.
LA - eng
KW - discontinuous function; generalized directional derivative; critical point; coercive functional; multiple solutions; Clarke subdifferential; Rayleigh quotient; first eigenvalue; p-Laplacian; elliptic inclusion; nonsmooth Palais-Smale condition; discontinuous data; existence theory
UR - http://eudml.org/doc/210732
ER -

References

top
  1. [1] Ambrosetti, A., Garcia Azorero, J. and Peral, I.: Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. Zbl0852.35045
  2. [2] Ambrosetti, A. and Rabinowitz, P.: Dual variational methods in critical point theory and applications, ibid. 14 (1973), 349-381. Zbl0273.49063
  3. [3] Chang, K. C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. Zbl0487.49027
  4. [4] Chiappinelli, R. and De Figueiredo, D.: Bifurcation from infinity and multiple solutions for an elliptic system, Differential Integral Equations 6 (1993), 757-771. Zbl0784.35008
  5. [5] Chiappinelli, R., Mawhin, J. and Nugari, R.: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. 18 (1992), 1099-1112. Zbl0780.35038
  6. [6] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Zbl0582.49001
  7. [7] De Figueiredo, D.: The Ekeland Variational Principle with Applications and Detours, Springer, Berlin, 1989. 
  8. [8] Hu, S. and Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997. Zbl0887.47001
  9. [9] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121-149. Zbl0292.35034
  10. [10] Kourogenis, N. C. and Papageorgiou, N. S.: Discontinuous quasilinear elliptic problems at resonance, Colloq. Math. 78 (1998), 213-223. Zbl0920.35061
  11. [11] Lindqvist, P.: On the equation d i v ( ∥ D x ∥ p - 2 D x ) + λ | x | p - 2 x = 0 , Proc. Amer. Math. Soc. 109 (1991), 157-164. 
  12. [12] Mawhin, J. and Schmitt, K.: Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math. 14 (1988), 138-146. Zbl0780.35043
  13. [13] Mawhin, J. and Schmitt, K.: Nonlinear eigenvalue problems with a parameter near resonance, Ann. Polon. Math. 51 (1990), 241-248. Zbl0724.34025
  14. [14] Rabinowitz, R.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, RI, 1986. Zbl0609.58002
  15. [15] Ramos, M. and Sanchez, L.: A variational approach to multiplicity in elliptic problems near resonance, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 385-394. Zbl0869.35041
  16. [16] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. Zbl0488.35017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.