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

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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

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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

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  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

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