# 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

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topKourogenis, 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] 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] Ambrosetti, A. and Rabinowitz, P.: Dual variational methods in critical point theory and applications, ibid. 14 (1973), 349-381. Zbl0273.49063
- [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] 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] 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] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Zbl0582.49001
- [7] De Figueiredo, D.: The Ekeland Variational Principle with Applications and Detours, Springer, Berlin, 1989.
- [8] Hu, S. and Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997. Zbl0887.47001
- [9] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121-149. Zbl0292.35034
- [10] Kourogenis, N. C. and Papageorgiou, N. S.: Discontinuous quasilinear elliptic problems at resonance, Colloq. Math. 78 (1998), 213-223. Zbl0920.35061
- [11] Lindqvist, P.: On the equation $div(\parallel Dx{\parallel}^{p}-2Dx)+\lambda {\left|x\right|}^{p}-2x=0$, Proc. Amer. Math. Soc. 109 (1991), 157-164.
- [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] Mawhin, J. and Schmitt, K.: Nonlinear eigenvalue problems with a parameter near resonance, Ann. Polon. Math. 51 (1990), 241-248. Zbl0724.34025
- [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] 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] 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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