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 from the left, being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.
@article{bwmeta1.element.bwnjournal-article-cmv81i1p89bwm,
author = {Nikolaos Kourogenis and Nikolaos Papageorgiou},
title = {Multiple solutions for nonlinear discontinuous elliptic problems near resonance},
journal = {Colloquium Mathematicae},
volume = {79},
year = {1999},
pages = {89-99},
zbl = {0933.35152},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv81i1p89bwm}
}
Kourogenis, Nikolaos; Papageorgiou, Nikolaos. Multiple solutions for nonlinear discontinuous elliptic problems near resonance. Colloquium Mathematicae, Tome 79 (1999) pp. 89-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv81i1p89bwm/
[000] [1] Ambrosetti, A., Garcia Azorero, J. and Peral, I.: Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. | Zbl 0852.35045
[001] [2] Ambrosetti, A. and Rabinowitz, P.: Dual variational methods in critical point theory and applications, ibid. 14 (1973), 349-381. | Zbl 0273.49063
[002] [3] Chang, K. C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. | Zbl 0487.49027
[003] [4] Chiappinelli, R. and De Figueiredo, D.: Bifurcation from infinity and multiple solutions for an elliptic system, Differential Integral Equations 6 (1993), 757-771. | Zbl 0784.35008
[004] [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. | Zbl 0780.35038
[005] [6] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983. | Zbl 0582.49001
[006] [7] De Figueiredo, D.: The Ekeland Variational Principle with Applications and Detours, Springer, Berlin, 1989.
[007] [8] Hu, S. and Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997. | Zbl 0887.47001
[008] [9] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121-149. | Zbl 0292.35034
[009] [10] Kourogenis, N. C. and Papageorgiou, N. S.: Discontinuous quasilinear elliptic problems at resonance, Colloq. Math. 78 (1998), 213-223. | Zbl 0920.35061
[010] [11] Lindqvist, P.: On the equation , Proc. Amer. Math. Soc. 109 (1991), 157-164.
[011] [12] Mawhin, J. and Schmitt, K.: Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math. 14 (1988), 138-146. | Zbl 0780.35043
[012] [13] Mawhin, J. and Schmitt, K.: Nonlinear eigenvalue problems with a parameter near resonance, Ann. Polon. Math. 51 (1990), 241-248. | Zbl 0724.34025
[013] [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. | Zbl 0609.58002
[014] [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. | Zbl 0869.35041
[015] [16] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. | Zbl 0488.35017