In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.
@article{bwmeta1.element.bwnjournal-article-cmv78z2p213bwm,
author = {Nikolaos Kourogenis and Nikolaos Papageorgiou},
title = {Discontinuous quasilinear elliptic problems at resonance},
journal = {Colloquium Mathematicae},
volume = {78},
year = {1998},
pages = {213-223},
zbl = {0920.35061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv78z2p213bwm}
}
Kourogenis, Nikolaos; Papageorgiou, Nikolaos. Discontinuous quasilinear elliptic problems at resonance. Colloquium Mathematicae, Tome 78 (1998) pp. 213-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv78z2p213bwm/
[000] [1] Ahmad, S., Lazer, A. and Paul, J., Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), 933-944. | Zbl 0351.35036
[001] [2] Ambrosetti, A. and Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. | Zbl 0273.49063
[002] [3] Benci, V., Bartolo, P. and Fortunato, D., Abstract critical point theorems and applications to nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 (1983), 961-1012. | Zbl 0522.58012
[003] [4] Browder, F. and Hess, P., Nonlinear mappings of monotone type, J. Funct. Anal. 11 (1972), 251-294. | Zbl 0249.47044
[004] [5] Chang, K. C., Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. | Zbl 0487.49027
[005] [6] Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley, New York, 1983. | Zbl 0582.49001
[006] [7] Lazer, A. and Landesman, E., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. | Zbl 0193.39203
[007] [8] Lindqvist, P., On the equation , Proc. Amer. Math. Soc. 109 (1991), 157-164.
[008] [9] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, R.I., 1986.
[009] [10] Thews, K., Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 119-129. | Zbl 0431.35040
[010] [11] Ward, J., Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math. 10 (1984), 291-305. | Zbl 0594.35037