# Discontinuous quasilinear elliptic problems at resonance

Nikolaos Kourogenis; Nikolaos Papageorgiou

Colloquium Mathematicae (1998)

- Volume: 78, Issue: 2, page 213-223
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topKourogenis, Nikolaos, and Papageorgiou, Nikolaos. "Discontinuous quasilinear elliptic problems at resonance." Colloquium Mathematicae 78.2 (1998): 213-223. <http://eudml.org/doc/210611>.

@article{Kourogenis1998,

abstract = {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.},

author = {Kourogenis, Nikolaos, Papageorgiou, Nikolaos},

journal = {Colloquium Mathematicae},

keywords = {compact embedding; Poincaré's inequality; Palais-Smale condition; critical point; variational method; Mountain Pass Theorem; subdifferential; problems at resonance; locally Lipschitz functional},

language = {eng},

number = {2},

pages = {213-223},

title = {Discontinuous quasilinear elliptic problems at resonance},

url = {http://eudml.org/doc/210611},

volume = {78},

year = {1998},

}

TY - JOUR

AU - Kourogenis, Nikolaos

AU - Papageorgiou, Nikolaos

TI - Discontinuous quasilinear elliptic problems at resonance

JO - Colloquium Mathematicae

PY - 1998

VL - 78

IS - 2

SP - 213

EP - 223

AB - 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.

LA - eng

KW - compact embedding; Poincaré's inequality; Palais-Smale condition; critical point; variational method; Mountain Pass Theorem; subdifferential; problems at resonance; locally Lipschitz functional

UR - http://eudml.org/doc/210611

ER -

## References

top- [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. Zbl0351.35036
- [2] Ambrosetti, A. and Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. Zbl0273.49063
- [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. Zbl0522.58012
- [4] Browder, F. and Hess, P., Nonlinear mappings of monotone type, J. Funct. Anal. 11 (1972), 251-294. Zbl0249.47044
- [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. Zbl0487.49027
- [6] Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Zbl0582.49001
- [7] Lazer, A. and Landesman, E., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609-623. Zbl0193.39203
- [8] Lindqvist, P., On the equation ${div\left(\right|Dx|}^{p-2}{Dx)+\lambda |x|}^{p-2}x=0$, Proc. Amer. Math. Soc. 109 (1991), 157-164.
- [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.
- [10] Thews, K., Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 119-129. Zbl0431.35040
- [11] Ward, J., Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math. 10 (1984), 291-305. Zbl0594.35037

## Citations in EuDML Documents

top## NotesEmbed ?

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