Discontinuous quasilinear elliptic problems at resonance

Nikolaos Kourogenis; Nikolaos Papageorgiou

Colloquium Mathematicae (1998)

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

Abstract

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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.

How to cite

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Kourogenis, 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

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  1. [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. [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. [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. [4] Browder, F. and Hess, P., Nonlinear mappings of monotone type, J. Funct. Anal. 11 (1972), 251-294. Zbl0249.47044
  5. [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. [6] Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley, New York, 1983. Zbl0582.49001
  7. [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. [8] Lindqvist, P., On the equation div ( | D x | p - 2 D x ) + λ | x | p - 2 x = 0 , Proc. Amer. Math. Soc. 109 (1991), 157-164. 
  9. [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. [10] Thews, K., Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 119-129. Zbl0431.35040
  11. [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

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