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Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities


Authors: Sophia Th. Kyritsi and Nikolaos S. Papageorgiou
Journal: Proc. Amer. Math. Soc. 133 (2005), 2369-2376
MSC (2000): Primary 35J20, 35J60, 35R05
DOI: https://doi.org/10.1090/S0002-9939-05-07864-0
Published electronically: March 15, 2005
MathSciNet review: 2138879
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Abstract | References | Similar Articles | Additional Information

Abstract: We examine a nonlinear strongly resonant elliptic problem driven by the $p$-Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the literature. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions.


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  • 1. D. Arcoya-A. Canada, The dual variational principle and discontinuous elliptic problems with strong resonance at infinity, Nonlin. Anal. 15 (1990), 1145-1154. MR 1082289 (91m:35230)
  • 2. M. Badiale, Semilinear elliptic problems in $\mathbb{R} ^N$with discontinuous nonlinearities, Atti Sem. Mat. Univ. Modena 43 (1995), 293-305.MR 1366063 (96m:35074)
  • 3. P. Bartolo-V. Benci-D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ``strong" resonance at infinity, Nonlin. Anal. 7 (1983), 981-1012. MR 0713209 (85c:58028)
  • 4. K.-C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. MR 0614246 (82h:35025)
  • 5. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983).MR 0709590 (85m:49002)
  • 6. D. Costa-J. Goncalves, Critical point theory for nondifferentiable functionals and applications, J. Math. Anal. Appl. 153 (1990), 470-485.MR 1080660 (91j:58034)
  • 7. Z. Denkowski-S. Migórski-N. S. Papageorgiou, An Introduction to Nonlinear Analysis:Theory, Kluwer Plenum, New York (2003). MR 2024162
  • 8. Z. Denkowski-S. Migórski-N. S. Papageorgiou, An Introduction to Nonlinear Analysis. Volume II:Applications, Kluwer Plenum, New York (2003).MR 2024161
  • 9. S. Hu-N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume II: Applications, Kluwer, Dordrecht, The Netherlands (2000).
  • 10. N. Kourogenis-N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Australian Math. Soc., Series A, 69 (2000), 245-271. MR 1775181 (2001m:35078)
  • 11. P. Lindqvist, On the equation $div(\Vert Dx\Vert^{p-2}Dx)+\lambda\vert x\vert^{p-2}x=0$, Proc. AMS 109 (1990), 157-164. MR 1007505 (90h:35088)
  • 12. C. Stuart, Maximal and minimal solutions of differential equations with discontinuities, Math. Z. 163 (1978), 239-249.
  • 13. A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Analyse Non Lineaire 3 (1986), 77-109.MR 0837231 (87f:49021)
  • 14. K. Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Royal Soc. Edinburg 85A (1980), 119-129. MR 0566069 (81c:35054)
  • 15. J. Ward, Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math. 10 (1984), 291-305.MR 0744915 (85i:35064)

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Additional Information

Sophia Th. Kyritsi
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Nikolaos S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

DOI: https://doi.org/10.1090/S0002-9939-05-07864-0
Keywords: $p$-Laplacian, strong resonance, nonsmooth critical point theory, generalized subdifferential, multiple solutions, discontinuous nonlinearity, generalized Ekeland variational principle
Received by editor(s): January 13, 2004
Published electronically: March 15, 2005
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society

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