Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities
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- by Sophia Th. Kyritsi and Nikolaos S. Papageorgiou PDF
- Proc. Amer. Math. Soc. 133 (2005), 2369-2376 Request permission
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.References
- D. Arcoya and A. Cañada, The dual variational principle and discontinuous elliptic problems with strong resonance at infinity, Nonlinear Anal. 15 (1990), no. 12, 1145â1154. MR 1082289, DOI 10.1016/0362-546X(90)90050-Q
- Marino Badiale, Semilinear elliptic problems in $\textbf {R}^N$ with discontinuous nonlinearities, Atti Sem. Mat. Fis. Univ. Modena 43 (1995), no. 2, 293â305. MR 1366063
- P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with âstrongâ resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981â1012. MR 713209, DOI 10.1016/0362-546X(83)90115-3
- Kung Ching Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102â129. MR 614246, DOI 10.1016/0022-247X(81)90095-0
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- D. G. Costa and J. V. A. Gonçalves, Critical point theory for nondifferentiable functionals and applications, J. Math. Anal. Appl. 153 (1990), no. 2, 470â485. MR 1080660, DOI 10.1016/0022-247X(90)90226-6
- ZdzisĆaw Denkowski, StanisĆaw MigĂłrski, and Nikolas S. Papageorgiou, An introduction to nonlinear analysis: theory, Kluwer Academic Publishers, Boston, MA, 2003. MR 2024162, DOI 10.1007/978-1-4419-9158-4
- ZdzisĆaw Denkowski, StanisĆaw MigĂłrski, and Nikolas S. Papageorgiou, An introduction to nonlinear analysis: applications, Kluwer Academic Publishers, Boston, MA, 2003. MR 2024161
- S. Hu-N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume II: Applications, Kluwer, Dordrecht, The Netherlands (2000).
- Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Austral. Math. Soc. Ser. A 69 (2000), no. 2, 245â271. MR 1775181
- Peter Lindqvist, On the equation $\textrm {div}\,(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157â164. MR 1007505, DOI 10.1090/S0002-9939-1990-1007505-7
- C. Stuart, Maximal and minimal solutions of differential equations with discontinuities, Math. Z. 163 (1978), 239-249.
- Andrzej Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. PoincarĂ© Anal. Non LinĂ©aire 3 (1986), no. 2, 77â109 (English, with French summary). MR 837231
- Klaus Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), no. 1-2, 119â129. MR 566069, DOI 10.1017/S0308210500011732
- James R. Ward Jr., Applications of critical point theory to weakly nonlinear boundary value problems at resonance, Houston J. Math. 10 (1984), no. 2, 291â305. MR 744915
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
- MR Author ID: 135890
- Received by editor(s): January 13, 2004
- Published electronically: March 15, 2005
- Communicated by: David S. Tartakoff
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2138879