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

Published electronically:
March 15, 2005

MathSciNet review:
2138879

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Abstract: We examine a nonlinear strongly resonant elliptic problem driven by the -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.

**1.**D. Arcoya and A. Cañada,*The dual variational principle and discontinuous elliptic problems with strong resonance at infinity*, Nonlinear Anal.**15**(1990), no. 12, 1145–1154. MR**1082289**, 10.1016/0362-546X(90)90050-Q**2.**Marino Badiale,*Semilinear elliptic problems in 𝑅^{𝑁} with discontinuous nonlinearities*, Atti Sem. Mat. Fis. Univ. Modena**43**(1995), no. 2, 293–305. MR**1366063****3.**P. Bartolo, V. Benci, and D. Fortunato,*Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity*, Nonlinear Anal.**7**(1983), no. 9, 981–1012. MR**713209**, 10.1016/0362-546X(83)90115-3**4.**Kung Ching Chang,*Variational methods for nondifferentiable functionals and their applications to partial differential equations*, J. Math. Anal. Appl.**80**(1981), no. 1, 102–129. MR**614246**, 10.1016/0022-247X(81)90095-0**5.**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****6.**D. G. Costa and J. V. A. Gonçalves,*Critical point theory for nondifferentiable functionals and applications*, J. Math. Anal. Appl.**153**(1990), no. 2, 470–485. MR**1080660**, 10.1016/0022-247X(90)90226-6**7.**Zdzisław Denkowski, Stanisław Migórski, and Nikolas S. Papageorgiou,*An introduction to nonlinear analysis: theory*, Kluwer Academic Publishers, Boston, MA, 2003. MR**2024162****8.**Zdzisław Denkowski, Stanisław Migórski, and Nikolas S. Papageorgiou,*An introduction to nonlinear analysis: applications*, Kluwer Academic Publishers, Boston, MA, 2003. MR**2024161****9.**S. Hu-N. S. Papageorgiou,*Handbook of Multivalued Analysis. Volume II: Applications*, Kluwer, Dordrecht, The Netherlands (2000).**10.**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, 245–271. MR**1775181****11.**Peter Lindqvist,*On the equation 𝑑𝑖𝑣(\vert∇𝑢\vert^{𝑝-2}∇𝑢)+𝜆\vert𝑢\vert^{𝑝-2}𝑢=0*, Proc. Amer. Math. Soc.**109**(1990), no. 1, 157–164. MR**1007505**, 10.1090/S0002-9939-1990-1007505-7**12.**C. Stuart,*Maximal and minimal solutions of differential equations with discontinuities*, Math. Z.**163**(1978), 239-249.**13.**Andrzej Szulkin,*Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems*, Ann. Inst. H. Poincaré Anal. Non Linéaire**3**(1986), no. 2, 77–109 (English, with French summary). MR**837231****14.**Klaus Thews,*Nontrivial solutions of elliptic equations at resonance*, Proc. Roy. Soc. Edinburgh Sect. A**85**(1980), no. 1-2, 119–129. MR**566069**, 10.1017/S0308210500011732**15.**James R. Ward Jr.,*Applications of critical point theory to weakly nonlinear boundary value problems at resonance*, Houston J. Math.**10**(1984), no. 2, 291–305. MR**744915**

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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