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Matematicheskie Zametki, 2017, Volume 101, Issue 2, Pages 247–261
DOI: https://doi.org/10.4213/mzm10743 (Mi mzm10743) 		 
This article is cited in 7 scientific papers (total in 8 papers)

Existence of Three Nontrivial Solutions of an Elliptic Boundary-Value Problem with Discontinuous Nonlinearity in the Case of Strong Resonance

V. N. Pavlenkoa, D. K. Potapovb

a Chelyabinsk State University
b Saint Petersburg State University
Full-text PDF (546 kB) Citations (8)
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DOI: https://doi.org/10.4213/mzm10743
Abstract: We consider a strongly resonant homogeneous Dirichlet problem for elliptic-type equations with discontinuous nonlinearity in the phase variable. Using the variational method, we prove an existence theorem for at least three nontrivial solutions of the problem under consideration; at least two of these are semiregular. The resulting theorem is applied to the eigenvalue problem for elliptic-type equations with discontinuous nonlinearity with positive spectral parameter. An example of a discontinuous nonlinearity satisfying all the assumptions of the theorem is given.
Keywords: elliptic boundary-value problem, strong resonance, discontinuous nonlinearity, nontrivial and semiregular solution.
Received: 16.02.2015
Revised: 24.06.2015
English version:
Mathematical Notes, 2017, Volume 101, Issue 2, Pages 284–296
DOI: https://doi.org/10.1134/S0001434617010333
Bibliographic databases:
Document Type: Article
UDC: 517.95
Language: Russian
Citation: V. N. Pavlenko, D. K. Potapov, “Existence of Three Nontrivial Solutions of an Elliptic Boundary-Value Problem with Discontinuous Nonlinearity in the Case of Strong Resonance”, Mat. Zametki, 101:2 (2017), 247–261; Math. Notes, 101:2 (2017), 284–296
Citation in format AMSBIB
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  • https://doi.org/10.4213/mzm10743
  • https://www.mathnet.ru/eng/mzm/v101/i2/p247
    • This publication is cited in the following 8 articles:
    1. V. N. Pavlenko, D. K. Potapov, “Semiregular solutions of elliptic boundary-value problems with discontinuous nonlinearities of exponential growth”, Sb. Math., 213:7 (2022), 1004–1019  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. V. N. Pavlenko, D. K. Potapov, “One class of quasilinear elliptic type equations with discontinuous nonlinearities”, Izv. Math., 86:6 (2022), 1162–1178  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. V. N. Pavlenko, D. K. Potapov, “Positive solutions of superlinear elliptic problems with discontinuous non-linearities”, Izv. Math., 85:2 (2021), 262–278  mathnet  crossref  crossref  zmath  adsnasa  isi  elib
    4. V. N. Pavlenko, D. K. Potapov, “On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity”, Izv. Math., 84:3 (2020), 592–607  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. V. N. Pavlenko, D. K. Potapov, “On the existence of three nontrivial solutions of a resonance elliptic boundary value problem with a discontinuous nonlinearity”, Differ. Equ., 56:7 (2020), 831–841  crossref  mathscinet  zmath  isi
    6. V. N. Pavlenko, D. K. Potapov, “Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity”, Sb. Math., 210:7 (2019), 1043–1066  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. V. Arutyunov, S. E. Zhukovskii, “Variational Principles in Nonlinear Analysis and Their Generalization”, Math. Notes, 103:6 (2018), 1014–1019  mathnet  crossref  crossref  mathscinet  isi  elib
    8. S. M. Voronin, S. F. Dolbeeva, O. N. Dementev, A. A. Ershov, M. G. Lepchinskii, S. V. Matveev, N. B. Medvedeva, D. K. Potapov, E. A. Rozhdestvenskaya, E. A. Sbrodova, I. M. Sokolinskaya, A. A. Solovev, V. I. Ukhobotov, V. E. Fedorov, “K 70-letiyu professora Vyacheslava Nikolaevicha Pavlenko”, Chelyab. fiz.-matem. zhurn., 2:4 (2017), 383–387  mathnet  elib
    Citing articles in Google Scholar: Russian citations, English citations
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