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Mat. Sb., 2015, Volume 206, Number 9, Pages 121–138 (Mi msb8427)  

This article is cited in 8 scientific papers (total in 9 papers)

The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities

V. N. Pavlenkoa, D. K. Potapovb*

a Chelyabinsk State University
b Saint Petersburg State University

Abstract: This paper is concerned with the existence of semiregular solutions to the Dirichlet problem for an equation of elliptic type with discontinuous nonlinearity and when the differential operator is not assumed to be formally self-adjoint. Theorems on the existence of semiregular (positive and negative) solutions for the problem under consideration are given, and a principle of upper and lower solutions giving the existence of semiregular solutions is established. For positive values of the spectral parameter, elliptic spectral problems with discontinuous nonlinearities are shown to have nontrivial semiregular (positive and negative) solutions.
Bibliography: 32 titles.

Keywords: spectral problems, equations of elliptic type, discontinuous nonlinearity, semiregular solutions, the method of upper and lower solutions.
* Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2015, 206:9, 1281–1298

Bibliographic databases:

UDC: 517.95
MSC: 35J25, 35J60, 35P30
Received: 01.10.2014 and 06.01.2015

Citation: V. N. Pavlenko, D. K. Potapov, “The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities”, Mat. Sb., 206:9 (2015), 121–138; Sb. Math., 206:9 (2015), 1281–1298

Citation in format AMSBIB
\by V.~N.~Pavlenko, D.~K.~Potapov
\paper The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities
\jour Mat. Sb.
\yr 2015
\vol 206
\issue 9
\pages 121--138
\jour Sb. Math.
\yr 2015
\vol 206
\issue 9
\pages 1281--1298

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    This publication is cited in the following articles:
    1. A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “Existence of solutions for second-order differential equations with discontinuous right-hand side”, Electron. J. Differential Equations, 2016 (2016), 124, 9 pp.  mathscinet  zmath  isi
    2. 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
    3. V. N. Pavlenko, D. K. Potapov, “Estimates for a spectral parameter in elliptic boundary value problems with discontinuous nonlinearities”, Siberian Math. J., 58:2 (2017), 288–295  mathnet  crossref  crossref  isi  elib  elib
    4. 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  adsnasa  isi  elib
    5. 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  isi  elib
    6. 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
    7. 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  isi  elib
    8. V. N. Pavlenko, D. K. Potapov, “Variational method for elliptic systems with discontinuous nonlinearities”, Sb. Math., 212:5 (2021), 726–744  mathnet  crossref  crossref  isi  elib
    9. V. N. Pavlenko, D. K. Potapov, “Existence of Semiregular Solutions of Elliptic Systems with Discontinuous Nonlinearities”, Math. Notes, 110:2 (2021), 226–241  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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