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On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity. / Pavlenko, V. N.; Potapov, D. K.

In: Differential Equations, Vol. 56, No. 7, 01.07.2020, p. 831-841.

Research output: Contribution to journal › Article › peer-review

Harvard

Pavlenko, VN & Potapov, DK 2020, 'On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity', Differential Equations, vol. 56, no. 7, pp. 831-841. https://doi.org/10.1134/S0012266120070034

APA

Pavlenko, V. N., & Potapov, D. K. (2020). On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity. Differential Equations, 56(7), 831-841. https://doi.org/10.1134/S0012266120070034

Vancouver

Pavlenko VN, Potapov DK. On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity. Differential Equations. 2020 Jul 1;56(7):831-841. https://doi.org/10.1134/S0012266120070034

Author

Pavlenko, V. N. ; Potapov, D. K. / On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity. In: Differential Equations. 2020 ; Vol. 56, No. 7. pp. 831-841.

BibTeX

@article{df7b8c5adaaa49c9aa9b237ec72539ba,
title = "On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity",
abstract = "Abstract: We study the homogeneous Dirichlet problem for a second-order elliptic equation with anonlinearity discontinuous in the state variable in the resonance case. A class of resonanceproblems that does not overlap with the previously investigated class of strongly resonanceproblems is singled out. Using the variational method, we establish a theorem on the existence ofat least three nontrivial solutions of the problem under study (the zero is its solution). In thiscase, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on thediscontinuities of the nonlinearity only on a set of measure zero. We give an example of anonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition isobtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest.Applications of the theorem to problems with a parameter are considered. The existence ofnontrivial (including semiregular) solutions of the problem with a parameter for an ellipticequation with a discontinuous nonlinearity for all positive values of the parameter is established.",
keywords = "NON-DIFFERENTIABLE FUNCTIONALS, EQUATIONS",
author = "Pavlenko, {V. N.} and Potapov, {D. K.}",
year = "2020",
month = jul,
day = "1",
doi = "10.1134/S0012266120070034",
language = "English",
volume = "56",
pages = "831--841",
journal = "Differential Equations",
issn = "0012-2661",
publisher = "Pleiades Publishing",
number = "7",

}

RIS

TY - JOUR

T1 - On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity

AU - Pavlenko, V. N.

AU - Potapov, D. K.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - Abstract: We study the homogeneous Dirichlet problem for a second-order elliptic equation with anonlinearity discontinuous in the state variable in the resonance case. A class of resonanceproblems that does not overlap with the previously investigated class of strongly resonanceproblems is singled out. Using the variational method, we establish a theorem on the existence ofat least three nontrivial solutions of the problem under study (the zero is its solution). In thiscase, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on thediscontinuities of the nonlinearity only on a set of measure zero. We give an example of anonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition isobtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest.Applications of the theorem to problems with a parameter are considered. The existence ofnontrivial (including semiregular) solutions of the problem with a parameter for an ellipticequation with a discontinuous nonlinearity for all positive values of the parameter is established.

AB - Abstract: We study the homogeneous Dirichlet problem for a second-order elliptic equation with anonlinearity discontinuous in the state variable in the resonance case. A class of resonanceproblems that does not overlap with the previously investigated class of strongly resonanceproblems is singled out. Using the variational method, we establish a theorem on the existence ofat least three nontrivial solutions of the problem under study (the zero is its solution). In thiscase, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on thediscontinuities of the nonlinearity only on a set of measure zero. We give an example of anonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition isobtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest.Applications of the theorem to problems with a parameter are considered. The existence ofnontrivial (including semiregular) solutions of the problem with a parameter for an ellipticequation with a discontinuous nonlinearity for all positive values of the parameter is established.

KW - NON-DIFFERENTIABLE FUNCTIONALS

KW - EQUATIONS

UR - http://www.scopus.com/inward/record.url?scp=85089084220&partnerID=8YFLogxK

U2 - 10.1134/S0012266120070034

DO - 10.1134/S0012266120070034

M3 - Article

AN - SCOPUS:85089084220

VL - 56

SP - 831

EP - 841

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 7

ER -

ID: 61342044