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Sbornik: Mathematics, 2019, Volume 210, Issue 7, Pages 1043–1066
DOI: https://doi.org/10.1070/SM9117
(Mi sm9117)
 

This article is cited in 4 scientific papers (total in 4 papers)

Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity

V. N. Pavlenkoa, D. K. Potapovb

a Chelyabinsk State University, Chelyabinsk, Russia
b Saint Petersburg State University, St. Petersburg, Russia
References:
Abstract: An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter $\lambda$ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum $\sigma$ of the nonlinear problem under consideration is closed ($\sigma$ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, $\lambda^*$. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that $\sigma=[\lambda^*,+\infty)$ and that for each $\lambda\in\sigma$ the problem has a nontrivial semiregular solution. If there exists a positive constant $M$ such that the sum of the nonlinearity and $Mu$ is a function which is nondecreasing in the phase variable $u$, then for any $\lambda\in\sigma$ the boundary value problem has a minimal nontrivial solution $u_\lambda(x)$. The required solution is semiregular, and $u_\lambda(x)$ is a decreasing mapping with respect to $\lambda$ on $[\lambda^*,+\infty)$. Applications of the results to the Gol'dshtik mathematical model for separated flows in an incompressible fluid are considered.
Bibliography: 37 titles.
Keywords: spectrum, elliptic boundary value problem, parameter, discontinuous nonlinearity, semiregular solution.
Received: 02.04.2018 and 01.06.2018
Bibliographic databases:
Document Type: Article
UDC: 517.95
MSC: Primary 35J65; Secondary 35R05
Language: English
Original paper language: Russian
Citation: 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
Citation in format AMSBIB
\Bibitem{PavPot19}
\by V.~N.~Pavlenko, D.~K.~Potapov
\paper Properties of the spectrum of an elliptic boundary value problem with a~parameter and a~discontinuous nonlinearity
\jour Sb. Math.
\yr 2019
\vol 210
\issue 7
\pages 1043--1066
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\crossref{https://doi.org/10.1070/SM9117}
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  • https://www.mathnet.ru/eng/sm9117
  • https://doi.org/10.1070/SM9117
  • https://www.mathnet.ru/eng/sm/v210/i7/p145
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    References:81
    First page:21
     
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