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 Izv. RAN. Ser. Mat., 2020, Volume 84, Issue 3, Pages 168–184 (Mi izv8847)

On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity

V. N. Pavlenkoa, D. K. Potapovb*

a Chelyabinsk State University
b Saint Petersburg State University

Abstract: We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let $\widetilde{u}(x)$ be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting $v(x)=u(x)-\widetilde{u}(x)$, we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function $v(x)=0$ is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray $[\lambda^*,+\infty)$, where $\lambda^*>0$. As an application, we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.

Keywords: elliptic boundary-value problem, problem with parameter, discontinuous non-linearity, iterative process, minimal solution, semiregular solution, spectrum, Gol'dshtik model.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/im8847

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English version:
Izvestiya: Mathematics, 2020, 84:3, 592–607

Bibliographic databases:

UDC: 517.95
PACS: N/A
MSC: 35J25, 35J60, 35P30
Revised: 25.06.2019

Citation: V. N. Pavlenko, D. K. Potapov, “On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity”, Izv. RAN. Ser. Mat., 84:3 (2020), 168–184; Izv. Math., 84:3 (2020), 592–607

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8847
• https://doi.org/10.4213/im8847
• http://mi.mathnet.ru/eng/izv/v84/i3/p168

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This publication is cited in the following articles:
1. V. N. Pavlenko, D. K. Potapov, “Positive solutions of superlinear elliptic problems with discontinuous non-linearities”, Izv. Math., 85:2 (2021), 262–278
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