TY - JOUR

T1 - On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity

AU - Pavlenko, V. N.

AU - Potapov, D. K.

PY - 2020/6

Y1 - 2020/6

N2 - 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. Letbe a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting, 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 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, where . 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.

AB - 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. Letbe a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting, 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 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, where . 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.

KW - elliptic boundary-value problem

KW - problem with parameter

KW - discontinuous non-linearity

KW - iterative process

KW - minimal solution

KW - semiregular solution

KW - spectrum

KW - Gol'dshtik model

KW - EQUATIONS

KW - EXISTENCE

UR - http://www.scopus.com/inward/record.url?scp=85090914101&partnerID=8YFLogxK

U2 - 10.1070/IM8847

DO - 10.1070/IM8847

M3 - Article

AN - SCOPUS:85090914101

VL - 84

SP - 592

EP - 607

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 3

ER -