TY - JOUR

T1 - Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity

AU - Pavlenko, V. N.

AU - Potapov, D. K.

PY - 2019/7

Y1 - 2019/7

N2 - An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ 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 σ of the nonlinear problem under consideration is closed (σ 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, λ∗. 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 σ = [λ∗, +∞) and that for each λ ∈ σ 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 λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ∗, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.

AB - An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ 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 σ of the nonlinear problem under consideration is closed (σ 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, λ∗. 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 σ = [λ∗, +∞) and that for each λ ∈ σ 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 λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ∗, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.

KW - Discontinuous nonlinearity

KW - Elliptic boundary value problem

KW - Parameter

KW - Semiregular solution

KW - Spectrum

KW - EXISTENCE

KW - NON-DIFFERENTIABLE FUNCTIONALS

KW - EIGENVALUE PROBLEMS

KW - SEMIREGULAR SOLUTIONS

KW - POSITIVE SOLUTIONS

KW - discontinuous nonlinearity

KW - NONTRIVIAL SOLUTIONS

KW - EQUATIONS

KW - semiregular solution

KW - spectrum

KW - elliptic boundary value problem

KW - parameter

UR - http://www.scopus.com/inward/record.url?scp=85073037520&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/properties-spectrum-elliptic-boundary-value-problem-parameter-discontinuous-nonlinearity

U2 - 10.1070/SM9117

DO - 10.1070/SM9117

M3 - Article

AN - SCOPUS:85073037520

VL - 210

SP - 1043

EP - 1066

JO - Sbornik Mathematics

JF - Sbornik Mathematics

SN - 1064-5616

IS - 7

ER -