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

V. N. Pavlenko ; D. K. Potapov

Sbornik. Mathematics, Tome 210 (2019) no. 7, pp. 1043-1066

Résumé

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.

Détail

Keywords: spectrum, elliptic boundary value problem, parameter, discontinuous nonlinearity, semiregular solution.

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