On spectral properties of the Sturm–Liouville operator with power nonlinearity
| Autor: | D. V. Valovik |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Comparison theorem
Pure mathematics 010505 oceanography General Mathematics Operator (physics) 010102 general mathematics Mathematical analysis Sturm–Liouville theory Mathematics::Spectral Theory Eigenfunction 01 natural sciences Prime (order theory) Nonlinear system Boundary value problem 0101 mathematics Eigenvalues and eigenvectors 0105 earth and related environmental sciences Mathematics |
| Zdroj: | Monatshefte für Mathematik. 188:369-385 |
| ISSN: | 1436-5081 0026-9255 |
| DOI: | 10.1007/s00605-017-1124-0 |
| Popis: | The paper considers the nonlinear eigenvalue problem for the equation $$y^{\prime \prime }(x) = \left( \lambda - \alpha |y(x)|^{2q}\right) y(x)$$ with boundary conditions $$y(0) = y(h) = 0$$ and $$y^{\prime }(0) = p$$ , where $$\alpha $$ , q, and p are positive constants, $$\lambda $$ is a real spectral parameter. It is proved that the nonlinear problem has infinitely many isolated negative as well as positive eigenvalues, whereas the corresponding linear problem (for $$\alpha = 0$$ ) has only an infinite number of negative eigenvalues. Negative eigenvalues of the nonlinear problem reduce to the solutions to the corresponding linear problem as $$\alpha \rightarrow +0$$ ; positive ‘nonlinear’ eigenvalues are nonperturbative. Asymptotical inequalities for the eigenvalues are found. Periodicity of the eigenfunctions is proved and the period is found, zeros of the eigenfunctions are determined, and a comparison theorem is proved. |
| Databáze: | OpenAIRE |
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