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On spectral properties of the Sturm-Liouville operator with power nonlinearity. (English) Zbl 1420.34099

The eigenvalue problem for the equation \[y''=(\lambda-\alpha |y|^{2q})y\] with boundary conditions \(y(0)=0=y(h)\) where \(y'(0)=p\) and \(\alpha, q, p>0\) is considered for real \(\lambda\). It is shown that there are infinitely many isolated negative as well as infinitely many isolated positive eigenvalues. Asymptotic approximations for the eigenvalues are given, periodicity of the eigenfunctions is proved and the zeros and periods of the eigenfunctions determined.

MSC:

34L05 General spectral theory of ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B24 Sturm-Liouville theory
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