On spectral properties of the Sturm-Liouville operator with power nonlinearity (Q1712584)

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.