On spectral properties of the Sturm–Liouville operator with power nonlinearity

Abstract

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.