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

The paper considers the nonlinear eigenvalue problem for the equation y(x)=-|y(x)|2qy(x) with boundary conditions y(0)=y(h)=0 and y(0)=p, where , q, and p are positive constants, 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 =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 +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.