On a Nonlinear Eigenvalue Problem Related to the Theory of Propagation of Electromagnetic Waves

The eigenvalue problem is studied for a quasilinear second-order ordinary differential equation on a closed interval with Dirichlet's boundary conditions (the corresponding linear problem has an infinite number of negative and no positive eigenvalues). An additional (local) condition imposed at one of the endpoints of the closed interval is used to determine discrete eigenvalues. The existence of an infinite number of (isolated) positive and negative eigenvalues is proved; their asymptotics is specified; a condition for the eigenfunctions to be periodic is established; the period is calculated; and an explicit formula for eigenfunction zeroes is provided. Several comparison theorems are obtained. It is shown that the nonlinear problem cannot be studied comprehensively with perturbation theory methods.