Perturbation of nonlinear operators in the theory of nonlinear multifrequency electromagnetic wave propagation

Abstract

The paper develops an original approach to study nonlinear multiparameter eigenvalue problems arising in the theory of nonlinear multifrequency electromagnetic wave propagation. The problem under consideration is a multiparameter eigenvalue problem that under some conditions degenerates into n nonlinear one-parameter eigenvalue problems. Further simplification reduces the one-parameter nonlinear problems to linear (one-parameter) eigenvalue problems. Each of the linear problems has a finite number of positive eigenvalues, whereas each of the nonlinear (one-parameter) problems has an infinite number of positive eigenvalues. Using the nonlinear one-parameter problems as ’nonperturbed’ ones, one can prove existence of eigentuples of the multiparameter problem that have no connections with solutions to the linear (one-parameter) problems even if the nonlinear terms have small factors.