Approximation of Positive Solutions to Symmetric Eigenvalue Problems with Nonlinear Dependence on the Spectral Parameter

A symmetric partial differential eigenvalue problem with nonlinear dependence on the spectral parameter arising in plasma physics is studied. New conditions for the existence of a positive eigenvalue and the corresponding positive eigenfunctions are proposed and justified. A finite element approximation for the problem preserving the property of positivity of solutions is constructed. The existence and convergence of approximate solutions are established.