Approximation of positive solutions of 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. We propose and justify new conditions for the existence of a positive eigenvalue and the corresponding positive eigenfunction. A finite element approximation of the problem preserving the property of positivity of solutions is constructed. The existence and convergence of approximate solutions are established.