COMPUTATIONAL METHODS FOR NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS

A basic problem is that of finding nontrivial solutions of the following nonlinear eigenvalue problem: $-\Delta$u = $\lambda$F$\prime$ (u) in D, u = 0 on $\partial$D, where D is a domain in R$\sp{\rm N}$, N = 1,2,3, and F$\prime$(u) is non-monotone. Problems of this kind arise, for example, in plasma physics, fluid dynamics, and astrophysics. In the first part of this thesis, an equivalent variational formulation is used to obtain an iterative procedure for solving the problem with a general function F(u). The global convergence of this procedure is established, i.e., convergence from any initial guess. The method is applied to a test problem with F(u) = $-$cos u. In the last part of this thesis, a problem of internal solitary waves in stratified fluids is studied. Attention is paid to establishing the range of validity of existing asymptotic theories. New large amplitude solutions are obtained numerically