| A basic problem is that of finding nontrivial solutions of the following nonlinear eigenvalue problem: {dollar}-Delta{dollar}u = {dollar}lambda{dollar}F{dollar}prime{dollar} (u) in D, u = 0 on {dollar}partial{dollar}D, where D is a domain in R{dollar}sp{lcub}rm N{rcub}{dollar}, N = 1,2,3, and F{dollar}prime{dollar}(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) = {dollar}-{dollar}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. |
