THE STRUCTURE OF RABINOWITZ GLOBAL BIFURCATING CONTINUA FOR PROBLEMS WITH WEAK NONLINEARITIES

Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenva lue (or characteristic value) of odd multiplicity of the linearized pr oblem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper consider s a class of such nonlinear eigenvalue problems having simple eigenval ues and a ''weak'' nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further diffe rentiability conditions it is shown that the continua are smooth, 1-di mensional curves and that there are no non-trivial solutions of the eq uation other than those lying on the bifurcating continua.