Multiplicity and eigenvalue intersecting nonlinearities

Summary Several results on the existence of multiple solutions are proved for nonlinear Sturm-Liouville equations with nonlinearities which cross asymptotically some eigenvalues of the linear operator (so-called jumping nonlinearities).By a change of variable the problem is transformed into a bifurcation equation. The bifurcation branches of this equation are seen to start in linear eigenvalues and to end asymptotically in nonlinear eigenvalues (split eigenvalues). This allows to deduce in a unified way several multiplicity results.