Multiple bifurcation from “simple” eigenvalues

摘要

Consider an operator equation G(u,λ) = 0 where λ is a real parameter. Suppose 0 is a “simple” eigenvalue of the Fréchet derivative Gu at (u0, λ0). We give a hierarchy of conditions which completely determines the solution structure of the operator equation. It will be shown that multiple bifurcation as well as simple bifurcation can occur. This extends the standard bifurcation theory from a simple eigenvalue in which only one branch bifurcates. We also discuss limit point bifurcations. Applications to semilinear elliptic equations and the homotopy method for the matrix eigenvalue problem are also given.