Note on a paper by H. S. Qin

In a recent paper entitled "Some sufficient conditions for occurring bifurcation" [1], H. S. Qin claims to have obtained sufficient conditions for bifurcation from multiple eigenvalues. The setting is that of an equation F(x,/~) = 0, (1) where F: R" x R--, R" is smooth, with F(0,/~) = 0, V# e R. It is assumed that for some Poe R we have rank DxF(O, ~) = n - rn, with 1 l, is misleading and gives a false impression that the results have to do with "higher multiplicities". What happens is that one looks for solutions in a subspace with codimension m - 1, while assuming that: (a) m - 1 of the n scalar equations in equation (1) are automatically satisfied for x in this subspace; (b) zero is a simple eigenvalue of DxF(O, ~o) when the nonrelevant dimensions are taken out of the picture. Condition (a) is especially severe when m > 1, except when it is a consequence of symmetry considerations (e.g. Ref. [3]). To be more precise let us formulate the following "generalized" version of the Crandall- Rabinowitz theorem. Theorem I Let X and Y be Banach spaces, X 0 a closed subspace of X and Y0 a closed subspace of Y. Let F: Y x R --* Y be a smooth mapping such that and