The boundary value problem
Δu + λu − u3= g on Ω, u|∂Ω = 0, where Ω ⊂Rn(n ≤ 4) is a bounded domain,
defines a real analytic map Aλ of the Sobolev space H = W01,2(Ω) onto itself. A
point u ∈ H is a fold point if Aλ at u is C∞ equivalent to f ×id: R × E → R × E,
where f(t) = t2. (1) There is a closed subset Γλ⊂ H such that (a) at each point of
Aλ−1(H − Γλ) the map Aλ is either locally a diffeomorphism or a fold, and (b) for
each nonempty connected open subset V ⊂ H, V − Γλ is nonempty and connected;
thus Γλ is nowhere dense in H and does not locally separate H. Suppose
that n ≤ 3 and the second eigenvalue λ2 of −Δu on Ω with u|∂Ω = 0 is
simple. Define A : H × R → H × R by A(u,λ) = (Aλ(u),λ). (2) There is a
connected open neighborhood V of (0,λ2) in H × R such that A−1(V ) has three
components U0, U1, U2 with A : Ui→ V a diffeomorphism for i = 1,2 and
A|U0: U0→ V C∞ equivalent to w ×id: R2× E → R2× E defined by
(w ×id)(t,λ,ν) = (t3− λt,λ,ν).
