A Bifurcation Theorem for Potential Operators with Infinite Dimensional Kernel

In this paper we discuss the abstract equation F(λ, x) = ƒ′(x) − λx = θ, F(λ, θ) = θ in a Hilbert space. Assuming the potential function ƒ of ƒ′ satisfies the Palais and Smale condition, we prove that if λ0 is an isolated eigenvalue of ƒ′′(θ), then (λ0, θ) is a bifurcation point of the abstract equation. The main tool is the Mountain Pass Lemma and its modified form.