A note on a question of Ricceri

Recently, Ricceri posed a question about the uniqueness of a parameter λ ∗ such that the functional x ↦ 1 2 ‖ x ‖ 2 + λ ∗ J ( x ) admits at least three critical points in a finite-dimensional Hilbert space X (under certain assumptions on the functional J ∈ C 1 ( X ) ). We give the answer, which is negative, even in the case of infinite-dimensional spaces. Moreover, we discuss the structure of the set of λ s such that the above situation occurs.