A note on a question of Ricceri

Recently, Ricceri posed a question about the uniqueness of a parameter λ∗λ∗ such that the functional x↦12‖x‖2+λ∗J(x) admits at least three critical points in a finite-dimensional Hilbert space XX (under certain assumptions on the functional J∈C1(X)J∈C1(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 λλs such that the above situation occurs.