Uniqueness properties of functionals with Lipschitzian derivative

Let X be a real Hilbert space and J a C functional on X . For x0 ∈ X , r > 0, set S(x0, r) = {x ∈ X : ‖x− x0‖ = r}. Also on the basis of the beautiful theory developed and applied by Schechter and Tintarev in [2], [3], [4] and [5], it is of particular interest to know when the restriction of J to S(0, r) has a unique maximum. The aim of the present paper is to offer a contribution along this direction. We show that such a uniqueness property holds (for suitable r) provided that J ′ is Lipschitzian and J (0) 6= 0. At the same time, we also show that (for suitable s) the set J(s) has a unique element of minimal norm. After proving the general result (Theorem 1), we present an application to a semilinear Dirichlet problem involving a Lipschitzian nonlinearity (Theorem 2).