A note on spherical maxima sharing the same Lagrange multiplier

In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let X be a real Hilbert space. For each r > 0, let S-r = {x is an element of X : parallel to x parallel to(2) = r}. Let J : X -> R be a sequentially weakly upper semicontinuous functional which is G teaux differentiable in X \ {0}. Assume that lim sup(x -> 0)J(x)parallel to x parallel to(2) = +infinity. Then, for each rho > 0, there exists an open interval I subset of]0,+infinity[and an increasing function phi : I ->] 0, rho[such that, for each lambda is an element of I, one has phi not equal {x is an element of S-phi(lambda) : J(x) = sup(S phi(lambda)) J} subset of {x is an element of X : x = lambda J'(x)}.