On the existence and uniqueness of minima and maxima on spheres of the integral functional of the calculus of variations

Given a bounded domain Ω⊂Rn, we prove that if is a C1 function whose gradient is Lipschitzian in Rn+1 and non-zero at 0, then, for each r>0 small enough, the restriction of the integral functional to the sphere has a unique global minimum and a unique global maximum.