On the existence and uniqueness of minima and maxima on spheres of the   integral functional of the calculus of variations
  We deal with the integral functional of the calculus of variations assuming that the gradient of the integrand is Lipschitzian. We then prove that if this gradient does not vanish at zero, then the functional has a unique minimum and a unique maximum on each sphere, centered at zero, with radius small enough. 
Biagio Ricceri