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 ⊂ R, we prove that if f : R → R is a C function whose gradient is Lipschitzian in R and non-zero at 0, then, for each r > 0 small enough, the restriction of the integral functional u → ∫ Ω f(u(x),∇u(x))dx to the sphere {u ∈ H(Ω) : ∫ Ω (|∇u(x)| + |u(x)|)dx = r} has a unique global minimum and a unique global maximum.