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 f :Rn+1 →R 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 u→ Ω f (u(x),∇u(x)) dx to the sphere {u ∈ H1(Ω): Ω(|∇u(x)|2 + |u(x)|2)dx = r} has a unique global minimum and a unique global maximum. © 2006 Elsevier Inc. All rights reserved