On Radially Symmetric Minima of Nonconvex Functionals

Abstract

We study a minimization problem in the space W1,10(BR) where BR is the ball of radius R with center at the origin; the functional considered is not necessarily convex. Under suitable assumptions, we prove the existence of a radially symmetric (decreasing) solution. By strengthening the assumptions we obtain uniqueness results. Finally, we study under which assumptions and in which sense the solutions found solve the corresponding Euler equation. The proofs are very direct and simple: they only make use of the functions T±n introduced by the author [Arch. Rational Mech. Anal., 1999]. © 2001 Academic Press.