Euler-Lagrange Inclusions and Existence of Minimizers for a Class of Non-Coercive Variational Problems

We prove the existence of radially symmetric minimizers, in the class of Sobolev vector-valued functions vanishing on the boundary of a ball, for convex non-coercive integral functionals. We associate to the functional a system of differential inclusions of Euler-Lagrange type, and we prove that the solvability of these inclusions is a necessary and sufficient condition for the existence of a radially symmetric minimizer.