Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems

We are concerned with integral functionals of the form J(v) = integral (n)(BR) [f(\x\, \delv(x)\) + h(\x\, v(x))] dx, defined on W-0(1,1)(B-R(n), R-m), where B-R(n) is the ball of R-n centered at the origin and with radius R > 0. We assume that the functional J is con vex, but the compactness of the sublevels of J is not required. We Drove th at, under suitable assumptions on J and h, there exists a radially symmetri c minimizer v is an element of W-0(1,1)(B-R, R-m) for J. Moreover, we assoc iate to the functional J a system of differential inclusions of the Euler-L agrange type, and we prove that the solvability of these inclusions is a ne cessary and sufficient condition for the existence of a radially symmetric minimizer for J.