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

We are concerned with integral functionals of the form \[ J(v)\doteq \int_{B_R^n} \pq{f\pt{\mod{x},\mod{\nabla v(x)}}+h(|x|,v(x))}\,dx,~~~~~~~~~{ } \] defined on $\Wuu(B_R^n, \R^m)$, 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 convex, but the compactness of the sublevels of $J$ is not required. We prove that, under suitable assumptions on $f$ and $h$, there exists a radially symmetric minimizer $v\in\Wuu(B_R, \R^m)$ for $J$. Moreover, we associate to the functional $J$ a system of differential inclusions of the 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 for $J$.