Existence, uniqueness and qualitative properties of minima to radially symmetric noncoercive nonconvex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem \[ \min_{u\in\Wuu(B_R)}\int_{B_R} \pq{f\pt{\mod{x},\mod{\nabla u(x)}}+h(|x|,u(x))}\,dx, ~~~~~~~~~{ } \] where $B_R$ is the ball of $\R^n$ centered at the origin and with radius $R>0$, the map $f\colon [0,R]\times[0,+\infty[\to\Re$ is a normal integrand, and $h\colon[0,R]\times\R\to\R$ is a convex function of the second variable. %Neither convexity nor growth conditions are made on $f$. %In particular, $f$ is allowed to grow sub-linearly at infinity. This kind of problems, with non-convex lagrangians with respect to $\nabla u$, arise in various fields of applied sciences, such as optimal design and nonlinear elasticity.