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.