  We consider the problem of minimizing variational integrals defined on
\cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The
main novelty is that the underlying energy features a non-uniformly elliptic
integrand exhibiting different polynomial growth conditions and no homogeneity.
We develop a few intrinsic methods aimed at proving partial regularity of
minima and providing techniques for treating larger classes of similar
constrained non-uniformly elliptic variational problems. In order to give
estimates for the singular sets we use a general family of Hausdorff type
measures following the local geometry of the integrand. A suitable comparison
is provided with respect to the naturally associated capacities.
