  We study optimal functions in a family of Caffarelli-Kohn-Nirenberg
inequalities with a power-law weight, in a regime for which standard
symmetrization techniques fail. We establish the existence of optimal
functions, study their properties and prove that they are radial when the power
in the weight is small enough. Radial symmetry up to translations is true for
the limiting case where the weight vanishes, a case which corresponds to a
well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based
on a concentration-compactness analysis and on a perturbation method which uses
a spectral gap inequality. As a consequence, we prove that optimal functions
are explicit and given by Barenblatt-type profiles in the perturbative regime.
