  We prove radial symmetry for bounded nonnegative solutions of a weighted
anisotropic problem. Given the anisotropic setting that we deal with, the term
"radial" is understood in the Finsler framework. In the whole space, J. Serra
obtained the symmetry result in the isotropic unweighted setting. In this case
we provide the extension of his result to the anisotropic setting. This
provides a generalization to the anisotropic setting of a celebrated result due
to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of
linear operators whenever the dimension is greater than 2. In proper cones, the
results presented are new even in the isotropic and unweighted setting for
suitable nonlinear cases. Even for the previously known case of unweighted
isotropic setting, the present paper provides an approach to the problem by
exploiting integral (in)equalities which is new for $N>2$: this complements the
corresponding symmetry result obtained via the moving planes method by
Berestycki-Pacella.
