  Using variational methods together with symmetries given by singular
Riemannian foliations with positive dimensional leaves, we prove the existence
of an infinite number of sign-changing solutions to Yamabe type problems, which
are constant along the leaves of the foliation, and one positive solution of
minimal energy among any other solution with these symmetries. In particular,
we find sign-changing solutions to the Yamabe problem on the round sphere with
new qualitative behavior when compared to previous results, that is, these
solutions are constant along the leaves of a singular Riemannian foliation
which is not induced neither by a group action nor by an isoparametric
function. To prove the existence of these solutions, we prove a Sobolev
embedding theorem for general singular Riemannian foliations, and a Principle
of Symmetric Criticality for the associated energy functional to a Yamabe type
problem.
