conformal to a given one and which has constant scalar curvature. From
the analytic point of view, this problem becomes a semilinear elliptic
PDE with critical (for the Sobolev embedding) power non-linearity. If we
study the problem in the Euclidean space, allowing the presence of
nonzero-dimensional singularities can be transformed into reducing the
non-linearity to a Sobolev-subcritical power. A quite recent notion of
non-local curvature gives rise to a parallel study which weakens the
geometric assumptions giving rise to a non-local semilinear elliptic
(Sobolev-critical) PDE.
In this talk, we will focus on the Euclidean space in the presence of
singularities of maximal possible dimension. In both cases, local and
non-local, we will construct singular solutions of a local/non-local
semilinear elliptic equation with superlinear nonlinearity which is
subritical for Sobolev embeddings, but it is critical for the existence
of singular solutions (below such threshold the singularity is removable).
This is a joint work with H. Chan.