In this talk, we will consider the issues of non-existence of solutions to a Yamabe type equation on bounded Euclidean domains (dim>2). The leading order terms of this equation are invariant under conformal transformations which leads to the classical Pohozaev identity. This in turn gives non-existence of solutions to the PDE when the domain is star-shaped with respect to the origin.
We show that this non-existence is surprisingly stable under perturbations, which includes situations not covered by the Pohozaev obstruction, if the boundary of the domain has a positive curvature. In particular, we show that there are no positive variational solutions to our PDE under $C^1$-perturbations of the potential when the domain is star-shaped with respect to the origin and the mean curvature of the boundary at the origin is positive. The proof of our result relies on sharp blow-up analysis. This is a joint work with Nassif Ghoussoub (UBC, Vancouver) and Frédéric Robert (Institut Élie Cartan, Nancy).