Supercritical elliptic problems in exterior domains

I shall discuss some results, old and new, about solvability of elliptic boundary value problems with a power non-linearity. Of particular interest is the case when the power lies above Sobolev’s critical exponent, where variational methods do not apply, and strong obstructions for solvability are present. I will concentrate on a new existence result for the classical Lane-Emden-Fowler equation in a exterior domain.