Poincar\'e-Einstein metrics and Yamabe invariants

In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincar\'e-Einstein metric on the ball $B^8$. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a suitable compactification).