Are there scaling solutions in the O(N)-models for large N in d > 4?

There have been some speculations about the existence of critical unitary O(N)-invariant scalar field theories in dimensions 4 < d < 6 and for large N. Using the functional renormalization group equation, we show that in the lowest order of the derivative expansion and assuming that the anomalous dimension vanishes for large N, the corresponding critical potentials are either unbounded from below or singular for some finite value of the field.