  In this note we prove the instability by blow-up of the ground state
solutions for a class of fourth order Schr\" odinger equations. This extends
the first rigorous results on blowing-up solutions for the biharmonic NLS due
to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from
\cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.
