  We establish linear profile decompositions for the fourth order Schr\"odinger
equation and for certain fourth order perturbations of the Schr\"odinger
equation, in dimensions greater than or equal to two. We apply these results to
prove dichotomy results on the existence of extremizers for the associated
Stein--Tomas/Strichartz inequalities; along the way, we also obtain lower
bounds for the norms of these operators.
