Fourth-order nonlinear Schrödinger equations in an exterior domain

We discuss necessary conditions on the power p>1 for the existence of weak solutions to the fourth-order nonlinear Schrödinger equation (Formula presented.). The equation is studied in an exterior domain (Formula presented.), where (Formula presented.) and (Formula presented.) is the unit closed ball. We show that no weak solution may exist for critical and subcritical powers p. In the first scenario, the critical power depends on an integral sign condition on the boundary conditions (Formula presented.) or (Formula presented.), for (Formula presented.). Moreover, the critical power is sharp, in the sense that (stationary) solutions may be constructed for suitable data for any supercritical power. In the second scenario, the critical power depends on an integral sign condition on the initial value (Formula presented.) but it becomes larger if a stronger sign condition on h is assumed as (Formula presented.).