Fourth-order nonlinear Schrodinger 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 Schrodinger equation iu(t)+gamma Delta(2)u=|u|p. The equation is studied in an exterior domain(t, x) is an element of [0,infinity) x B (c), where B (c)=R-N\ B and Bis 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 u=f(x) or Delta u=g(x), for x is an element of partial derivative B. Moreover, the critical power is sharp, in the sense that (stationary) solutions may be constructed for suit-able data for any supercritical power. In the second scenario, the critical power depends on an integral sign condition on the initial value u(0,x)= h(x)but it becomes larger if a stronger sign condition on his assumed as |x|->infinity.