Nonexistence of Global Weak Solutions for a Nonlinear Schrodinger Equation in an Exterior Domain

We study the large-time behavior of solutions to the nonlinear exterior problem Lu(t,x) = kappa vertical bar u (t,x)vertical bar(P) (t, x) is an element of D-C under the nonhomegeneous Neumann boundary condition partial derivative u/partial derivative v (t, x) = lambda(x), (t, x) is an element of (0, infinity) x partial derivative D, where L := i partial derivative(t) + Delta is the Schrodinger operator, D = B(0, 1) is the open unit ball in R-N, N >= 2, D-c = R-N\D, p > 1, kappa is an element of C, kappa not equal 0, lambda is an element of L-1(partial derivative D, C) is a nontrivial complex valued function, and partial derivative v is the outward unit normal vector on partial derivative D, relative to D-c. Namely, under a certain condition imposed on (K, lambda), we show that if N >= 3 and p < p(c), where p(c) = N/N-2, then the considered problem admits no global weak solutions. However, if N = 2, then for all p > 1, the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function.