LAUSR

We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = κ | u ( t , x ) | p , ( t , x ) ∈ ( 0 , ∞ ) × D c under the nonhomegeneous Neumann boundary condition ∂ u ∂ ν ( t , x ) = λ ( x ) , ( t , x ) ∈ ( 0 , ∞ ) × ∂ D , where L : = i ∂ t + Δ is the Schrödinger operator, D = B ( 0 , 1 ) is the open unit ball in R N , N ≥ 2 , D c = R N ∖ D , p > 1 , κ ∈ C , κ ≠ 0 , λ ∈ L 1 ( ∂ D , C ) is a nontrivial complex valued function, and ∂ ν is the outward unit normal vector on ∂ D , relative to D c . Namely, under a certain condition imposed on ( κ , λ ) , 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.