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

We study the large-time behavior of solutions to the nonlinear exterior problem $\mathcal{L}u(t,x)={\kappa |u(t,x)|}^p,(t,x)\in (0,\infty )\times D^c$ under the nonhomegeneous Neumann boundary condition $\frac{\partial u}{\partial \nu }(t,x)=\lambda (x),(t,x)\in (0,\infty )\times \partial D,$ where $\mathcal{L}:=i{\partial }_t+\Delta$ is the Schrödinger operator, $D=B(0,1)$ is the open unit ball in ${\mathbb{R}}^N$ , $N\ge 2$ , $D^c={\mathbb{R}}^N\setminus D$ , $p>1$ , $\kappa \in \mathbb{C}$ , $\kappa \ne 0$ , $\lambda \in L^1(\partial D,\mathbb{C})$ is a nontrivial complex valued function, and $\partial \nu$ is the outward unit normal vector on $\partial D$ , relative to $D^c$ . Namely, under a certain condition imposed on $(\kappa ,\lambda )$ , we show that if $N\ge 3$ and $p<p_c$ , where $p_c=\frac{N}{N-2},$ 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. View Full-Text