Scattering below the ground-state threshold for the focusing energy-critical nonlinear Schrödinger equation

Let $d\geq 3$, let $I$ be the lifespan interval, and let $u:I\times\mathbf{R}^{d}\to\mathbf{C}$ solve the focusing energy-critical nonlinear Schrödinger equation with $p=\frac{4}{d-2}$. Define

$$W(x)=\frac{1}{\left(1+\frac{|x|^{2}}{d(d-2)}\right)^{\frac{d-2}{2}}},$$

where $W\in\dot{H}^{1}(\mathbf{R}^{d})$ is the ground-state solution of $\Delta W+|W|^{\frac{4}{d-2}}W=0$. Scattering-below-threshold conjecture. If

$$\|u_{0}\|_{\dot{H}^{1}(\mathbf{R}^{d})}<\|W\|_{\dot{H}^{1}(\mathbf{R}^{d})}$$

and

$$E(u_{0})<E(W),$$

then

$$\int_{I}\int |u(t,x)|^{\frac{2(d+2)}{d-2}}\,dx\,dt\leq C\bigl(\|u_{0}\|_{\dot{H}^{1}},E(u_{0})\bigr)<\infty.$$

This spacetime bound yields global existence and scattering below the ground-state threshold, while the ground state shows that scattering cannot hold for arbitrary data. The conjecture is presented as the focusing analogue of the corresponding mass-critical result.