Construction of two-bubble solutions for the energy-critical NLS

Abstract

We construct pure two-bubbles for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N\ge 7$. The constructed solution is global in (at least) one time direction and approaches a superposition of two stationary states both centered at the origin, with the ratio of their length scales converging to $0$. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t{|}^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle.