CONSTRUCTION OF TWO-BUBBLE SOLUTIONS FOR THE ENERGY-CRITICAL NLS

We construct pure two-bubbles for the energy-critical focusing nonlinear Schrodinger equation in space dimension N >= 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 vertical bar t vertical bar-2/N-6. The phases of the two bubbles form the right angle.