Construction of two-bubble solutions for the energy-critical Hartree equation

We construct a pure two-bubble solution for the focusing, energy-critical Hartree equation in space dimension $N \geq 7$. The constructed solution is spherically symmetric, global in (at least) the negative time direction and asymptotically behaves as a superposition of two ground states (or bubbles) both centered at the origin, with the ratio of their length scales converging to $0$ and the phases of the two bubbles form the right angle. The main arguments are the modulation analysis, the bootstrap argument and the topological argument. The main novelty with respect to existing constructions of pure two-bubble solutions is the nonlocal interaction, which is more complex to analyze.