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

TL;DR

This paper constructs a special two-bubble solution for the energy-critical Hartree equation in high dimensions, revealing complex nonlocal interactions and asymptotic behavior of superposed ground states.

Contribution

It introduces the first construction of two-bubble solutions for the energy-critical Hartree equation with nonlocal interactions in high dimensions.

Findings

Solution is spherically symmetric and global in negative time.

The two bubbles' scales converge to zero with phases forming a right angle.

Analysis handles complex nonlocal interactions unique to the Hartree equation.

Abstract

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.