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

TL;DR

This work constructs a pure two-bubble solution to the focusing energy-critical Hartree equation in dimension $N\ge 7$, yielding a global radial solution that asymptotically behaves as a superposition of two ground states with decoupled scales and a right-angle phasing. The authors adapt modulation analysis, a bootstrap framework, and a topological shooting argument to handle the intrinsic nonlocality of the Hartree nonlinearity, proving a precise energy decomposition with $E(u)=2E(W)$ and a two-bubble configuration with $\lambda(t)\sim |t|^{-2/(N-6)}$ tending to zero. A key technical novelty is the detailed coercivity and spectral analysis of the linearized and interaction operators in the nonlocal setting, together with a virial-type correction that enables closing the bootstrap and controlling the phase. The results advance understanding of multi-soliton dynamics for nonlocal dispersive equations and lay groundwork for soliton-resolution-type questions in energy-critical Hartree dynamics.

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.

Theorems & Definitions (42)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1: LLTX:Nondegeneracy
• Lemma 2.2
• Lemma 2.3
• proof