Construction of 2-bubbles for the energy critical bi-harmonic Schrödinger equation

TL;DR

This work constructs, for the energy-critical focusing biharmonic NLS in $\mathbb{R}^N$ with $N\ge 13$, a global radially symmetric solution that decomposes into two bubbles as time goes to $-\infty$: one bubble remains at scale $1$ while the other decays like $|t|^{-2/(N-12)}$, with the two phases in a right-angle configuration. The authors implement a precise modulation analysis around a two-bubble ansatz, derive sharp evolution laws for the scales, and exploit energy coercivity near bubbles together with a localized virial correction to control the phase. A Brouwer fixed-point argument handles linear instabilities, enabling a bootstrap that yields a global radial two-bubble solution and a weak limit converging to $-iW+W_{\tilde C|t|^{-2/(N-12)}}$ as $t\to -\infty$. This constitutes the first constructive blow-up result of this type for the energy-critical 4NLS and emphasizes potential connections to soliton resolution phenomena for higher-order Schrödinger equations.

Abstract

We construct a blowing-up solution for the energy critical focusing biharmonic nonlinear Schrödinger equation in infinite time in dimension $N\geq 13$. Our solution is radially symmetric and converges asymptotically to the sum of two bubbles. The scale of one of the bubble is of order $1$ whereas the other one is of order $|t|^{-\frac{2}{N-12}}$. Moreover, the phase between the two bubbles form a right angle.

Theorems & Definitions (33)

• Theorem 1.1
• Lemma 2.1
• Proposition 3.1
• proof
• Lemma 3.1
• proof
• Proposition 4.1
• Lemma 4.1
• proof
• Lemma 4.2