Concentric bubble-tree blow-up in the three-dimensional energy-critical NLS

Construct concentric bubble-tree solutions with two or more concentrating bubbles for the focusing energy-critical nonlinear Schrödinger equation in dimensions d≥3.

Background

The paper distinguishes single-bubble type II blow-up from more complicated bubble-tree dynamics. Although pure two-bubble solutions with one concentrating bubble have been constructed in higher dimensions, the existence of concentric bubble trees involving two or more concentrating bubbles is not established in any dimension d≥3. This remains an unresolved construction problem beyond the single-bubble regime studied in the paper.

References

However, the existence of concentric bubble trees with two or more concentrating bubbles remains open for all $d\ge3$.

— Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation  (2610.26127 - Kim, 19 Oct 2026) in Section 1, Introduction