The Choquard equation describes particles interacting through a nonlocal potential — each point in the solution is influenced not just by its neighbors but by the entire domain through a convolution kernel. Below a critical exponent, the equation admits single-bubble solutions: concentrated, radially symmetric peaks that approximate a delta function as a parameter approaches zero. These bubbles are the building blocks of more complex solution families.
Gao (arXiv:2603.24100) proves that crossing the critical exponent from below to above destroys single-bubble solutions entirely. For the slightly supercritical Choquard equation, there is no family of positive solutions concentrating at a single point as the supercritical parameter approaches zero.
The sharpness is what matters. Below the threshold, bubbles exist. Above it, they don't. The transition isn't gradual — bubbles don't become harder to construct or less stable as you approach criticality from above. They simply cease to exist. The critical exponent acts as a topological wall, not a performance gradient.
The contrast with the subcritical case is precise: Chen and Wang (2024) constructed single-bubble solutions for the slightly subcritical Choquard equation using standard concentration-compactness methods. The same methods fail above the critical exponent because the nonlocal interaction changes character. Below criticality, the convolution kernel allows concentrated solutions to balance their own interaction energy. Above criticality, the interaction grows too strong — any attempt to concentrate the solution amplifies the nonlocal term faster than the local terms can compensate.
For nonlinear analysis, the result maps the boundary of a solution family's existence. Single-bubble solutions are not merely difficult in the supercritical regime — they are provably absent. Whatever solutions exist above the critical exponent, they must have fundamentally different structure: multi-peaked, delocalized, or otherwise non-concentrating.