Existence of ring and non‑axisymmetric fully localized patterns in 3D Swift–Hohenberg

Prove the existence of ring‑type axisymmetric localized steady states and fully localized steady states with non‑trivial angular dependence for the three‑dimensional Swift–Hohenberg equation.

Background

For the 3D Swift–Hohenberg equation, axisymmetric spot solutions at onset are known, but there are no existence proofs for ring solutions or for fully localized non‑axisymmetric structures.

Such patterns are relevant for soft‑matter crystals and potential 3D generalizations of planar localized patches.