Existence of ring solutions in higher dimensions for the Swift–Hohenberg equation

Establish the existence of ring‑type axisymmetric localized steady states bifurcating from the Turing point μ = 0 in the n‑dimensional Swift–Hohenberg equation for n > 2, or determine conditions precluding such bifurcations.

Background

For n=2, ring solutions bifurcating from the Turing point have been constructed using radial spatial dynamics and computer‑assisted proofs for a related Ginzburg–Landau reduction.

For n>2, only spot‑type solutions are established; whether analogous ring states exist at onset remains unresolved.