Non-degeneracy of the constructed planar doubling nodal solutions

Determine whether the two families of finite-energy sign-changing solutions u0 and u1 to the critical Yamabe equation −Δu = γ |u|^{p−1} u in R^n, each concentrating along two concentric circles in the x1x2-plane (the planar doubling configurations defined by the ansatz u_{*,m} with m ∈ {0,1}), are non-degenerate; that is, ascertain whether the kernel of the linearized operator L_{u_m} = −Δ − γ p |u_m|^{p−2} u_m coincides with the tangent space T_{u_m}(Orb_{u_m}(Mob(n))) generated by Möbius symmetries.

Background

The paper constructs two new families of nodal solutions to the Yamabe equation in R^n that concentrate along two planar circles (“planar doubling” of the equator). One family (m = 0) is Kelvin invariant and the other (m = 1) is a twisted variant that is not Kelvin invariant. In dimension 3, these solutions attain maximal rank. This advances prior constructions such as the crown solutions and the doubling solutions arranged on prisms or grids.

A central structural question for such solutions is non-degeneracy, meaning that the kernel of the linearized operator at the solution equals the space generated by the Möbius symmetries. Non-degeneracy is crucial for stability analyses and for applications in dynamical problems (e.g., in the sense of Duyckaerts–Kenig–Merle). While non-degeneracy of certain earlier constructions (e.g., the crown solution) is known, whether the new planar doubling families u0 and u1 are non-degenerate remains unresolved in this work.