Geometric significance of the critical GJMS Kirchhoff–Boussinesq equation

Determine whether the Kirchhoff–Boussinesq equation involving the GJMS operator mathscr{P}_g^m u pm (Delta_g)_p u=f_q(u)+|u|^{r-2}u on a closed Einstein manifold of positive scalar curvature has geometric significance when the nonlinearity has the critical Sobolev exponent r=2_{m,d}^*.

Background

The paper applies its multiplicity theorem to equations involving the higher-order conformal GJMS operator on closed Einstein manifolds with positive scalar curvature. Although these operators are important in conformal geometry, the authors explicitly distinguish the analytical existence result from the unresolved question of whether the resulting Kirchhoff–Boussinesq equation has an underlying geometric interpretation in the critical Sobolev regime.

The question concerns the equation with the critical power r=2_{m,d}^*, rather than the subcritical cases for which the paper establishes ground states and infinitely many equivariant solutions. No geometric interpretation is established in the paper.