Existence of positive radial solutions for general boundary parameters

Determine whether positive radial solutions exist for the fourth-order boundary value problem on the unit ball with prescribed boundary value, prescribed nonlinear B^3_i-condition, and prescribed nonlinear B^3_3-condition for general boundary parameters, beyond the small-boundary-volume regimes established in the paper.

Background

The classification theorem reduces finite-volume solutions of the conformally invariant fourth-order problem on the upper half-space to positive radial solutions of a fourth-order boundary value problem on the unit ball. The resulting radial problem involves a nonlinear interior equation and nonlinear boundary conditions associated with the conformal boundary operators B3_i and B3_3.

The paper proves only partial existence results: for i=1, existence under a small-boundary-volume condition, and for i=2, existence for sufficiently small boundary values, corresponding to sufficiently large values of the third-order boundary-curvature parameter. The authors indicate that existence outside these regimes, as well as uniqueness and explicit descriptions in general—particularly when c_1 is nonzero—remains difficult and is not settled by the results presented.

References

The next natural question is therefore whether such positive radial solutions actually exist. In general, the uniqueness and explicit description of solutions to BVP_U-1 appear to be difficult, especially when $c_1\ne0$.

— Conformal Metrics on the unit Ball with Constant $Q$-Curvature, Constant $T$-Curvature, and Minimal Boundary  (2608.23106 - Sun et al., 24 Aug 2026) in Section 1, subsection “Main results” (discussion immediately preceding Theorem 1.2/Theorem \ref{thm:Existence})