Eigenfunction convergence for all fixed ratios

Prove that, for every fixed ratio \(\alpha>1\), the first radial \((k,\alpha k)\)-eigenfunctions \(u_{k,\alpha k}\) converge to the limiting function \(v_\alpha\) as \(k\to\infty\), extending the established result from \(1<\alpha<2e\).

Background

The paper proves convergence of the first radial (k,αk)(k,\alpha k)-eigenfunctions to the explicitly defined function vαv_\alpha only when 1<α<2e1<\alpha<2e. Numerical experiments are reported as evidence that the same convergence statement should remain valid for every α>1\alpha>1, but the paper does not establish this extension.

References

The numerical data suggests the following two statements should be true: Theorem \ref{eigenfunction theorem intro} should hold for all $\alpha 1$.

— Asymptotics for the $k$-Hessian Eigenvalue on the Unit Ball  (2609.05277 - McCleerey et al., 4 Sep 2026) in Section 7, Numerical Results and Further Questions