Geometric uniqueness of positive radial solutions for singular p-Laplacian equations on annuli

We prove uniqueness results for positive radial Dirichlet problems for the p -Laplacian on annuli in the singular range 1 < p < 2 and when p = 2 . For the Lane–Emden equation − Δ p u = u q − 1 , uniqueness holds for every q ≥ 2 with q > p , up to and including the critical Sobolev exponent when n > p . We also show that uniqueness persists under sufficiently small weighted C 1 perturbations of the pure power in the strictly subcritical range. The proof is geometric: varying the initial momentum at the inner boundary produces a shooting curve in the ( u , w ) -plane, and uniqueness follows by proving that this curve meets the outer boundary condition transversely with the same orientation. For the pure power, transversality follows from an exact scaling determinant and a Pohozaev-type trapping estimate. Under perturbation, a quantitative determinant bound preserves the same geometric mechanism.