We prove uniqueness results for positive radial Dirichlet problems for the p-Laplacian on annuli in the singular range 1 p, up to and including the critical Sobolev exponent when n>p. We also show that uniqueness persists under sufficiently small weighted C1 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.
