Sharp concentration estimates near criticality for radial sign-changing solutions of Dirichlet and Neumann problems

We consider radial solutions of the slightly subcritical problem (Formula presented.) either on (Formula presented.) ((Formula presented.)) or in a ball (Formula presented.) satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as (Formula presented.)) of all local minima and maxima of (Formula presented.) and of the value of the derivative (Formula presented.) at the zeros of the solution. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem