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

Consider the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb{R}^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. For radial solutions, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. 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.
Moreover, we analyse the nonradial case with Neumann boundary conditions, namely the existence of least energy solutions and their dependence on the exponent $p$ up to the Sobolev critical exponent.
These are joint works with Alberto Saldaña and Massimo Grossi.