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

TL;DR

This paper derives sharp asymptotic estimates for radial sign-changing solutions of a slightly subcritical elliptic problem near criticality, detailing their local extrema and boundary behavior.

Contribution

It provides precise asymptotic rates and constants for solutions' extrema and derivatives, extending understanding of solution behavior near criticality for Dirichlet and Neumann problems.

Findings

Sharp asymptotic rates for solution extrema as epsilon approaches zero

Explicit constants describing solution behavior near criticality

Extension of bubble tower approximation to Neumann boundary conditions

Abstract

We consider radial solutions of 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. In particular, 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. 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.