On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem

TL;DR

This paper analyzes the asymptotic behavior and structure of least energy sign-changing radial solutions to the fractional Brezis-Nirenberg problem in a ball, revealing sign-change patterns, nodal set properties, and bubble concentration phenomena as parameters vary.

Contribution

It provides new insights into the nodal set structure, sign-change bounds, and bubble profiles of solutions for the fractional Brezis-Nirenberg problem near the linear limit.

Findings

Solutions vanish everywhere if they vanish at the center.

Bound on the number of sign-changes of solutions.

Solutions change sign exactly once when s is close to 1 and λ is small.

Abstract

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $\lambda$ is the (positive) parameter in the linear part in $u$, and we consider $\lambda$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $\lambda$ such solutions change sign exactly once. Moreover, for any $s \in (0,1)$ and $\lambda$ sufficiently small we prove that the number of connected components of the complement of the nodal set corresponds to the number of sign-changes plus one. In addition, for any $s \in (\frac{1}{2},1)$, we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as $\lambda \to 0^+$, i.e. the positive and negative parts concentrate at the same point (which is the center of the ball) with different concentration speeds.