# Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

## TL;DR

This paper investigates the asymptotic behavior of sign-changing solutions to fractional elliptic problems, revealing a tower of bubbles phenomenon and analyzing the solutions' nodal sets as the perturbation parameter approaches zero.

## Contribution

It introduces the concept of a tower of bubbles for fractional elliptic solutions and characterizes their concentration behavior and nodal set structure.

## Key findings

- Solutions form a tower of bubbles as epsilon approaches zero.
- Positive and negative parts concentrate at the same point with different speeds.
- Provides detailed description of the nodal set of solutions.

## Abstract

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-\Delta)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-\Delta)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.