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

Abstract

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-Δ)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s\in (0,1)$, $(-\Delta {)}^s$ is the s-Laplacian, $B_R$ is a ball of ${\mathbb{R}}^n$, $2_s^{\ast }:=\frac{2n}{n-2s}$ is the critical Sobolev exponent and $\epsilon >0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $\epsilon \to 0^{+}$, 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.