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.
Categories: math.AP