A new type of bubble solutions for a critical fractional Schrödinger equation

We consider the following critical fractional Schrödinger equation \begin{equation*} (-Δ)^s u+V(|y’|,y’')u = u^{2_s^-1},\quad u>0,\quad y =(y’,y’') \in $\mathbb{R}$^3\times$\mathbb{R}$^{N-3}, \end{equation} where $N\ge 3,s\in (0,1)$, $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid },y^{\mathrm{\prime }\mathrm{\prime }})$ is a bounded non-negative function in ${\mathbb{R}}^3\times {\mathbb{R}}^{N-3}$. If $r^{2s}V(r,y^{\mathrm{\prime }\mathrm{\prime }})$ has a stable critical point $(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})$ with $r_0>0$ and $V(r_0,y_0^{\mathrm{\prime }\mathrm{\prime }})>0$, by using a finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y^{\mathrm{\prime }\mathrm{\prime }})$. We have to overcome some difficulties caused by the non-localness of the fractional Laplacian.