Infinitely many bubbling solutions and non-degeneracy results to fractional prescribed curvature problems

Abstract

We consider the following fractional prescribed curvature problem $$(-\Delta {)}^{s}u=K(y)u^{2_s^{\ast }-1},u>0,y\in {\mathbb{R}}^N,(0.1)$$ where $s\in (0,\frac{1}{2})$ for $N=3$, $s\in (0,1)$ for $N⩾4$ and $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ has a local maximum point in $r\in (r_0-\delta ,r_0+\delta )$. First, for any sufficient large $k$, we construct a $2k$ bubbling solution to (0.1) of some new type, which concentrate on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.