Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$ R N

Abstract

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: 0.1 $$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$ For $$N\ge 4$$ and $$\sigma \in (\frac{1}{2}, 1),$$ we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).