Multi-bump solutions for fractional Nirenberg problem

Abstract

We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-\Delta)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where $s\in (0,1)$ and $n>2+2s$. If $K$ is a periodic function in some $k$ variables with $1\leq k<\frac{n-2s}2$, we proved that \eqref{01} has multi-bump solutions with bumps clustered on some lattice points in $\mathbb{R}^k$ via Lyapunov-Schmidt reduction. It is also established that the equation \eqref{01} has an infinite-many-bump solutions with bumps clustered on some lattice points in $\mathbb{R}^n$ which is isomorphic to $\mathbb{Z}_+^k$.