In this paper, we investigate a class of fractional Schrödinger equation with slowly decaying potentials 0.1 \(\begin{aligned} (-\Delta )^s u+V(|x|)u =Q(|x|) u^p, \quad u>0\,\ \hbox {in}\,\ {\mathbb {R}}^{N}, \,\ u\in H^{s}({\mathbb {R}}^{N}), \end{aligned}\) where \(s\in (0,1),\,N\ge 3,\,p\in (1,\frac{N+2s}{N-2s})\) and \(V,\,Q\) are continuous radial functions satisfying \(\begin{aligned}V(|x|)={V_0}+\frac{a}{|x|^n}+O\left( \frac{1}{|x|^{n+\kappa }}\right) , \quad Q(|x|)={Q_0}+\frac{b}{|x|^m}+O\left( \frac{1}{|x|^{m+\theta }}\right) ,\end{aligned}\) as \(|x| \rightarrow \infty ,\) where \(V_0,\, Q_0,\, \kappa ,\, \theta ,\, a>0,\,b\in {\mathbb {R}}.\) By introducing the Miranda theorem and some delicate analysis, via the Lyapunov-Schmit finite dimensional reduction method, we construct infinitely many multi-bump solutions of this equation when \(\frac{N+2s}{2(N+2s)+1}< n< m<N+2s\) and \(b\in {\mathbb {R}},\) or \(\frac{N+2s}{2(N+2s)+1}<m\le n<N+2s\) and \(b\le 0.\) This result not only supplements the existing literature on multi-bump solutions for slower decay rates of potential functions at infinity but also extends the range of decay rates from \(m,\,n\in \left( \frac{N+2s}{N+2s+1},N+2s\right) \) to \(m,\,n\in \left( \frac{N+2s}{2(N+2s)+1},N+2s\right) \) , which partially answers the open problem proposed in [Wang L, Zhao C. arXiv, 2014.]
