In the present paper, we investigate the existence, multiplicity and concentration of normalized solutions to the following fractional Schrödinger equation with potential \( {\left\{ \begin{array}{ll} (-\Delta )^s u+V(\varepsilon x)u+\lambda u=f(u),~ x\in \mathbb {R}^N,\\ \displaystyle \int \limits _{\mathbb {R}^N}|u|^2\mathrm{{d}}x=a^2, \end{array}\right. } \) where \(0<s<1\) , \(N\ge 2\) , \(a, \ \varepsilon >0\) , \(V \in C(\mathbb {R}^N, \mathbb {R})\) , \(\lambda \) is an unknown parameter that will appear as a Lagrange multiplier, f is a mass subcritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f and a global condition about V, with the aid of minimization techniques and Lusternik–Schnirelmann category theory, we study the relation between the numbers of normalized solutions and the topology of the set where the potential V attains its minimum value. In addition, we obtain the decay behavior of normalized solutions. Finally, by using of the cut-off technique we also consider the Sobolev supercritical case that has not been considered about the study of normalized solutions.
