In this paper, we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations (P) \(\begin{cases}(-\Delta)^{s} u+\lambda u=|u|^{p-2}u-|u|^{q-2}u,\ \ x\in \mathbb{R}^{N},\\ \displaystyle \int_{\mathbb{R}^{N}}|u|^{2}{\rm d}x=c>0,\\\end{cases}\) where N ≥ 2, s ∈ (0, 1), \(2+{{4s}\over{N}}<p<q\leq 2_{s}^{*}={{2N}\over{N-2s}}\) , (−Δ)s represents the fractional Laplacian operator of order s, and the frequency λ ∈ ℝ is unknown and appears as a Lagrange multiplier. Specifically, we show that there exists a ĉ > 0 such that if c > ĉ, then the problem (P) has at least two normalized solutions, including a normalized ground state solution and a mountain pass type solution. We mainly extend the results in [Commun Pure Appl Anal, 2022, 21: 4113–4145], which dealt with the problem (P) for the case \(2<p<q<2+{{4s}\over{N}}\) .
