Blow-up profile of normalized solutions for fractional nonlinear Schrödinger equation with negative potentials

We investigate the existence and blow-up profile of normalized solutions to the fractional nonlinear Schrödinger equation$ \begin{align} \begin{cases} (-\Delta)^su+V(x)u+\lambda u = |u|^{\frac{4s}{N}}u, \ \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x = \alpha, \end{cases} \end{align} $with $ N\ge 2, s\in(0, 1), \alpha>0 $, $ \lambda\in\mathbb{R} $, and negative potentials $ V(x) $. Firstly, we prove the existence and nonexistence of normalized solutions for negative potentials $ V\in L^p(\mathbb{R}^N)+L^{q}(\mathbb{R}^N) $ with $ \frac{N}{2s}\le p