Normalized solutions for a fractional Schrödinger equation with potentials

In this paper, we study the fractional Schrödinger equation with potentials: \(\begin{aligned} (-\Delta )^{s}u=\lambda u+P(x)|u|^{p-2}u+Q(x)|u|^{q-2}u, \quad \text{ in }\,\,{\mathbb {R}}^{N},\,\,N> 2s, \end{aligned}\) $$\begin{array}{c}\hfill {(-\mathrm{\Delta })}^{s}u=\lambda u+{P(x)|u|}^{p-2}u+Q(x){|u|}^{q-2}u,\text{in}{\mathbb{R}}^N,N>2s,\end{array}$$ having prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^{N}}|u|^{2}dx=a, \,\,\text {with}\,\, a>0, \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=a,\text{with}a>0,\end{array}$$ where \(s\in (0,1)\) $s\in (0,1)$ , \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ is unknown and appears as a Lagrange multiplier and \(2\le p<q<2^{*}_{s}=\frac{2N}{N-2s}\) $2\le p<q<2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. Under certain assumptions on the potentials P(x) and Q(x), we prove several existence results, by recovering the compactness for a minimizing sequence on a suitable manifold and comparing the essential influences due to the nonconstant potentials.