On the existence of normalized solutions to a class of fractional Choquard equation with potentials

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$(-\mathrm{\Delta }{)}^{s}u+V(x)u=\lambda u+f(x)(I_{\alpha }\ast \left(f\mathrm{\mid }u{\mathrm{\mid }}^q\right))\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+g(x)(I_{\alpha }\ast \left(g\mathrm{\mid }u{\mathrm{\mid }}^p\right))\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,x\in {\mathbb{R}}^N$$ subject to the mass constraint $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a>0,$$ where $N>2s,s\in (0,1),\alpha \in (0,N)$, and $\frac{N+\alpha }{N}\le q<p\le \frac{N+\alpha +2s}{N}$. Here, the parameter $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x),f(x)$, and $g(x)$, we establish several existence results for normalized solutions.