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: (-Δ)ˢ u+V(x) u=λu+f(x)(I_α*(f|u|q))|u|q-2 u+g(x)(I_α*(g|u|ᵖ))|u|ᵖ⁻² u, x ∈ RN subject to the mass constraint ∫RN|u|² d x=a>0, where N>2 s, s ∈(0,1), α∈(0, N), and N+α/N ≤ q<p ≤ N+α+2 s/N. Here, the parameter λ∈ 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.