Normalized solutions to a class of Choquard-type equations with potential

In this paper, we study the existence and nonexistence of solutions to the following Choquard-type equation {equation*} -Δ u+(V+λ)u=(I_α*F(u))f(u) R^N, {equation*} having prescribed mass ∫RNu²=a, where λ will arise as a Lagrange multiplier, N≥ 3, α∈(0,N), I_α is Riesz potential. Under suitable assumptions on the potential function V and the nonlinear term f, a₀∈[0,∞) exists such that the above equation has a positive ground state normalized solution if a∈(a₀,∞) and one has no ground state normalized solution if a∈(0,a₀) when a₀> 0 by comparison arguments. Moreover, we obtain sufficient conditions for a₀=0.