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* - u+ (V+) u= (I_*F (u) ) f (u) RN, equation* having prescribed mass ₑ₍u²=a, where will arise as a Lagrange multiplier, N 3, (0, N), I_ 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.