Normalized solutions to a class of Choquard-type equations with potential - Long, Lei; Feng, Xiaojing (Topological Methods in Nonlinear Analysis, 2024)
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)in R^N, equation* having prescribed mass _R^Nu^2=a, where λR 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, a_0[0,∞) exists such that the above equation has a positive ground state normalized solution if a(a_0,∞) and one has no ground state normalized solution if a(0,a_0) when a_0> 0 by comparison arguments. Moreover, we obtain sufficient conditions for a_0=0.