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

Abstract

In this paper, we study the existence and nonexistence of solutions to the following Choquard-type equation \begin{equation*} -\Delta u+(V+\lambda)u=(I_\alpha*F(u))f(u)\quad\text{in } \mathbb{R}^N, \end{equation*} having prescribed mass $\int_{\mathbb{R}^N}u^2=a$, where $\lambda\in\mathbb{R}$ will arise as a Lagrange multiplier, $N\geq 3$, $\alpha\in(0,N)$, $I_\alpha$ is Riesz potential. Under suitable assumptions on the potential function $V$ and the nonlinear term $f$, $a_0\in[0,\infty)$ exists such that the above equation has a positive ground state normalized solution if $a\in(a_0,\infty)$ and one has no ground state normalized solution if $a\in(0,a_0)$ when $a_0> 0$ by comparison arguments. Moreover, we obtain sufficient conditions for $a_0=0$.