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 solutionsto 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 assumptionson 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 $a_0=0$.