Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials

\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation \begin{equation*}\label{1} -Δu=λu+μ(|x|^{-α}\ast|u|^{2})u+(|x|^{-β}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint $\int_{{\mathbb{R}^N}} {{u}^2}=c>0,$ where $N\geq 3$, $μ\!>\!0$, $α$, $β\in (0,N)$, and the frequency $λ\in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the well studied case $α=β$, the solution set of the above problem with different width of two body potentials $α\neqβ$ is much richer. Under different assumptions on $c$, $α$ and $β$, we prove several existence, multiplicity and asymptotic behavior of solutions to the above problem. In addition, the stability of the corresponding standing waves for the related time-dependent problem is discussed.

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