\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schrodinger 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} -\Delta u=\lambda u+\mu (|x|^{-\alpha}\ast|u|^{2})u+(|x|^{-\beta}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint \[\int_{{\mathbb{R}^N}} {{u}^2}=c>0,\] where $N\geq 3$, $\mu\!>\!0$, $\alpha$, $\beta\in (0,N)$, and the frequency $\lambda\in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the well studied case $\alpha=\beta$, the solution set of the above problem with different width of two body potentials $\alpha\neq\beta$ is much richer. Under different assumptions on $c$, $\alpha$ and $\beta$, 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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具有范德瓦尔斯型势的薛定谔方程具有规定质量的驻波

\begin{abstract} 在本文中，我们关注具有范德瓦尔斯型势的薛定谔方程的规定质量的驻波，即不同宽度的二体势。这导致研究以下非局部椭圆方程 \begin{equation*}\label{1} -\Delta u=\lambda u+\mu (|x|^{-\alpha}\ast|u|^{2 })u+(|x|^{-\beta}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} 在规范化约束下 \[\int_ {{\mathbb{R}^N}} {{u}^2}=c>0,\] 其中 $N\geq 3$, $\mu\!>\!0$, $\alpha$, $ \beta\in (0,N)$，频率 $\lambda\in \mathbb{R}$ 未知，显示为拉格朗日乘数。与很好研究的情况 $\alpha=\beta$ 相比，上述问题具有不同宽度的两个体势 $\alpha\neq\beta$ 的解集要丰富得多。在对 $c$、$\alpha$ 和 $\beta$ 的不同假设下，我们证明了上述问题解的几个存在性、多重性和渐近行为。此外，讨论了相关时间相关问题的相应驻波的稳定性。