In this paper, we focus on the existence and multiplicity of normalized solutions for a Schrödinger equation with van der Waals type potentials: \(\begin{aligned}\left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle -\Delta u=\lambda u+\mu (I_\alpha *|u|^p)|u|^{p-2}u+(I_\beta *|u|^q)|u|^{q-2}u,~~~~x\in \mathbb {R}^N,\\ \displaystyle u\in H^1(\mathbb {R}^N),~~~\int _{\mathbb {R}^N}|u|^2dx=a>0,\\ \end{array}\right. \end{array}\right. \end{aligned}\) where \(N\ge 3\) , \(\mu <0\) , \(\frac{N+\alpha +2}{N}< p<q<2_\beta ^*=(N+\beta )/(N-2)\) , \(0<\beta<\alpha <N\) , \(I_\alpha \) and \(I_\beta \) are the Riesz potentials. We deal with the case, where the associated functional is not bounded below on the \(L^2\) -unit sphere and obtain that the Schrödinger equation has normalized ground states, which are also mountain-pass type solutions using Pohožaev constraint and mountain pass theorem. In addition, we apply some properties of cohomological index to prove the problem has infinitely many radial solutions. For \(N=4\) or \(N\ge 6\) , the problem has infinitely many non-radial solutions.
