In this paper, we focus on the multiplicity of normalized solutions for a Hardy–Littlewood–Sobolev upper critical Schrödinger equation with van der Waals type potentials (two-body potentials with different width, see (Cao et al. in J Differ Equ 276:228–263, 2021)) \(\begin{aligned} -\Delta u=\lambda u+\mu (I_\alpha *|u|^p)|u|^{p-2}u+(I_\beta *|u|^{2_\beta ^*})|u|^{2_\beta ^*-2}u~\text {in}~ {\mathbb {R}}^N,~~\displaystyle \int \limits _{{\mathbb {R}}^N}|u|^2=a>0, \end{aligned}\) where \(N\ge 3\) , \(\mu >0\) , \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\) , \(0<\beta<\alpha <N\) , \(I_\alpha \) and \(I_\beta \) are the Riesz potentials. We show that the equation admits a normalized ground state, which is a real-valued, positive, radially symmetric, radially non-increasing function. Also, we prove that the associated energy functional has a second critical point, which is located at positive energy level. Moreover, we briefly give the qualitative property and asymptotic behavior of the second solution. The study in this paper is the counterpart of Brezis–Nirenberg-type problem for Schrödinger equation with van der Waals type potentials in the context of normalized solutions. In addition, we improve on the method employed by Ye et al. (J Geom Anal 32:44, 2022) by substantially reducing the additional conditions required to prove the existence of the second solution.
