In this paper, we study the normalized solutions for the following Chern–Simons–Schrödinger system with the Choquard type nonlinearity and the local nonlinear perturbation: \(\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} - \Delta u + \lambda u + \left( {A_0} + \sum \limits _{j = 1}^2 {A_j^2}\right) u = (I_\alpha *|u|^{\frac{\alpha }{2} + 1})|u|^{\frac{\alpha }{2}-1}u + \mu {\left| u \right| ^{p - 2}}u,~~ x\in {\mathbb {R}}^2,\\ {\partial _1}{A_2} - {\partial _2}{A_1} = - \frac{1}{2}{\left| u \right| ^2},{\partial _1}{A_1} + {\partial _2}{A_2} = 0,\\ {\partial _1}{A_0} = {A_2}{\left| u \right| ^2},{\partial _2}{A_0} = - {A_1}{\left| u \right| ^2},\\ \int _{{{\mathbb {R}}^2}} {\left| u \right| }^2\text {d}x = {c}>0, \end{array} \right. \end{aligned} \end{aligned}\) where \(\lambda \in \mathbb {R}\) is known as the Lagrange multiplier, \(\mu >0\) , \(2<p<+\infty \) , and \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0,2)\) . Under different assumptions on \(\mu \) , c, and p, we obtain several existence and nonexistence results of the above Chern–Simons–Schrödinger system. We also prove the relationship between minimizers and ground state solutions under the Pohožaev–Nehari manifold, which seems to be a new result for the considered Chern–Simons–Schrödinger system.
