This paper investigates normalized solutions of the Chern–Simons–Schrödinger equations with a combined Choquard-type nonlocal nonlinearity and a local nonlinear perturbation \(\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} -\Delta u {+} \lambda u {+} \Big ( \dfrac{h^2(|x|)}{|x|^2} {+} \displaystyle \int _{|x|}^{+\infty } \dfrac{h(s)}{s}u^2(s)\,\textrm{d}s \Big ) u {=} (I_\alpha * |u|^{\frac{\alpha }{2} + 1}) |u|^{\frac{\alpha }{2}-1} u\\ ~ + \mu |u|^{p-2}u,x \in {\mathbb R}^2, \\ \displaystyle \int _{{\mathbb R}^2} |u|^2\,\textrm{d}x = c > 0, \end{array} \right. \end{aligned} \end{aligned}\) where \(\lambda \in {\mathbb R}\) is an unknown Lagrange multiplier, \(\mu >0\) , \(2<p<+\infty \) , and \(I_\alpha \) denotes the Riesz potential of order \(\alpha \in (0,2)\) . Under various assumptions on \(\mu \) , c and p, we establish several existence and nonexistence results. To the best of our knowledge, these results are novel for the Chern–Simons–Schrödinger equations.
