In this article, we study the existence and the limit behavior of solutions with a prescribed \(L^2\) -norm for a class of Chern–Simons–Schrödinger equations with combined nonlinearities with \(0<\alpha \le 1\) , \(0<\beta \le \frac{p-2}{2}\) and \(\omega >0\) . To obtain such solutions, we look for critical points of the energy functional on the mass constraint. When \(q=4<p<\infty \) , by the general minimax theorem and constraint minimization approach, we prove the existence of a positive critical point \(u^*\) for the energy functional, and this positive solution is the ground state normalized solution. When \(2<q<p,\,4<p<\infty \) , by the blow-up analysis and constraint minimization approach, on the premise that there is a radial Mountain Pass type normalized solution \({\hat{u}}\) to the Chern–Simons–Schrödinger equations, we get the limit behavior: the radial Mountain Pass type normalized solutions converge to the ground state normalized solution \(u^*\) as q tending to 4.
