We are concerned with the existence of normalized solutions for a class of generalized Chern–Simons–Schrödinger type problems with supercritical exponential growth \( \left\{ \begin{array}{ll} \displaystyle -\Delta u +\lambda u+A_0 u+\sum \limits _{j=1}^2A_j^2 u=f(u), \\ \displaystyle \partial _1A_2-\partial _2A_1=-\frac{1}{2}|u|^2,~\partial _1A_1+\partial _2A_2=0,\\ \displaystyle \partial _1A_0=A_2|u|^2,~ \partial _2A_0=-A_1|u|^2,\\ \displaystyle \int _{\mathbb {R}^2}|u|^2dx=a^2, \end{array} \right. \) where \(a\ne 0\) , \(\lambda \in \mathbb {R}\) is known as the Lagrange multiplier and \(f\in \mathcal {C}^1(\mathbb {R})\) denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger–Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regular theory, we obtain that the problem has at least a ground state solution.
