Normalized solutions to planar Schrödinger equation with exponential critical nonlinearity

This paper is concerned with the following planar Schrödinger equation \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda u = f(u),&x \in {\mathbb {R}}^{2},\\&\mathop \int \limits _{{\mathbb {R}}^2}u^2dx=c,&\lambda \in {\mathbb {R}}^+. \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& -\mathrm{\Delta }u+\lambda u=f(u),\hfill & \hfill x\in {\mathbb{R}}^2,\\ \hfill & \underset{{\mathbb{R}}^2}{\int }{u}^{2}dx=c,\hfill & \hfill \lambda \in {\mathbb{R}}^{+}.\end{array}\right)\end{array}$$ where \(f \in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ is of critical exponential growth. We obtain the existence of ground state normalized solutions \((u,\lambda )\) $(u,\lambda )$ under general assumptions, and here \(\lambda \) $\lambda$ stands for a Lagrange multiplier. Our theorems extend the results of Alves, Ji and Miyagaki (Calc Var 61:18, 2022) and Chang, Liu and Yan (J Geom Anal 33:83, 2023), where f satisfies a strong global assumption. In particular, some new estimates and approaches are introduced to overcome the lack of compactness resulting from the critical growth of f(u).