On prescribed mass solutions for a kind of nonlinear elliptic system

This paper is devoted to studying the following nonlinear Schr\(\ddot{\mathrm{o}}\)dinger equation \( \left\{ \begin{array}{lr} -\Delta{u}+{\lambda}u={\mu}f(u)+h(x),\ \ x\in \mathbb{R}^2;\\ u\in H^{1}(\mathbb{R}^{2}),\int_{\mathbb{R}^2}u^2dx=\sigma\notag, \end{array} \right. \) where \(\sigma>0\) is given, \(\mu>0\), \(h(x)\) acts as a perturbation, \(f\) satisfies an exponential critical growth, \(\lambda \in \mathbb{R}\) is a Lagrange multiplier. Without taking into account the Ambrosetti-Rabinowitz condition, we prove the existence of normalized ground state solutions in two cases.