On prescribed mass solutions for a kind of nonlinear elliptic system

This paper is devoted to studying the following nonlinear Schr\(ö\)dinger equation\(\{{array}{lr}-Δ{u}+{λ}u={μ}f(u)+h(x),\ \ x∈ R^2;\\ u∈ H¹(R²),∫R^2u^2dx=σ,{array}.\)where \(σ>0\) is given, \(μ>0\), \(h(x)\) acts as a perturbation, \(f\) satisfies an exponential critical growth, \(λ ∈ R\) is a Lagrange multiplier. Without taking into account the Ambrosetti-Rabinowitz condition, we prove the existence of normalized ground state solutions in two cases.