Ground state solutions of magnetic Schrödinger equations with exponential growth

In this paper, we investigate the following nonlinear magnetic Schrödinger equation with exponential growth: style='text-indent:20px;'> \begin{document}$ (-i\nabla+A(x))^2 u+V(x)u = f(x, |u|^2)u\ \mbox{in}\; \mathbb{R}^2 , $\end{document} style='text-indent:20px;'>where \begin{document}$ V $\end{document} is electric potential and id="M2">\begin{document}$ A potential. We prove existence of ground state solutions both in indefinite case subcritical growth definite critical growth. In order to overcome difficulty brings from presence field, by using subtle estimates establishing a new energy estimate inequality complex weaken Ambrosetti-Rabinowitz type condition strict monotonicity condition, which are commonly used case. Furthermore, case, introduce Moser function involving some analytical techniques, can also be applied related elliptic equations. Our results extend complement present ones literature.