Ground state solutions of planar indefinite Schrödinger equation with supercritical exponential growth

This work is concerned with the planar indefinite Schrödinger equation with nonlinearity $ f(x, t) $ has supercritical exponential growth on $ t $, which behaviors as $ \exp\left[\left(\beta+|x|^{\alpha}\right) t^{2}\right] $ with $ \beta $, $ \alpha>0 $. Previous results in this framework required $ f $ to be subcritical or critical exponential growth, thus missing the paramount case of supercritical exponential growth. The purpose of this paper is to fill this gap and obtain a ground state solution.