Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass case

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schrödinger systems $$\{\begin{array}{l}{\displaystyle -\mathrm{\Delta }u+A_{0}u+\sum _{j=1}^2{A}_j^{2}u=f(u)-a(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,}\\ {\displaystyle {\mathrm{\partial }}_1{A}_2-{\mathrm{\partial }}_2{A}_1=-\frac{1}{2}\mathrm{\mid }u{\mathrm{\mid }}^2,{\mathrm{\partial }}_1{A}_1+{\mathrm{\partial }}_2{A}_2=0,}\\ {\displaystyle {\mathrm{\partial }}_1{A}_0=A_2\mathrm{\mid }u{\mathrm{\mid }}^2,{\mathrm{\partial }}_2{A}_0=-A_1\mathrm{\mid }u{\mathrm{\mid }}^2,}\end{array}$$ where $a:{\mathbb{R}}^2\to {\mathbb{R}}^{+}$ is an external potential, $p\in (1,2)$ and $f\in \mathcal{C}(\mathbb{R})$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method.