Generalized Chern–Simons–Schrödinger system with critical exponential growth: The zero-mass case

Abstract

We consider the existence of ground state solutions for a class of zero-mass Chern–Simons–Schrödinger systems $$\begin{array}{r}\{\begin{array}{l}-\mathrm{\Delta }u+A_{0}u+\sum _{j=1}^2{A}_j^{2}u=f(u)-a(x)|u|{}^{p-2}u,\\ {\mathrm{\partial }}_1{A}_2-{\mathrm{\partial }}_2{A}_1=-{\displaystyle \frac{1}{2}|u{|}^2,{\mathrm{\partial }}_1{A}_1+{\mathrm{\partial }}_2{A}_2=0,}\\ {\mathrm{\partial }}_1{A}_0=A_2|u{|}^2,{\mathrm{\partial }}_2{A}_0=-A_1|u{|}^2,\end{array}\end{array}$$ where $a:{\mathbb{R}}^2\to {\mathbb{R}}^{+}$ is an external potential, $p\in (1,2)$ and $f\in \mathcal{C}(\mathbb{R})$ denotes the 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 approached in (J. Differential Equations 393 (2024) 204–237), we are able to investigate the existence of positive ground state solutions for the given system using variational method.