Planar Schrödinger-Poisson system with steep potential well: Supercritical exponential case

We study a class of planar Schrödinger-Poisson systems$ \left\{ \begin{array}{ll} -\Delta u+\lambda V(x)u+\phi u = f(u), & x\in \mathbb{R}^2, \\ \Delta \phi = u^2, & x\in \mathbb{R}^2, \ \end{array} \right. $where $ \lambda>0 $ is a parameter, $ V\in \mathcal{C}( \mathbb{R}^2, \mathbb{R}^+) $ has a potential well $ \Omega : = \text{int}\, V^{-1}(0) $, and the nonlinearity $ f $ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regularity theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $ \lambda $.