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

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