Groundstates for Planar Schrödinger–Poisson System Involving Convolution Nonlinearity and Critical Exponential Growth

This paper is concerned with a planar Schrödinger–Poisson system involving Stein–Weiss nonlinearity 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=\frac{1}{|x|^\beta } \left( \int _{\mathbb {R}^2}\frac{F(u(y))}{|x-y|^\mu |y|^\beta }dy\right) f(u),\ &{}x\in \mathbb {R}^2,\\ \Delta \phi =u^2, \ {} &{}x\in \mathbb {R}^2,\\ \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+V(x)u+\varphi u=\frac{1}{{|x|}^{\beta }}\left({\int }_{{\mathbb{R}}^2},\frac{F(u(y))}{{|x-y|}^{\mu }{|y|}^{\beta }},d,y\right)f(u),\hfill & x\in {\mathbb{R}}^2,\hfill \\ \mathrm{\Delta }\varphi =u^2,\hfill & x\in {\mathbb{R}}^2,\hfill \end{array}\right)\end{array}$$ and its degenerate case 0.2 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\phi u= \left( \int _{\mathbb {R}^2}\frac{F(u(y))}{|x-y|^\mu }dy\right) f(u),\ {} &{}x\in \mathbb {R}^2,\\ \Delta \phi =u^2, \ {} &{}x\in \mathbb {R}^2,\\ \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+\varphi u=\left({\int }_{{\mathbb{R}}^2},\frac{F(u(y))}{{|x-y|}^{\mu }},d,y\right)f(u),\hfill & x\in {\mathbb{R}}^2,\hfill \\ \mathrm{\Delta }\varphi =u^2,\hfill & x\in {\mathbb{R}}^2,\hfill \end{array}\right)\end{array}$$ where \(\beta \ge 0,\) $\beta \ge 0,$ \(0<\mu <2\) $0<\mu <2$ , \(2\beta +\mu <2\) $2\beta +\mu <2$ , \(V\in \mathcal {C}(\mathbb {R}^2,\mathbb {R})\) $V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R})$ and f is of exponential critical growth. By combining variational methods, Stein–Weiss inequality and some delicate analysis, we derive the existence of ground state solution for the first system. Under some mild assumptions, we introduce the Pohozaev identity of the equivalent equation of the second system and use Jeanjean’s monotonicity method to achieve the existence of nontrivial solution for the second system.