Non-radial ground state solutions for fractional Schrödinger–Poisson systems in \(\mathbb {R}^{2}\)

In this paper, we study the fractional Schrödinger–Poisson system with a general nonlinearity as follows: \(\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u+u+ l(x)\phi u=f(u) &{} \text { in }\mathbb {R}^{2}, \\ (-\Delta )^{t}\phi =l(x)u^{2} &{} \text { in }\mathbb {R}^{2}, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u+u+l(x)\varphi u=f(u)\hfill & \text{in}{\mathbb{R}}^2,\hfill \\ {(-\mathrm{\Delta })}^t\varphi =l(x)u^2\hfill & \text{in}{\mathbb{R}}^2,\hfill \end{array}\right)\end{array}$$ where \(\frac{1}{2}<t\le s<1\) $\frac{1}{2}<t\le s<1$ , the potential \(l\in C(\mathbb {R}^{2},\mathbb {R}^{+})\) $l\in C({\mathbb{R}}^2,{\mathbb{R}}^{+})$ and \(f\in C(\mathbb {R},\mathbb {R})\) $f\in C(\mathbb{R},\mathbb{R})$ does not require the classical (AR)-condition. When \(l(x)\equiv \mu >0\) $l(x)\equiv \mu >0$ is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of \(\mu \) $\mu$ . As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on l(x). When l(x) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on l(x), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed.