In this paper, we study the following Schrödinger-Newton system 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u=\Phi u, \ & x\in \mathbb {R}^3,\\ -\Delta \Phi =u^2+\rho (x), \ & x\in \mathbb {R}^3, \end{array} \right. \end{aligned}\) where \(\rho \in L^{\frac{6}{5}}(\mathbb {R}^3)\) is a doping profile function. The existence and nonexistence results of a ground state solution are established by using variational methods. Interestingly, whether a ground state solution exists is significantly affected by the sign of \(\rho (x)\) . Specifically, we demonstrate that system (0.1) possesses a positive ground state solution if \(\rho (x)\ge 0\) and \(\Vert \rho \Vert _{L^{\frac{6}{5}}(\mathbb {R}^3)}\) is appropriately small. Conversely, system (0.1) has no ground state solution if \(\rho (x) \le 0\) . However, by employing a global compactness lemma and a general minimax principle, we succeed in showing that a high energy positive solution exists for system (0.1) if \(\rho (x) \le 0\) and \(\Vert \rho \Vert _{L^{\frac{6}{5}}(\mathbb {R}^3)}\) is suitably small.
