Symmetry breaking and multiple solutions for the Schrödinger–Poisson–Slater equation

We study the Schrödinger–Poisson–Slater equation where \(p\in (2,3)\) $p\in (2,3)$ , \(\lambda >0\) $\lambda >0$ , \(V(x)\in C(\mathbb {R}^3, \mathbb {R}^+)\) $V(x)\in C({\mathbb{R}}^3,{\mathbb{R}}^{+})$ and \(I_2(x):=(4\pi |x|)^{-1}\) $I_2(x):={(4\pi |x|)}^{-1}$ is the Newtonian potential. We prove the nonexistence of nontrivial solutions, existence of positive nonradial (and radial) ground-state solutions, and mountain-pass-type solutions to (SPS), depending on the values of parameters p and \(\lambda \) $\lambda$ . To our knowledge, this is the first study of the existence of ground-state solutions at positive energy levels when \(p\in (2,3)\) $p\in (2,3)$ . Furthermore, we show that a symmetry breaking occurs for the ground-state solutions, which is a purely nonlocal phenomenon that cannot be observed in the local prototype case.