In this paper, we consider the following Schrödinger-Poisson systemWe investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions.In particular, the nonlinear term includes the power-type nonlinearity f (u) = |u| p−2 u for the well-studied case p ∈ (4, 6), and the less-studied case p ∈ (3, 4), and for the latter case few existence results are available in the literature.

Abstract:We consider a fractional Schrödinger-Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

In 1983, Berestycki and Lions [Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal.82 (1983) 313–346] studied the following elliptic problem: [Formula: see text] where N ≥ 3, g is subcritical at infinity. They proved the existence of a ground state under some appropriate growth restrictions on g. In the present paper, we improve this result by showing that under the critical growth assumption on g the problem admits a ground state. In addition we study a mountain pass characterization of the least energy solutions of the problem. Without the assumption of the monotonicity of the function [Formula: see text], we show that the mountain pass value gives the least energy level.