Positive solutions for the Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system [Formula: see text] where [Formula: see text] are real parameters and [Formula: see text]. Suppose that [Formula: see text] represents a potential well with the bottom [Formula: see text], the system has been widely studied in the case [Formula: see text]. In contrast, no existence result of solutions is available for the case [Formula: see text] due to the presence of the nonlocal term [Formula: see text]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Then we obtain the nonexistence of nontrivial solutions for [Formula: see text] large and [Formula: see text] large in the case [Formula: see text]. Finally, we explore the decay rate of the positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text].