Abstract In this paper, we study the following singularly perturbed Schrodinger-Poisson system { − e 2 △ u + V ( x ) u + ϕ u = f ( u ) + u 5 , x ∈ R 3 , − e 2 △ ϕ = u 2 , x ∈ R 3 , where e is a small positive parameter, V ∈ C ( R 3 , R ) and f ∈ C ( R , R ) satisfies neither the usual Ambrosetti-Rabinowitz type condition nor any monotonicity condition on f ( u ) / u 3 . By using some new techniques and subtle analysis, we prove that there exists a constant e 0 > 0 determined by V and f such that for e ∈ ( 0 , e 0 ] the above system admits a semiclassical ground state solution v ˆ e with exponential decay at infinity. We also study the asymptotic behavior of { v ˆ e } as e → 0 . In particular, our results can be applied to the nonlinearity f ( u ) ∼ | u | q − 2 u for q ∈ [ 3 , 4 ] , and extend the previous work that only deals with the case in which f ( u ) ∼ | u | q − 2 u for q ∈ ( 4 , 6 ) .

中文翻译：

---

具有较低扰动的临界薛定谔-泊松系统的半经典基态解

摘要 在本文中，我们研究了以下奇异摄动薛定谔-泊松系统 { − e 2 △ u + V ( x ) u + ϕ u = f ( u ) + u 5 , x ∈ R 3 , − e 2 △ ϕ = u 2 , x ∈ R 3 ，其中 e 是一个小的正参数，V ∈ C ( R 3 , R ) 和 f ∈ C ( R , R ) 既不满足通常的 Ambrosetti-Rabinowitz 类型条件，也不满足 f ( u ) / u 3 . 通过使用一些新技术和微妙的分析，我们证明存在由 V 和 f 确定的常数 e 0 > 0 使得对于 e ∈ ( 0 , e 0 ] 上述系统承认半经典基态解 v ˆ e 指数在无穷远处衰减。我们还研究了 { v ˆ e } as e → 0 的渐近行为。特别是，我们的结果可以应用于非线性 f ( u ) ∼ | u | q − 2 u for q ∈ [ 3 , 4 ] , 并扩展之前仅处理 f ( u ) ∼ | 情况的工作。你| q − 2 u 对于 q ∈ ( 4 , 6 ) 。