In this paper, we study the following nonlinear Schrödinger–Bopp–Podolsky system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+q\phi u=f(u)&{}\\ -\Delta \phi +a^2\Delta ^2\phi =4\pi u^2 \end{array} \right. \ \,\mathrm{in }\,{\mathbb {R}}^3, \end{aligned}$$

where \(a>0\), \(q>0\) and \(V\in {\mathcal {C}}({\mathbb {R}}^3,{\mathbb {R}})\). By means of the variational methods, we prove the existence of infinitely many nontrivial solutions, the existence of a ground state solution for \(f(u)=|u|^{p-2}u+h(u)\) with \(p\in [4,6)\) and the existence of at least one positive solution for \(f(u)=P(x)u^5+\mu |u|^{p-2}u\) with \(p\in (2,6)\) under some certain assumptions.

中文翻译：

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非线性作用下的Schrödinger–Bopp–Podolsky方程

在本文中，我们研究以下非线性Schrödinger–Bopp–Podolsky系统

$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta u + V（x）u + q \ phi u = f（u）＆{} \\-\ Delta \ phi + a ^ 2 \ Delta ^ 2 \ phi = 4 \ pi u ^ 2 \ end {array} \ right。\ \，\ mathrm {in} \，{\ mathbb {R}} ^ 3，\ end {aligned} $$

其中\（a> 0 \），\（q> 0 \）和\（V \ in {\ mathcal {C}}（{\ mathbb {R}} ^ 3，{\ mathbb {R}}）\）。通过变分法手段，我们证明无穷多个非平凡解的存在，对于一个基态溶液的存在\（F =（u）的| U | ^ {P-2} U + H（U）\）与\（[4,6）中的p \），并且至少存在一个正解，用于\（f（u）= P（x）u ^ 5 + \ mu | u | ^ {p-2} u \ ）在某些特定假设下具有\（p \ in（2,6）\）。