We consider the following Schrödinger–Bopp–Podolsky problem: $$ \textstyle\begin{cases} -\Delta u+V(x) u+\phi u=\lambda f(u)+ \vert u \vert ^{4}u,& \text{in } \mathbb{R}^{3}, \\ -\Delta \phi +\Delta ^{2}\phi = u^{2}, & \text{in } \mathbb{R}^{3}. \end{cases} $$ We prove the existence result without any growth and Ambrosetti–Rabinowitz conditions. In the proofs, we apply a cut-off function, the mountain pass theorem, and Moser iteration.

中文翻译：

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具有临界增长的Schrödinger-Bopp-Podolsky系统的非平凡解的存在性。

我们考虑以下Schrödinger–Bopp–Podolsky问题：$$ \ textstyle \ begin {cases}-\ Delta u + V（x）u + \ phi u = \ lambda f（u）+ \ vert u \ vert ^ {4} u，＆\ text {in} \ mathbb {R} ^ {3}，\\-\ Delta \ phi + \ Delta ^ {2} \ phi = u ^ {2}，＆\ text {in} \ mathbb { R} ^ {3}。\ end {cases} $$我们证明了存在结果没有任何增长和Ambrosetti–Rabinowitz条件。在证明中，我们应用了截断函数，山口定理和Moser迭代。