The paper considers the following Schrödinger–Maxwell system with supercritical nonlinearitie,$$ \textstyle\begin{cases} -\Delta u+K(x) \phi u =|u|^{p-1}u+h(x), \quad \mbox{in} \ \varOmega , \\ -\Delta \phi = K(x)u^{2}, \quad \mbox{in} \ \varOmega , \\ \phi = u=0, \quad \mbox{in} \ \partial \varOmega , \end{cases} $$(0.1)where \(\varOmega \subset \mathbb{R}^{3}\) is a bounded domain with smooth boundary, \(1< p\ \mbox{and}\ K\), \(h\in {L^{\infty }} (\varOmega )\). We prove the existence of at least one non-trivial weak solution. This result is already known for the subcritical case. In this paper, we extend it to the supercritical values of \(p\) as well. We use a new variational principle to prove our result.

中文翻译：

---

关于具有超临界非线性Schrödinger-Maxwell系统存在性结果的注释

本文考虑了以下具有超临界非线性的Schrödinger-Maxwell系统，$$ \ textstyle \ begin {cases}-\ Delta u + K（x）\ phi u = | u | ^ {p-1} u + h（x） ，\ quad \ mbox {in} \ \ varOmega，\\-\ Delta \ phi = K（x）u ^ {2}，\ quad \ mbox {in} \ \ varOmega，\\ \ phi = u = 0， \ quad \ mbox {in} \ \ partial \ varOmega，\ end {cases} $$（0.1）其中\（\ varOmega \ subset \ mathbb {R} ^ {3} \）是具有平滑边界的有界域，\ （1 <p \ \ mbox {and} \ K \），\（h \ in {L ^ {\ infty}}（\ varOmega）\）。我们证明至少存在一个非平凡的弱解。对于亚临界情况，此结果是已知的。在本文中，我们将其扩展到\（p \）的超临界值也一样 我们使用新的变分原理来证明我们的结果。