In this paper, we study the following critical fractional Schrödinger equations with the magnetic field: ${\epsilon }^{2s}{(-\mathrm{\Delta })}_{A/\epsilon }^{s}u+V(x)u=\lambda f(|u|)u+|u{|}^{2_s^{*}-2}u\text{in}{\mathbb{R}}^N$, where ɛ and λ are positive parameters and $V:{\mathbb{R}}^N\to \mathbb{R}$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are continuous electric and magnetic potentials, respectively. Under a global assumption on the potential V, by applying the method of Nehari manifold, Ekeland’s variational principle, and Ljusternick–Schnirelmann theory, we show the existence of ground state solution and multiplicity of non-negative solutions for the above equation for all sufficiently large λ and small ɛ. In this problem, f is only continuous, which allows us to study larger classes of nonlinearities.

中文翻译：

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具有临界增长的分数阶磁性薛定谔方程的存在性和多重性结果

在本文中，我们研究了以下具有磁场的临界分数薛定谔方程： ${\epsilon }^{2秒}{(-\mathrm{\Delta })}_{\mathrm{一种}/\epsilon }^{秒}你+伏(X)你=\lambda F(|你|)你+|你{|}^{2_{秒}^{*}-2}你\text{在}{\mathbb{电阻}}^N$, 其中ɛ和λ是正参数并且$伏：{\mathbb{电阻}}^N\to \mathbb{电阻}$ 和 $\mathrm{一种}：{\mathbb{电阻}}^N\to {\mathbb{电阻}}^N$分别是连续的电势和磁势。在对势V的全局假设下，通过应用 Nehari 流形、Ekeland 变分原理和 Ljusternick-Schnirelmann 理论的方法，我们证明了对于所有足够大的上述方程，基态解的存在性和非负解的多重性λ和小ɛ。在这个问题中，f只是连续的，这使我们能够研究更大的非线性类别。