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On the Fractional Schrödinger Equations with Critical Nonlinearity

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Abstract

In this paper, we study the existence and multiplicity results for the following critical fractional Schrödinger equation with magnetic field

$$\begin{aligned} \varepsilon ^{2 s}(-\Delta )_{{\mathcal {A}}/\varepsilon }^{s}q+{\mathcal {V}}(x)q -\lambda f(|q|)q-{\mathcal {K}}(x)|q|^{2_{s}^{*}-2} q=0,{} & {} \quad \text{ in } {\mathbb {R}}^{N}, \end{aligned}$$

where \(\varepsilon>0, \lambda >0, s \in (0,1), N \ge 3,\) \(2_{s}^{*}=\frac{2N}{N-2s},\) \({\mathcal {V}}: {\mathbb {R}}^{N} \rightarrow {\mathbb {R}} \text{ and } {\mathcal {A}}: {\mathbb {R}}^{N} \rightarrow {\mathbb {R}}^{N}\) are the electric and magnetic potentials respectively. The methods used here are based on the Nehari manifold method, Ekeland’s variational principle and the Ljusternick–Schnirelmann category theory.

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The author would like to express their sincere thanks to the referees for their valuable comments and suggestions.

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  1. Department of Mathematics, Preparatory Institute for Engineering Studies, University of Kairouan, Kairouan, Tunisia

    Khaled Khachnaoui

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Khachnaoui, K. On the Fractional Schrödinger Equations with Critical Nonlinearity. Results Math 78, 68 (2023). https://doi.org/10.1007/s00025-023-01849-y

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