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Multiplicity and Concentration Results for Fractional Schrödinger-Poisson Equations with Magnetic Fields and Critical Growth

  • Vincenzo AmbrosioEmail author
Article

Abstract

We deal with the following fractional Schrödinger-Poisson equation with magnetic field
$$\varepsilon^{2s}(-{\Delta})_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{{2}_{s}^{*}-2}u \quad \text{ in } \mathbb{R}^{3}, $$
where ε > 0 is a small parameter, \(s\in (\frac {3}{4}, 1)\), t ∈ (0, 1), \({2}_{s}^{*}=\frac {6}{3-2s}\) is the fractional critical exponent, \((-{\Delta })^{s}_{A}\) is the fractional magnetic Laplacian, \(V:\mathbb {R}^{3}\rightarrow \mathbb {R}\) is a positive continuous potential, \(A:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}\) is a smooth magnetic potential and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a subcritical nonlinearity. Under a local condition on the potential V, we study the multiplicity and concentration of nontrivial solutions as \(\varepsilon \rightarrow 0\). In particular, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.

Keywords

Fractional magnetic operators Schrödinger-Poisson equation Critical exponent Variational methods 

Mathematics Subject Classification (2010)

35A15 35R11 35S05 58E05. 

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Notes

Acknowledgments

The author would like to thank the anonymous referee for her/his careful reading of the manuscript and valuable suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche, Informatiche e FisicheUniversità di UdineUdineItaly

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