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Direct Methods for Pseudo-relativistic Schrödinger Operators

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Abstract

In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schrödinger operators \((-\Delta +m^{2})^{s}\) with \(s\in (0,1)\) and mass \(m>0\). As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators \((-\Delta +m^{2})^{s}\) in bounded or unbounded domains with certain geometrical structures (e.g., coercive epigraph and epigraph), including pseudo-relativistic Schrödinger equations, 3D boson star equations and the equations with De Giorgi-type nonlinearities. When \(m=0\) and \(s=1\), equations with De Giorgi-type nonlinearities are related to De Giorgi conjecture connected with minimal surfaces and the scalar Ginzburg–Landau functional associated to harmonic map.

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Acknowledgements

The authors are grateful to the referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University (BUAA), Beijing, 100083, People’s Republic of China

    Wei Dai

  2. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Guolin Qin

  3. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Guolin Qin

  4. College of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Dan Wu

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  1. Wei Dai
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Wei Dai is supported by the NNSF of China (No. 11971049) and the Fundamental Research Funds for the Central Universities. Dan Wu was supported by the NSFC (Grant No. 11901183), NSF of Hunan Province (No. 2017JJ3028) and Young Teachers Program of Hunan University (531118040104).

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Dai, W., Qin, G. & Wu, D. Direct Methods for Pseudo-relativistic Schrödinger Operators. J Geom Anal 31, 5555–5618 (2021). https://doi.org/10.1007/s12220-020-00492-1

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