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Classification of nonnegative solutions to static Schrödinger–Hartree and Schrödinger–Maxwell equations with combined nonlinearities

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Abstract

In this paper, we are concerned with static Schrödinger–Hartree and Schrödinger–Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution u in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorems 1.1, 1.3). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in Chen et al. (J Funct Anal 272(10):4131–4157, 2017). The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., Narrow region principle (Theorems 2.3, 3.1).

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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 Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100083, People’s Republic of China

    Wei Dai

  2. LAGA, Université Paris 13 (UMR 7539), Paris, France

    Wei Dai

  3. School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang, 330038, People’s Republic of China

    Zhao Liu

  4. Department of Mathematics, Yeshiva University, New York, NY, USA

    Zhao Liu

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  1. Wei Dai
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  2. Zhao Liu
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Correspondence to Zhao Liu.

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Communicated by A. Chang.

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Wei Dai was supported by the NNSF of China (No. 11501021), the Fundamental Research Funds for the Central Universities and the State Scholarship Fund of China (No. 201806025011); Zhao Liu was supported by the NNSF of China (No. 11801237) and the State Scholarship Fund of China (No. 201808360005).

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Dai, W., Liu, Z. Classification of nonnegative solutions to static Schrödinger–Hartree and Schrödinger–Maxwell equations with combined nonlinearities. Calc. Var. 58, 156 (2019). https://doi.org/10.1007/s00526-019-1595-z

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