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On fractional Schrödinger equations with Hartree type nonlinearities

  • Received: 10 August 2021 Accepted: 17 November 2021 Published: 30 December 2021
  • MSC : 35B38, 35B40, 35J20, 35Q40, 35Q55, 35R09, 35R11, 45M05

  • Goal of this paper is to study the following doubly nonlocal equation

    $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $

    in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu > 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].

    Citation: Silvia Cingolani, Marco Gallo, Kazunaga Tanaka. On fractional Schrödinger equations with Hartree type nonlinearities[J]. Mathematics in Engineering, 2022, 4(6): 1-33. doi: 10.3934/mine.2022056

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  • Goal of this paper is to study the following doubly nonlocal equation

    $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $

    in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu > 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].


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