On fractional Schrödinger equations with Hartree type nonlinearities

Silvia Cingolani; Marco Gallo; Kazunaga Tanaka; Silvia Cingolani; Marco Gallo; Kazunaga Tanaka

doi:10.3934/mine.2022056

Mathematics in Engineering

2022, Volume 4, Issue 6: 1-33. doi: 10.3934/mine.2022056

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On fractional Schrödinger equations with Hartree type nonlinearities

1.
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy
2.
Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shijuku-ku, Tokyo 169-8555, Japan

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].
- nonlinear Schrödinger equation,
- double nonlocality,
- fractional Laplacian,
- Choquard nonlinearity,
- Hartree term,
- symmetric solutions,
- regularity,
- asymptotic decay
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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Abstract

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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