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On fractional Schrödinger equations with Hartree type nonlinearities
Mathematics in Engineering 2022, 4(6): 1-33
Published: 15 December 2022
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Goal of this paper is to study the following doubly nonlocal equation

( Δ ) s u + μ u = ( I α F ( u ) ) F ( u ) i n R N ( P )

in the case of general nonlinearities F C 1 ( R ) of Berestycki-Lions type, when N 2 and μ > 0 is fixed. Here ( Δ ) s , s ( 0 , 1 ), denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential I α , α ( 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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