Calculus of Variations and Geometric Measure Theory

S. Cingolani - M. Gallo - K. Tanaka

On fractional Schrödinger equations with Hartree type nonlinearities

created by gallo on 10 May 2022
modified on 13 Nov 2022


Published Paper

Inserted: 10 may 2022
Last Updated: 13 nov 2022

Journal: Mathematics in Engineering
Volume: 4
Number: 6
Pages: 1-33
Year: 2021
Doi: 10.3934/mine.2022056

ArXiv: 2110.07530 PDF
Links: AIMS Press


Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eqabstract} (- \Delta)s u + \mu u = (I\alphaF(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} 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 \eqref{eqabstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in 25, 65.

Keywords: regularity, fractional Laplacian, Asymptotic decay, symmetric solutions, nonlinear Schrödinger equation, Double nonlocality, Choquard nonlinearity, Hartree term