Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} in the case of general nonlinearities $F \in C^1(\R)$ 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{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in \cite{DSS1, MS2}.

Cingolani, S., Gallo, M., Tanaka, K., On fractional Schrödinger equations with Hartree type nonlinearities, <<MATHEMATICS IN ENGINEERING>>, 2022; 4 (6): 1-33. [doi:10.3934/mine.2022056] [https://hdl.handle.net/10807/229090]

On fractional Schrödinger equations with Hartree type nonlinearities

Gallo, Marco;
2022

Abstract

Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} in the case of general nonlinearities $F \in C^1(\R)$ 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{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in \cite{DSS1, MS2}.
2022
Inglese
Cingolani, S., Gallo, M., Tanaka, K., On fractional Schrödinger equations with Hartree type nonlinearities, <<MATHEMATICS IN ENGINEERING>>, 2022; 4 (6): 1-33. [doi:10.3934/mine.2022056] [https://hdl.handle.net/10807/229090]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/229090
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