On fractional Schrödinger equations with Hartree type nonlinearities

Abstract

Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- Δ)^s u + μu = (I_α*F(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 $N\ge 2$ 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 [25, 65].

arXiv commentTo be published in Mathematics in Engineering

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Cited by in corpus (2)

• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• Asymptotic decay of solutions for sublinear fractional Choquard equations