On fractional Schrödinger equations with Hartree type nonlinearities
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2021
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| _version_ | 1866912443198865408 |
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| author | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| author_facet | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| contents | 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 $N \geq 2$ and $μ>0$ is fixed. Here $(-Δ)^s$, $s \in (0,1)$, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $I_α$, $α\in (0,N)$. 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_07530 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On fractional Schrödinger equations with Hartree type nonlinearities Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga Analysis of PDEs 35B38, 35B40, 35J20, 35Q40, 35Q55, 35R09, 35R11, 45M05 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 $N \geq 2$ and $μ>0$ is fixed. Here $(-Δ)^s$, $s \in (0,1)$, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $I_α$, $α\in (0,N)$. 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]. |
| title | On fractional Schrödinger equations with Hartree type nonlinearities |
| topic | Analysis of PDEs 35B38, 35B40, 35J20, 35Q40, 35Q55, 35R09, 35R11, 45M05 |
| url | https://arxiv.org/abs/2110.07530 |