In this paper we investigate some qualitative properties of the solutions to 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} where $N \geq 2$, $s\in (0,1)$, $\alpha \in (0,N)$, $\mu>0$ is fixed, $(-\Delta)^s$ denotes the fractional Laplacian and $I_{\alpha}$ is the Riesz potential. Here $F \in C^1(\mathbb{R})$ stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of \eqref{eq_abstract}. Then, by assuming $F$ odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation \eqref{eq_abstract}. In particular, we extend some results contained in \cite{DSS1}. Similar qualitative properties of the ground states are obtained in the limiting case $s=1$, generalizing some results by Moroz and Van Schaftingen in \cite{MS2} when $F$ is odd.

Cingolani, S., Gallo, M., On some qualitative aspects for doubly nonlocal equations, <<DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S>>, 2022; 15 (12): 3603-3620. [doi:10.3934/dcdss.2022041] [https://hdl.handle.net/10807/229086]

On some qualitative aspects for doubly nonlocal equations

Gallo, Marco
2022

Abstract

In this paper we investigate some qualitative properties of the solutions to 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} where $N \geq 2$, $s\in (0,1)$, $\alpha \in (0,N)$, $\mu>0$ is fixed, $(-\Delta)^s$ denotes the fractional Laplacian and $I_{\alpha}$ is the Riesz potential. Here $F \in C^1(\mathbb{R})$ stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of \eqref{eq_abstract}. Then, by assuming $F$ odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation \eqref{eq_abstract}. In particular, we extend some results contained in \cite{DSS1}. Similar qualitative properties of the ground states are obtained in the limiting case $s=1$, generalizing some results by Moroz and Van Schaftingen in \cite{MS2} when $F$ is odd.
2022
Inglese
Cingolani, S., Gallo, M., On some qualitative aspects for doubly nonlocal equations, <<DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S>>, 2022; 15 (12): 3603-3620. [doi:10.3934/dcdss.2022041] [https://hdl.handle.net/10807/229086]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/229086
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