TY - JOUR

T1 - On some qualitative aspects for doubly nonlocal equations

AU - Cingolani, Silvia

AU - Gallo, Marco

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Nonlinear Schrödinger equation

KW - Double nonlocality

KW - Choquard equations

KW - Hartree type term

KW - Radial symmetry

KW - Qualitative properties of the solutions

KW - Regularity

KW - Sign of the ground states

KW - Positivity

KW - Fractional Laplacian

KW - Nonlinear Schrödinger equation

KW - Double nonlocality

KW - Choquard equations

KW - Hartree type term

KW - Radial symmetry

KW - Qualitative properties of the solutions

KW - Regularity

KW - Sign of the ground states

KW - Positivity

KW - Fractional Laplacian

UR - http://hdl.handle.net/10807/229086

UR - https://www.aimsciences.org/article/doi/10.3934/dcdss.2022041?viewtype=html

U2 - 10.3934/dcdss.2022041

DO - 10.3934/dcdss.2022041

M3 - Article

SN - 1937-1632

VL - 15

SP - 3603

EP - 3620

JO - DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S

JF - DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES S

ER -