On some qualitative aspects for doubly nonlocal equations

Silvia Cingolani; Marco Gallo

doi:10.3934/dcdss.2022041

Article Contents

2022, Volume 15, Issue 12: 3603-3620. Doi: 10.3934/dcdss.2022041

This issue Previous Article Approximate solutions to second-order parabolic equations: Evolution systems and discretization Next Article Transmission problems for the fractional $ p $-Laplacian across fractal interfaces

On some qualitative aspects for doubly nonlocal equations

Silvia Cingolani^, and
Marco Gallo

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

^*Corresponding author: Silvia Cingolani
^*Corresponding author: Silvia Cingolani

Dedicated to the memory of Professor Rosella Mininni, a brilliant mathematician, and a very nice person

Received: October 2021

Early access: February 23, 2022

Published online: February 23, 2022

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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 \text{in } \mathbb{R}^N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(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 (P). 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 (P). In particular, we extend some results contained in [23]. Similar qualitative properties of the ground states are obtained in the limiting case $ s = 1 $, generalizing some results by Moroz and Van Schaftingen in [52] when $ F $ is odd.

Keywords:

Nonlinear Schrödinger equation,
double nonlocality,
Choquard equations,
Hartree type term,
fractional Laplacian,
qualitative properties of the solutions,
regularity,
sign of the ground states,
positivity,
radial symmetry.

Mathematics Subject Classification: 35B06, 35B09, 35B38, 35B65, 35Q40, 35Q55, 35R09, 35R11, 45M20.

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