On a class of mixed Choquard-Schrödinger-Poisson systems

Marius Ghergu; Gurpreet Singh

doi:10.3934/dcdss.2019021

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2019, Volume 12, Issue 2: 297-309. Doi: 10.3934/dcdss.2019021

This issue Previous Article Robin problems for the p-Laplacian with gradient dependence Next Article Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

On a class of mixed Choquard-Schrödinger-Poisson systems

Marius Ghergu^, and
Gurpreet Singh

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

* Corresponding author: Marius Ghergu
* Corresponding author: Marius Ghergu

Received: April 30, 2017

Revised: October 31, 2017

Published: April 2019

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Abstract

We study the system

$\left\{\begin{split}-\Delta u+u&=(I_\alpha*|u|^p)|u|^{p-2}u+K(x) \phi |u|^{q-2}u & \qquad \mbox{ in }\mathbb{R}^N,\\-\Delta \phi&=K(x)|u|^q& \qquad \mbox{ in }\mathbb{R}^N,\end{split}\right.$

where $N≥ 3$, $α∈ (0,N)$, $p,q>1$ and $K≥ 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,α$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.

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Mathematics Subject Classification: Primary: 35J60, 35Q40, 35J20.

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