Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term

Gui-Dong Li; Chun-Lei Tang

doi:10.3934/cpaa.2019015

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2019, Volume 18, Issue 1: 285-300. Doi: 10.3934/cpaa.2019015

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Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term

Gui-Dong Li and
Chun-Lei Tang^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author
* Corresponding author

Received: November 2017

Revised: November 2017

Published: August 2018

The research is supported by National Natural Science Foundation of China (No.11471267).

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Abstract

In this paper, we investigate the following a class of Choquard equation

$\begin{equation*} -Δ u+u = (I_α*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N,\end{equation*}$

where $N≥ 3,~α∈ (0,N),~I_α$ is the Riesz potential and $F(s) = ∈t_{0}^{s}f(t)dt$. If $f$ satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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