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# Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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Volume 5, Issue 4

# Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

Claudianor O. Alves
• Departamento de Matemática, Universidade Federal de Campina Grande, CEP: 58429-900, Campina Grande - Pb, Brazil
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Giovany M. Figueiredo
/ Minbo Yang
Published Online: 2015-12-17 | DOI: https://doi.org/10.1515/anona-2015-0123

## Abstract

We study the following nonlinear Choquard equation:

$-\mathrm{\Delta }u+V\left(x\right)u=\left(\frac{1}{{|x|}^{\mu }}\ast F\left(u\right)\right)f\left(u\right)\mathit{ }\text{in}{ℝ}^{N},$

where $0<\mu , $N\ge 3$, V is a continuous real function and F is the primitive function of f. Under some suitable assumptions on the potential V, which include the case $V\left(\mathrm{\infty }\right)=0$, that is, $V\left(x\right)\to 0$ as $|x|\to +\mathrm{\infty }$, we prove the existence of a nontrivial solution for the above equation by the penalization method.

MSC 2010: 35J20; 35J60; 35A15

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## About the article

Received: 2015-09-07

Revised: 2015-11-09

Accepted: 2015-11-13

Published Online: 2015-12-17

Published in Print: 2016-11-01

Award identifier / Grant number: 301807/2013-2

Award identifier / Grant number: 301292/2011-9

Award identifier / Grant number: 552101/2011-7

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571317

Award identifier / Grant number: 11101374

Award identifier / Grant number: 11271331

C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2. G. M. Figueiredo was partially supported by CNPq/Brazil 301292/2011-9 and 552101/2011-7. M. Yang was supported by NSFC (11571317, 11101374, 11271331) and ZJNSF (LY15A010010)

Citation Information: Advances in Nonlinear Analysis, Volume 5, Issue 4, Pages 331–345, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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