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Advances in Nonlinear Analysis

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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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/ 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:

-Δu+V(x)u=(1|x|μF(u))f(u)in N,

where 0<μ<N, N3, 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()=0, that is, V(x)0 as |x|+, we prove the existence of a nontrivial solution for the above equation by the penalization method.

Keywords: Nonlinear Choquard equation; nonlocal nonlinearities; vanishing potential; variational methods

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


Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

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, DOI: https://doi.org/10.1515/anona-2015-0123.

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