Geodesic
Ordinary differential equations
Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well
[Résultats d'existence et de concentration pour une classe d'équations de Kirchhoff fractionnaires avec non-linéarité de type Hartree et puits de potentiel abrupt]

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Shao, Liuyang1 ; Chen, Haibo1
1 School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, PR China
Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 489-497.

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In this paper, we study the following fractional Kirchhoff equations

{(a+b∫RN|(−△)α2u|2dx)(−△)αu+λV(x)u=(|x|−μ⁎G(u))g(u),u∈Hα(RN),N≥3,
where a,b>0 are constants, and (−△)α is the fractional Laplacian operator with α∈(0,1),2<2α,μ⁎=2N−μN−2α≤2α⁎=2NN−2α, 0<μ<2α, λ>0, is real parameter. 2α⁎ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Pohožaev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity g(u)=|u|p−1u with p∈(2,6−α).

Dans ce texte, nous étudions les équations de Kirchhoff fractionnaires suivantes :

{(a+b∫RN|(−△)α2u|2dx)(−△)αu+λV(x)u=(|x|−μ⁎G(u))g(u),u∈Hα(RN),N≥3,
où a,b>0 sont des constantes et (−Δ)α est l'opérateur laplacien fractionnaire avec α∈(0,1), 2<2α,μ⁎=2N−μN−2α≤2α⁎=2NN−2α, 0<μ<2α et λ>0 des paramètres réels. Ici, 2α⁎ désigne l'exposant de Sobolev critique et g satisfait une condition de type Berestycki–Lions (voir [2]). En utilisant l'identité de Pohozaev et la théorie de concentration–compacité, nous montrons que le problème ci-dessus a au moins une solution non triviale. De plus, nous explorons le phénomène de concentration des solutions. Nos résultats complètent ceux de Lü (voir [8]) sur la non-linéarité de type Hartree g(u)=|u|p−1, avec p∈(2,6−α).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.03.008

Shao, Liuyang 1 ; Chen, Haibo 1

1 School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, PR China 
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Shao, Liuyang; Chen, Haibo. Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well. Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 489-497. doi : 10.1016/j.crma.2018.03.008. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2018.03.008/

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☆ This work is partially supported by Natural Science Foundation of China 11671403, by the Fundamental Research Funds for the Central Universities of Central South University 2017zzts056.