Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

Shao, Liuyang; Chen, Haibo

doi:10.1016/j.crma.2018.03.008

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]

Présenté par : the Editorial Board

Shao, Liuyang ¹ ; Chen, Haibo ¹

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.

Résumé
Abstract

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

{\begin{matrix} (a + b \int_{R^{N}} | {(- △)}^{\frac{α}{2}} u |^{2} d x) {(- △)}^{α} u + λ V (x) u = (| x |^{- μ} ⁎ G (u)) g (u), \\ u \in H^{α} (R^{N}), N \geq 3, \end{matrix}

où

a, b > 0

sont des constantes et

{(- Δ)}^{α}

est l'opérateur laplacien fractionnaire avec

α \in (0, 1)

2 < 2_{α, μ}^{⁎} = \frac{2 N - μ}{N - 2 α} \leq 2_{α}^{⁎} = \frac{2 N}{N - 2 α}

0 < μ < 2 α

λ > 0

des paramètres réels. Ici,

2_{α}^{⁎}

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 \in (2, 6 - α)

In this paper, we study the following fractional Kirchhoff equations

{\begin{matrix} (a + b \int_{R^{N}} | {(- △)}^{\frac{α}{2}} u |^{2} d x) {(- △)}^{α} u + λ V (x) u = (| x |^{- μ} ⁎ G (u)) g (u), \\ u \in H^{α} (R^{N}), N \geq 3, \end{matrix}

where

a, b > 0

are constants, and

{(- △)}^{α}

is the fractional Laplacian operator with

α \in (0, 1), 2 < 2_{α, μ}^{⁎} = \frac{2 N - μ}{N - 2 α} \leq 2_{α}^{⁎} = \frac{2 N}{N - 2 α}

0 < μ < 2 α

λ > 0

, is real parameter.

2_{α}^{⁎}

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 - 1} u

with

p \in (2, 6 - α)

Reçu le : 2017-12-09
Accepté le : 2018-03-14
Publié le : 2018-03-26

DOI : 10.1016/j.crma.2018.03.008

Affiliations des auteurs :

Shao, Liuyang ¹ ; Chen, Haibo ¹

¹ School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, PR China

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     author = {Shao, Liuyang and Chen, Haibo},
     title = {Existence and concentration result for a class of fractional {Kirchhoff} equations with {Hartree-type} nonlinearities and steep potential well},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {489--497},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2018.03.008},
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     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.03.008/}
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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://www.numdam.org/articles/10.1016/j.crma.2018.03.008/

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Cité par Sources :

^☆ 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.