Multiple solutions for critical Choquard-Kirchhoff type equations
In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, where > 0, ≥ 0, 0 < < , ≥ 3, and are positive real parameters, is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, ∈ (ℝ ),...
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| Published in | Advances in nonlinear analysis Vol. 10; no. 1; pp. 400 - 419 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
De Gruyter
01.01.2021
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| Subjects | |
| Online Access | Get full text |
| ISSN | 2191-9496 2191-950X |
| DOI | 10.1515/anona-2020-0119 |
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| Abstract | In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,
where
> 0,
≥ 0, 0 <
<
,
≥ 3,
and
are positive real parameters,
is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality,
∈
(ℝ
), with
= 2
/(2
−
) if 1 <
< 2
and
= ∞ if
≥ 2
. According to the different range of
, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting. |
|---|---|
| AbstractList | In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,
where
> 0,
≥ 0, 0 <
<
,
≥ 3,
and
are positive real parameters,
is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality,
∈
(ℝ
), with
= 2
/(2
−
) if 1 <
< 2
and
= ∞ if
≥ 2
. According to the different range of
, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting. In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a + b ∫ R N | ∇ u | 2 d x Δ u = α k ( x ) | u | q − 2 u + β ∫ R N | u ( y ) | 2 μ ∗ | x − y | μ d y | u | 2 μ ∗ − 2 u , x ∈ R N , $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N , N ≥ 3, α and β are positive real parameters, 2 μ ∗ = ( 2 N − μ ) / ( N − 2 ) $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ L r (ℝ N ), with r = 2 ∗ /(2 ∗ − q ) if 1 < q < 2 * and r = ∞ if q ≥ 2 ∗ . According to the different range of q , we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting. In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, |
| Author | Pucci, Patrizia Liang, Sihua Zhang, Binlin |
| Author_xml | – sequence: 1 givenname: Sihua surname: Liang fullname: Liang, Sihua email: liangsihua@163.com organization: College of Mathematics and Informatics, Fujian Normal University, Qishan Campus, Fuzhou, 350108, P.R. China – sequence: 2 givenname: Patrizia surname: Pucci fullname: Pucci, Patrizia email: patrizia.pucci@unipg.it organization: Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, via Vanvitelli 1, 06123, Perugia, Italy – sequence: 3 givenname: Binlin surname: Zhang fullname: Zhang, Binlin email: zhangbinlin2012@163.com organization: College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, P.R. China |
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| Snippet | In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,
where
> 0,
≥ 0, 0... In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a + b ∫ R N | ∇... In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, |
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| SubjectTerms | 35A15 35B33 35J20 35J60 Choquard nonlinearity Concentraction compactness principle Hardy-Littlewood-Sobolev critical exponent Kirchhoff equation |
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| Title | Multiple solutions for critical Choquard-Kirchhoff type equations |
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