In:
Advances in Nonlinear Analysis, Walter de Gruyter GmbH, Vol. 10, No. 1 ( 2020-08-22), p. 400-419
Abstract:
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.
Type of Medium:
Online Resource
ISSN:
2191-950X
,
2191-9496
DOI:
10.1515/anona-2020-0119
Language:
English
Publisher:
Walter de Gruyter GmbH
Publication Date:
2020
detail.hit.zdb_id:
2645915-2
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