Multiple solutions for critical Choquard-Kirchhoff type equations

Advances in Nonlinear Analysis

article

Sihua Liang¹  Patrizia Pucci³  Binlin Zhang⁴ 

[1] College of Mathematics, Changchun Normal University;College of Mathematics and Informatics, Fujian Normal University, Qishan Campus;Dipartimento di Matematica e Informatica, Università degli Studi di Perugia;College of Mathematics and Systems Science, Shandong University of Science and Technology

关键词: Kirchhoff equation;    Hardy-Littlewood-Sobolev critical exponent;    Choquard nonlinearity;    Concentraction compactness principle;   

DOI  :  10.1515/anona-2020-0119

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a+b∫ RN|∇ u|2dxΔ u=α k(x)|u|q− 2u+β ∫ RN|u(y)|2μ ∗ |x− y|μ dy|u|2μ ∗ − 2u,x∈ RN, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N , N ≥ 3, α and β are positive real parameters, 2μ ∗ =(2N− 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.

【 授权许可】

CC BY