数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (4): 1321-1332.doi: 10.1007/s10473-021-0418-4

• 论文 • 上一篇    下一篇

EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES

Nemat NYAMORADI, Abdolrahman RAZANI   

  1. 1. Department of Mathematics, Razi University, Kermanshah, Iran;
    2. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, 34149-16818, Qazvin, Iran
  • 收稿日期:2020-07-14 修回日期:2020-08-10 出版日期:2021-08-25 发布日期:2021-09-01
  • 通讯作者: Nemat NYAMORADI E-mail:nyamoradi@razi.ac.ir,neamat80@yahoo.com
  • 作者简介:Abdolrahman RAZANI,E-mail:razani@sci.ikiu.ac.ir

EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES

Nemat NYAMORADI, Abdolrahman RAZANI   

  1. 1. Department of Mathematics, Razi University, Kermanshah, Iran;
    2. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, 34149-16818, Qazvin, Iran
  • Received:2020-07-14 Revised:2020-08-10 Online:2021-08-25 Published:2021-09-01
  • Contact: Nemat NYAMORADI E-mail:nyamoradi@razi.ac.ir,neamat80@yahoo.com

摘要: In this paper, we consider the following new Kirchhoff-type equations involving the fractional $p$-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:\begin{eqnarray*} && \left (a+ b\iint _{\mathbb{R}^{2N}} \frac{|u (x)-u (y)|^p}{|x-y|^{N + ps}} {\rm d}x {\rm d}y\right)^{p-1} (-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u\\ &=& \bigg(\int_{\mathbb{R}^N} \frac{|u|^{p^*_{\mu,s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{p^*_{\mu,s}-2}u, \; x \in \mathbb{R}^N, \end{eqnarray*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian with $0 < s < 1 < p$, $0 < \mu < N$, $N > ps$, $a,b>0$, $\lambda>0$ is a parameter, $V:\mathbb{R}^N \to \mathbb{R}^+$ is a potential function, $\theta \in[1, 2^*_{\mu,s})$ and $p^*_{\mu,s}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii's genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the $p$-Laplacian case.

关键词: Hardy-Littlewood-Sobolev inequality, concentration-compactness principle, variational method, Fractional $p$-Laplacian operators, multiple solutions

Abstract: In this paper, we consider the following new Kirchhoff-type equations involving the fractional $p$-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:\begin{eqnarray*} && \left (a+ b\iint _{\mathbb{R}^{2N}} \frac{|u (x)-u (y)|^p}{|x-y|^{N + ps}} {\rm d}x {\rm d}y\right)^{p-1} (-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u\\ &=& \bigg(\int_{\mathbb{R}^N} \frac{|u|^{p^*_{\mu,s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{p^*_{\mu,s}-2}u, \; x \in \mathbb{R}^N, \end{eqnarray*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian with $0 < s < 1 < p$, $0 < \mu < N$, $N > ps$, $a,b>0$, $\lambda>0$ is a parameter, $V:\mathbb{R}^N \to \mathbb{R}^+$ is a potential function, $\theta \in[1, 2^*_{\mu,s})$ and $p^*_{\mu,s}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii's genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the $p$-Laplacian case.

Key words: Hardy-Littlewood-Sobolev inequality, concentration-compactness principle, variational method, Fractional $p$-Laplacian operators, multiple solutions

中图分类号: 

  • 35B33