Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 702-722.

### Existence and Multiplicity of Solutions for a Fractional Choquard Equation with Critical or Supercritical Growth

1. 1 School of Preparatory Education, Yunnan Minzu University, Kunming 650500
2 School of Mathematics and Statistics, Central South University, Changsha 410205
3 School of Mathematics, Yunnan Normal University, Kunming 650500
• Received:2020-03-31 Online:2021-06-01 Published:2021-06-09
• Contact: Xianhua Tang E-mail:ynyangxianyong@163.com;tangxh@mail.csu.edu.cn;guangzegu@163.com
• Supported by:
the NSFC(11971485);the NSFC(11661083);the NSFC(11861078);the NSFC(11771385);the NSFC(11901345);the Hunan Provincial Innovation Foundation for Postgraduate and the Fundamental Research Funds for the Central Universities of Central South University

Abstract:

We consider the following fractional Choquard equation with critical or supercritical growth where $s \in (0, 1)$, $\mu\in (0, N)$, $N>2s$, $q\geq 2_{\mu, s}^\ast$, $f$ is a continuous function. It is well-known that $2_{\mu, s}^\ast=\frac{2N-\mu}{N-2s}$ and $2_{\mu, s}=\frac{2N-\mu}{N}$ are critical exponents for the above equation in the sense of Hardy-Littlewood-Sobolev inequality. Many existence results have been established for $q \in[2_{\mu, s}, 2_{\mu, s}^\ast]$ in recent years. Here we are interested in critical or supercritical case for the above equation. Under some assumptions of $f$, the existence and multiplicity of solutions for the above equation can be obtained by applying some analytical techniques.

CLC Number:

• O175.2
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