Existence of nontrivial solutions for fractional Choquard equations with critical or supercritical growth

ABSTRACT

In this paper, we study the following fractional Choquard equation with critical or supercritical growth $(-\mathrm{\Delta }{)}^{s}u+V(x)u=[|x{|}^{-\mu }\ast F\left(u\right)\left]f\right(u)+\lambda [|x{|}^{-\mu }\ast |u{|}^p]p|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ where 0<s<1, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order s, N>2s, $0<\mu <2s$ and $p\ge 2_{\mu ,s}^{\ast }:=(2N-\mu )/(N-2s)$. Under some suitable conditions, we prove that the equation has a nontrivial solution for small $\lambda >0$ by the variational method.

Keywords:

• Fractional Choquard equation
• critical or supercritical growth
• variational method

1991 Mathematics Subject Classifications:

• 35J20
• 35J70
• 35P05
• 35P30
• 34B15
• 58E05
• 47H04

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported in part by the National Natural Science Foundation of China [11801153; 11901514; 11501403; 11701322; 11561072] and the Honghe University Doctoral Research Programs [XJ17B11, XJ17B12] and the Yunnan Province Applied Basic Research for Youths [2018FD085; 2018FD084] and the Yunnan Province Local University (Part) Basic Research Joint Project [2017FH001-013] and the Yunnan Province Applied Basic Research for General Project [2019FB001; 2019FD091] and Technology Innovation Team of University in Yunnan Province.