Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

Xianyong Yang; Qing Miao; Xianyong Yang; Qing Miao

doi:10.3934/math.2021228

AIMS Mathematics

2021, Volume 6, Issue 4: 3838-3856. doi: 10.3934/math.2021228

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Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

Xianyong Yang ^1,2,
Qing Miao ^{1
,
,}

1.
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, 650500, P. R. China
2.
School of Mathematics and Statistics, Central south University, Changsha, 410205, P. R. China

Received: 09 December 2020 Accepted: 17 January 2021 Published: 29 January 2021
MSC : 35J50, 35Q40, 58E05

In this paper, we are concerned with the following fractional Choquard equation with critical growth:

$ (-\Delta)^s u+\lambda V(x)u = (|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u \; \hbox{in}\; \mathbb{R}^N, $

where $ s\in (0, 1) $, $ N > 2s $, $ \mu\in (0, N) $, $ 2^*_s = \frac{2N}{N-2s} $ is the fractional critical exponent, $ V $ is a steep well potential, $ F(t) = \int_0^tf(s)ds $. Under some assumptions on $ f $, the existence and asymptotic behavior of the positive ground states are established. In particular, if $ f(u) = |u|^{p-2}u $, we obtain the range of $ p $ when the equation has the positive ground states for three cases $ 2s < N < 4s $ or $ N = 4s $ or $ N > 4s $.
- fractional Choquard equation,
- critical growth,
- ground states,
- asymptotic behavior
Citation: Xianyong Yang, Qing Miao. Asymptotic behavior of ground states for a fractional Choquard equation with critical growth[J]. AIMS Mathematics, 2021, 6(4): 3838-3856. doi: 10.3934/math.2021228

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Abstract

In this paper, we are concerned with the following fractional Choquard equation with critical growth:

$ (-\Delta)^s u+\lambda V(x)u = (|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u \; \hbox{in}\; \mathbb{R}^N, $

where $ s\in (0, 1) $, $ N > 2s $, $ \mu\in (0, N) $, $ 2^*_s = \frac{2N}{N-2s} $ is the fractional critical exponent, $ V $ is a steep well potential, $ F(t) = \int_0^tf(s)ds $. Under some assumptions on $ f $, the existence and asymptotic behavior of the positive ground states are established. In particular, if $ f(u) = |u|^{p-2}u $, we obtain the range of $ p $ when the equation has the positive ground states for three cases $ 2s < N < 4s $ or $ N = 4s $ or $ N > 4s $.

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