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

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 4s $.