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 = (vertical bar x vertical bar(-mu) * F(u))f(u) + vertical bar u vertical bar(2s)*(-2) u in R-N, where s is an element of (0, 1), N > 2s, mu is an element of (0, N), 2(s)* = 2N/N-2s is the fractional critical exponent, V is a steep well potential, F(t) = R integral(t)(0) f(s)ds. Under some assumptions on f, the existence and asymptotic behavior of the positive ground states are established. In particular, if f(u) = vertical bar u vertical bar(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.