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In present paper, we study the fractional Choquard equation$$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\astF(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter,$s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<\mu<\min\{2s,N-2s\}$. Undersuitable assumption on $V$ and $f$, we prove this problem has a nontrivialnonnegative ground state solution. Moreover, we relate the number of nontrivialnonnegative solutions with the topology of the set where the potential attainsits minimum values and their's concentration behavior.

Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent