| Summary: | This paper is concerned with the following nonlocal problem with combined critical nonlinearities
$$
(-\Delta)^{s} u=-\alpha|u|^{q-2} u+\beta{u}+\gamma|u|^{2_{s}^{*}-2}u \quad \text{in}~\Omega, \quad \quad
u=0 \quad \text{in}~\mathbb{R}^{N} \backslash \Omega,
$$
where $s\in(0,1)$, $N>2s$, $\Omega\subset\mathbb{R}^N$ is a bounded $C^{1,1}$ domain with Lipschitz boundary, $\alpha$ is a positive parameter, $q \in(1,2)$, $\beta$ and $\gamma$ are positive constants, and $2_{s}^{*}=2 N /(N-2 s)$ is the fractional critical exponent. For $\gamma>0$, if $N\geqslant 4s$ and $0<\beta<\lambda_{1,s}$, or $N>2s$ and $\beta\geqslant\lambda_{1,s}$, we show that the problem possesses a ground state solution when $\alpha$ is sufficiently small.
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