Ground state solution for fractional problem with critical combined nonlinearities

This paper is concerned with the following nonlocal problem with combined critical nonlinearities ( − Δ ) s u = − α | u | q − 2 u + β u + γ | u | 2 s ∗ − 2 u in   Ω , u = 0 in   R N ∖ Ω , where s ∈ ( 0 , 1 ) , N > 2 s , Ω ⊂ R N is a bounded C 1 , 1 domain with Lipschitz boundary, α is a positive parameter, q ∈ ( 1 , 2 ) , β and γ are positive constants, and 2 s ∗ = 2 N / ( N − 2 s ) is the fractional critical exponent. For γ > 0 , if N ⩾ 4 s and 0 < β < λ 1 , s , or N > 2 s and β ⩾ λ 1 , s , we show that the problem possesses a ground state solution when α is sufficiently small.