Ground state solution for fractional problem with critical combined nonlinearities

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|(2s*-2) u in Omega, u = 0 in R-N\Omega, where s is an element of (0, 1), N > 2s, Omega subset of R-N is a bounded C-1,C-1 domain with Lipschitz boundary, alpha is a positive parameter, q is an element of ( 1, 2), beta and gamma are positive constants, and 2(s)(*) = 2N/(N - 2s) is the fractional critical exponent. For gamma > 0, if N >= 4s and 0 < beta <lambda(1,s), or N > 2s and ss >= lambda(1,s), we show that the problem possesses a ground state solution when alpha is sufficiently small.