Normalized Solutions for a Critical Fractional Choquard Equation with a Nonlocal Perturbation

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: (-Delta)u=lambda u+alpha(I divided by u divided by)divided by u divided by u+ (I divided by u divided by)divided by u divided by s q q - 2 2 2 2 mu,s- mu mu u,in IRN, * * * * mu,shaving prescribed mass integral u dx = c , 2 2 RN where s is an element of (0, 1), N > 2s, 0 < mu < N,alpha > 0, c > 0, and I mu(x) is the Riesz potential given by 2N mu] 2N- and < q < 2* = N mu,s mu N- 2s A mu mu Gamma 2 I x mu ( ) divided by divided by = with A = mu mu x , - /2N-mu 2N mu N pi Gamma 2 is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L2-subcri-2 tical perturbation ( divided by divided by )divided by divided by - alpha I mu u q u q 2 u N - mu 2 N - mu + 2 s * with exponent < q < N N , we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L2-critical and L2-supercritical cases 2N-mu+2s 2N- <= q< N mu N-2s , we also prove that the equation has ground states of mountain-pass-type.