Normalized Solutions to the Critical Choquard-type Equations with Weakly Attractive Potential and Nonlocal Perturbation

In this paper, we look for solutions to the following Choquard-type equation -Delta u | (V | lambda)u = (I alpha * |u|(p))|u|(p-2)u + mu(I-a * |u|(q))|u|(q-2)u in R-N, having a prescribed mass integral u(2) = a > 0, where lambda is an element of R will arise as a Lagrange multiplier, N >= 3, I-alpha is the Riesz potential, alpha is an element of(0, N), p is an element of ( (alpha) over bar, 2(alpha)(*)], q is an element of( (alpha) over bar, 2(alpha)(*)), (alpha) over bar = (N + alpha + 2)/N is the mass critical exponent, 2(alpha)(*) = (N + alpha)/(N - 2) is the Hardy-Littlewood-Sobolev upper critical exponent and mu > 0 is a constant. Under suitable conditions on the potential V, the above Choquard-type equation admits a positive ground state normalized solution by comparison arguments, in particular, when p = 2(alpha)(*), mu needs to be larger and the Hardy-Littlewood-Sobolev subcritical approximation method is used. At the end of this paper, a new result on the regularity of solutions and Pohozaev identity to a more general Choquard-type equation is established.