Existence and concentration of ground states to a critical Choquard-type equation involving steep potential well

We study the Choquard-type equation with a critical local term and steep potential well {-Delta u+(1+lambda b(x))u = (I-alpha*F(u))f(u)+u(2*-1), x is an element of R-N, u is an element of H-1(R-N), where N >= 3, alpha is an element of (0, N), lambda > 0, 2* = 2N/(N-2), F(s) = integral(s)(0) f(t)dt, and I-alpha is the Riesz potential. Some results about the existence of ground states to this problem are obtained. Furthermore, we establish the concentrate behavior of the ground states as lambda -> +infinity.