In this paper, we study the following elliptic problem with critical exponent and a Hardy potential: where Ω is a smooth open bounded domain in () which contains the origin and is the critical Sobolev exponent. We show that, if and , this problem has a ground state solution for each fixed . Moreover, we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions. If and , this problem has infinitely many sign-changing solutions for each fixed .