On an elliptic problem with critical exponent and Hardy potential

In this paper, we study the following elliptic problem with critical exponent and a Hardy potential:−Δu−μ|x|2u=λu+|u|2⁎−2u,u∈H01(Ω), where Ω   is a smooth open bounded domain in RNRN (N⩾3N⩾3) which contains the origin and 2⁎2⁎ is the critical Sobolev exponent. We show that, if N⩾5N⩾5 and μ∈(0,(N−22)2−(N+2N)2), this problem has a ground state solution for each fixed λ>0λ>0. Moreover, we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions. If N⩾7N⩾7 and μ∈(0,(N−22)2−4), this problem has infinitely many sign-changing solutions for each fixed λ>0λ>0.