On the elliptic problems involving multi-singular inverse square potentials and critical Sobolev exponent

Assume that Ω⊂RN(N≥3) is a smooth bounded domain, ai are different points in Ω, i=1,2,…,k,k≥2,λ>0,2≤q<2∗,μ<μ̄≔(N−22)2,2∗=2NN−2 is the critical Sobolev exponent. We study the existence and asymptotic behavior of solutions to the multi-singular critical elliptic problem −Δu−μu∏i=1k|x−ai|2=|u|2∗−2u+λ|u|q−2u with Dirichlet boundary condition. The results depend crucially on the parameters λ,μ and q.