Solutions to critical elliptic equations with multi-singular inverse square potentials

Let ΩΩ be an open-bounded domain in RN(N⩾3)RN(N⩾3) with smooth boundary ∂Ω∂Ω. We are concerned with the multi-singular critical elliptic problem -Δu-∑i=1kμiu|x-ai|2=Q(x)|u|2*-2u,u∈H01(Ω),(*)where μi∈R,2*=2NN-2,ai∈Ω(1⩽i⩽k) and Q(x)Q(x) is a positive bounded function on Ω¯. Using Moser iteration, we prove the asymptotic behavior of solutions for (*)(*) at points ai(1⩽i⩽k). By exploiting the effect of the coefficient of the critical nonlinearity, we, by means of a variational method, establish the existence of positive and sign-changing solutions for problem (*)(*).