Existence of solutions for singular elliptic systems with critical exponents

Let Ω∋0Ω∋0 be an open bounded domain in RN(N≥3)RN(N≥3) and 2∗=2NN−2. We consider the following elliptic system of two equations in H01(Ω)×H01(Ω): equation(∗ )−Δu−tu|x|2=2αα+β|u|α−2u|v|β+λu,−Δv−tv|x|2=2βα+β|u|α|v|β−2v+μv, where λ,μ>0λ,μ>0 and α,β>1α,β>1 satisfy α+β=2∗α+β=2∗. Using Moser iteration, we prove the asymptotic behavior of solutions for (∗) at the origin. By exploiting the mountain pass theorem, we establish the existence of solutions to (∗). We also discuss sign-changing solutions for (∗) in this paper.