An elliptic equation with combined critical Sobolev–Hardy terms

Let ΩΩ be a bounded domain in RN(N≥3) with the origin 0∈Ω0∈Ω, μ<((N−2)/2)2μ<((N−2)/2)2, 2∗(s)=2(N−s)/(N−2)2∗(s)=2(N−s)/(N−2); K(x)≥0K(x)≥0 and Q(x)≥0Q(x)≥0 are two smooth functions on ΩΩ. In this paper, we investigate the singular elliptic equation −Δu=μu|x|2+K(x)u2∗(s)−1|x|s+Q(x)u2∗(t)−1|x−x0|t+f(x,u) with Dirichlet boundary conditions. We study the limit behavior of the (P.S.) sequence of the corresponding energy functional and give a global compactness theorem, and then give some existence results.