A global compactness result for an elliptic equation with double singular terms

Abstract

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem −Δu−μ|x|2u=1|x|s|u|2s∗−2u+λu,x∈Ω,u=0on∂Ω,where Ω⊂Rn, n≥3, is a bounded smooth domain with 0∈Ω, μ∈[0,(n−2)2∕4), s∈[0,2) and λ∈R are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems.