A global compactness result for singular elliptic problems involving critical Sobolev exponent

Abstract

Let Ω ⊂ R N \Omega \subset R^N be a bounded domain such that 0 ∈ Ω , N ≥ 3 , 2 ∗ = 2 N N − 2 , λ ∈ R , ϵ ∈ R 0 \in \Omega , N \geq 3,2^*=\frac {2N}{N-2},\lambda \in R, \epsilon \in R . Let { u n } ⊂ H 0 1 ( Ω ) \{u_n\}\subset H_0^1(\Omega ) be a (P.S.) sequence of the functional E λ , ϵ ( u ) = 1 2 ∫ Ω ( | ∇ u | 2 − λ u 2 | x | 2 − ϵ u 2 ) − 1 2 ∗ ∫ Ω | u | 2 ∗ E_{\lambda ,\epsilon }(u)=\frac {1}{2}\int _{\Omega }(|\nabla u|^{2}-\frac {\lambda u^2}{|x|^2}-\epsilon u^2)-\frac {1}{2^*}\int _{\Omega } |u|^{2^*} . We study the limit behaviour of u n u_n and obtain a global compactness result.