Positive solutions for critical inhomogeneous elliptic problems in non-contractible domains

In this paper, we investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non-contractible domains. We show that for f∈H−1f∈H−1 satisfying a suitable condition and 0≤f≢00≤f≢0 the Dirichlet problem equation(0.1){−Δu=|u|2∗−2u+f,x∈Ω,u=0,x∈∂Ω admits at least four positive solutions, where Ω⊃B1ρ∖Bρ(0)¯, Ω¯⁄⊃Bρ(0) and ρρ are sufficiently small.