Non-existence results about on non-starshaped domains

We show in this paper that there exist domains Ω, which are not conformally equivalent to starshaped domains, such that the Dirichlet problem: , u>0 in Ω; u=0 on ∂Ω has no solution. Some nonexistence results about the Dirichlet problem: −Δu=λu+u5, u>0 in Ω; u=0 on ∂Ω for certain λ>0 over three dimensional non-starshaped domains are also obtained.