Non-existence results about [formula omitted] on non-starshaped domains

Abstract

We show in this paper that there exist domains Ω, which are not conformally equivalent to starshaped domains, such that the Dirichlet problem: − Δ u = u n + 2 n − 2 , u > 0 in Ω; u = 0 on ∂ Ω has no solution. Some nonexistence results about the Dirichlet problem: − Δ u = λ u + u 5 , u > 0 in Ω; u = 0 on ∂ Ω for certain λ > 0 over three dimensional non-starshaped domains are also obtained.