On the supercritical mean field equation on pierced domains

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We consider the problem (P) Δu +λeᵘ∫Ω B(ξ,)eᵘ=0inΩ B(ξ,), u =0on∂ Ω B(ξ,), where Ω is a smooth bounded open domain in ² which contains the point ξ. We prove that if λ>8π, problem (P) has a solutions u such that u(x)→ 8π+ λ2 G(x,ξ) uniformly on compact sets of Ω{ξ} as goes to zero. Here G denotes Green's function of Dirichlet Laplacian in Ω. If λ∈ 8πN we will not make any symmetry assumptions on Ω, while if λ∈ 8πN we will assume that Ω is invariant under a rotation through an angle 8π² around the point ξ.