Quantization property of n-Laplacian mean field equation and sharp Moser–Onofri inequality

In this paper, we are concerned with the following n-Laplacian mean field equation (Formula presented.) where Ω is a smooth bounded domain of Rn(n≥2) and -Δnu=-div(|∇u|n-2∇u). We first establish the quantization property of solutions to the above n-Laplacian mean field equation. As an application, combining the Pohozaev identity and the capacity estimate, we obtain the sharp constant C(n) of the Moser–Onofri inequality in the n-dimensional unit ball Bn:=Bn(0,1), (Formula presented.) which extends the result of Caglioti in (Commun Math Phys 143:501–525, 1992) to the case of n-dimensional ball. Here Cn=n2n-1n-1ωn-1 and ωn-1 is the surface measure of Bn. For the Moser–Onofri inequality in a general bounded domain of Rn, we apply the technique of n-harmonic transplantation to give the optimal concentration level of the Moser–Onofri inequality and obtain the criterion for the existence and non-existence of extremals for the Moser–Onofri inequality.