A note on minimizing the Moser–Trudinger functional of high order derivatives

Let ΩΩ be a bounded smooth domain in R2nR2n (n≥2n≥2). In this note, we consider the functional Jρ(u)=12∫Ω|(−Δ)n/2u(x)|2dx−ρlog∫Ωh(x)eu(x)dx. Suppose h(x)h(x) is a smooth function with 0