A sharp Adams inequality in dimension four and its extremal functions

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Let Ω be a smooth oriented bounded domain in R⁴, H₀²(Ω) be the Sobolev space, and λ₁(Ω)= {Δu₂²: u∈ H₀²(Ω), u₂ =1} be the first eigenvalue of the bi-Laplacian operator Δ² on Ω. For α∈ [0,λ₁(Ω)), we define u₂,α² = Δu₂² - αu₂², for u ∈ H₀²(Ω). In this paper, we will prove the following inequality ᵤ∈ ₕ₀₂₍Ω₎, ᵤ₂,α ≤ ₁ ∫Ω e³² π² ᵘ⁽ˣ⁾² dx < ∞. This strengthens a recent result of Lu and Yang LuYang. We also show that there exists a function u*∈ H₀²(Ω)∩ C⁴(Ω) such that u*₂,α =1 and the supremum above is attained by u*. Our proofs are based on the blow-up analysis method.