Adams ’ inequalities for bi - Laplacian and extremal functions in dimension four ✩

Let Ω ⊂ R4 be a smooth oriented bounded domain, H 2 0 (Ω) be the Sobolev space, and λ(Ω) = inf u∈H 2 0 (Ω),‖u‖2=1 ‖ u‖ 2 2 be the first eigenvalue of the bi-Laplacian operator 2. Then for any α: 0 α < λ(Ω), we have sup u∈H 2 0 (Ω),‖ u‖2=1 ∫ Ω e32π u(1+α‖u‖2) dx <+∞ and the above supremum is infinity when α λ(Ω). This strengthens Adams’ inequality in dimension 4 [D. Adams, A sharp inequality of J. Moser for high order derivatives, Ann. of Math. 128 (1988) 365–398] where he proved the above inequality holds for α = 0. Moreover, we prove that for sufficiently small α an extremal function for the above inequality exists. As a special case of our results, we thus show that there exists u∗ ∈H 2 0 (Ω)∩C4(Ω) with ‖ u∗‖22 = 1 such that ∫ Ω e32π 2u∗2 dx = sup u∈H 2 0 (Ω), ∫ Ω | u|2 dx=1 ∫