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

Let Ω⊂R4Ω⊂R4 be a smooth oriented bounded domain, H02(Ω) be the Sobolev space, and λ(Ω)=infu∈H02(Ω),‖u‖22=1‖Δu‖22 be the first eigenvalue of the bi-Laplacian operator Δ2Δ2. Then for any α  : 0⩽α<λ(Ω)0⩽α<λ(Ω), we havesupu∈H02(Ω),‖Δu‖22=1∫Ωe32π2u2(1+α‖u‖22)dx<+∞ and the above supremum is infinity when α⩾λ(Ω)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=0α=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*∈H02(Ω)∩C4(Ω¯) with ‖Δu*‖22=1 such that∫Ωe32π2u*2dx=supu∈H02(Ω),∫Ω|Δu|2dx=1∫Ωe32π2u2dx. This establishes the existence of an extremal function of the original Adams inequality in dimension 4.