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

Abstract

Let Ω ⊂ R 4 be a smooth oriented bounded domain, H 0 2 ( Ω ) be the Sobolev space, and λ ( Ω ) = inf u ∈ H 0 2 ( Ω ) , ‖ u ‖ 2 2 = 1 ‖ Δ u ‖ 2 2 be the first eigenvalue of the bi-Laplacian operator Δ 2 . Then for any α: 0 ⩽ α < λ ( Ω ) , we have sup u ∈ H 0 2 ( Ω ) , ‖ Δ u ‖ 2 2 = 1 ∫ Ω e 32 π 2 u 2 ( 1 + α ‖ u ‖ 2 2 ) d x < + ∞ 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 0 2 ( Ω ) ∩ C 4 ( Ω ¯ ) with ‖ Δ u * ‖ 2 2 = 1 such that ∫ Ω e 32 π 2 u * 2 d x = sup u ∈ H 0 2 ( Ω ) , ∫ Ω | Δ u | 2 d x = 1 ∫ Ω e 32 π 2 u 2 d x . This establishes the existence of an extremal function of the original Adams inequality in dimension 4.