Desigualdade de Adams do tipo Adimurthi-Druet em todo o espaço Rn

Abstract

Let Wm, n m (R n ) with 1 ≤ m < n be the standard higher order derivative Sobolev space in the critical exponential growth threshold. We investigate a new Adams-Adimurthi-Druet type inequality on the whole space R n which is strongly influenced by the vanishing phenomenon. Specifically, we prove sup u∈Wm, n m (Rn) ∥∇mu∥ n m n m +∥u∥ n m n m ≤1

Rn Φ   β   1 +α∥u∥ n m n m 1−γα∥u∥ n m n m   m n−m |u| n n−m   dx < +∞, where 0 ≤ α < 1, 0 < γ < 1 α − 1 for α > 0, ∇mu is the m-th order gradient for u, 0 ≤ β ≤ β0, with β0 being the Adams critical constant, and Φ(t) = et −∑ jm,n−2 j=0 t j j! with jm,n = min{j ∈ N : j ≥ n/m}. In addition, we prove that the constant β0 is sharp. In the subcritical case β < β0, we investigate both the existence and non-existence of extremal functions for the case n = 2m. In the critical case β = β0, attainability is established for n = 4 and m = 2. Our approach relies on blow-up analysis, together with a truncation method recently introduced by DelaTorre-Mancini [16], and incorporates ideas from Chen-Lu-Zhu [10], who studied the critical Adams inequality in R 4 . Moreover, we generalize these attainability results to arbitrary even dimensions by combining blow-up techniques with a suitable polyharmonic truncation and the construction of refined test functions, in the spirit of [11].