2025/05/30 by Macedo, Abiel Costa, de Oliveira, José Francisco, Rocha, Fábio Sodré
#26D10 #31A30 #35B33 #35J30 #35J60 #35J91 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2505.24297
Let Wm,(n)/(m)(ℝn) 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 ℝn which is strongly influenced by the vanishing phenomenon. Specifically, we prove sup_\underset‖∇m u‖(n)/(m)^(n)/(m)+‖u‖(n)/(m)(n)/(m) ≤ 1u∈ Wm,(n)/(m)(ℝn) ∫ℝnΦ(β(\frac1+α‖u‖(n)/(m)(n)/(m)1-γα‖u‖(n)/(m)(n)/(m))(m)/(n-m)|u|(n)/(n-m)) dxlt;+∞. where 0≤ α<1, 0<γ<\frac1α-1 for α>0, ∇m u is the m-th order gradient for u, 0≤β≤ β0, with β0 being the Adams critical constant, and Φ(t) = et-∑j=0^jm,n-2\fractjj! 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, the existence and non-existence of extremal function are investigated for n=2m and attainability is proven for n=4 and m=2 in the critical case β=β0. Our method to analyze the extremal problem is based on blow-up analysis, a truncation argument recently introduced by DelaTorre-Mancini \citeDelaTorre and some ideas by Chen-Lu-Zhu \citeluluzhu20, who studied the critical Adams inequality in ℝ4.