The weighted Moser–Trudinger inequalities of Adimurthi–Druet type in RN. (June 2020) - Nguyen, Van Hoang
Abstract: Let N ≥ 2, γ ≤ β < N and λ ∈ [ 0, 1 ), we show that sup ‖ ∇ u ‖ N N + ‖ u ‖ N, γ N ≤ 1 ∫ R N Φ N ( α ( 1 + λ ‖ u ‖ N, γ N ) 1 N − 1 | u | N N − 1 ) | x | − β d x < ∞ where ‖ u ‖ N, γ N = ∫ R N | u | N | x | − γ d x and Φ N ( t ) = e t − ∑ i = 0 N − 2 t i i ! if and only if α ≤ α ¯ N, β = ( 1 − β N ) α N if  β ∈ [ 0, N ) α N ∕ ( 1 − β ∕ N ) 1 ∕ ( N − 1 ) if  β < 0,  with  α N = N ( 2 π N ∕ 2 ∕ Γ ( N ∕ 2 ) ) 1 ∕ ( N − 1 ) . This value of α ¯ N, β is new for β < 0 or γ < 0 ≤ β . Especially, for 0 ≤ γ ≤ β < N, we obtain an improved singular Moser–Trudinger inequality in the spirit of Adimurthi–Druet. We also study the attainability of the supremum above for α = α ¯ N, β .