Abstract
We investigate the subcritical anisotropic Trudinger–Moser inequality in the entire space ℝ N , obtain the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger–Moser inequalities on the entire Euclidean spaces, and provide a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger–Moser inequalities. Furthermore, we can prove critical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction and obtain a similar relationship with the supremums of subcritical anisotropic Trudinger–Moser inequalities.
Highlights
Let Ω be a domain with finite measure in RN and W10,p(Ω) represent the completion of C∞(Ω) in the norm
N-function made more precise by Moser [5] and obtained the following inequality: sup‖∇u‖N
Dx is replaced by Sobolev norm precisely, they proved that sup ( u∈W1,N RN)
Summary
Let Ω be a domain with finite measure in RN and W10,p(Ω) represent the completion of C∞(Ω) in the norm N-function made more precise by Moser [5] and obtained the following inequality: sup‖∇u‖N In 2000, Adachi-Tanaka [6] obtained a sharp Trudinger–Moser inequality on RN: sup u∈W1,N (RN ), Dx is replaced by Sobolev norm precisely, they proved that sup ( u∈W1,N RN)
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