Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces

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.

In 2000, Adachi-Tanaka [6] obtained a sharp Trudinger-Moser inequality on R N : where Φ N (t): � e t − N− 2 i�0 t i /i!. Note that inequality (3) has the subcritical form, that is, α < α N . Later, in [7,8], Li and Ruf showed that the best exponent α N becomes admissible if the Dirichlet norm R N |∇u| N dx is replaced by Sobolev norm R N (|u| N + |∇u| N )dx. More precisely, they proved that e proofs of both critical and subcritical Trudinger-Moser inequalities (3) and (4) rely on the Pólya-Szegö inequality and the symmetrization argument. Lam and Lu [9,10] developed a symmetrization-free method to establish the critical Trudinger-Moser inequality (see also Li,Lu,and Zhu [11]) in settings such as the Heisenberg group where the Pólya-Szegö inequality fails. Such an argument also provides an alternative proof of both critical and subcritical Trudinger-Moser inequalities (3) and (4) in the Euclidean space. In fact, the equivalence and relationship between the supremums of critical and subcritical Trudinger-Moser inequalities have been established by Lam, Lu, and Zhang [12].
In 2012, Wang and Xia [21] investigated a sharp Trudinger-Moser inequality involving the anisotropic Dirichlet norm ( Ω F N (∇u)) 1/N dx on W 1,N 0 (Ω) for N ≥ 2: Here, k N is the volume of a unit Wulff ball W F : � x ∈ R N |F 0 (x) ≤ 1 , F is convex and homogeneous of degree 1, and its polar F 0 represents a Finsler metric on R N . Similar to B. Ruf's work [8], when anisotropic Dirichlet norm ( Ω F N (∇u)dx) 1/N is replaced by full anisotropic Sobolev norm ( Ω (F N (∇u) + |u| N )dx) 1/N , Zhou [22] extended the results of Wang and Xia [21] to the entire space, provided λ ≤ λ N , and the integral above will tend to infinity for any λ > λ N . In this paper, we will establish the Adachi-Tanaka-type subcritical Trudinger-Moser inequality and the equivalence relationship between the supremums of critical and subcritical Trudinger-Moser inequalities involving the anisotropic norm restriction similar as in [12].
Our main results can be stated as follows.
If λ is close enough to λ N , then there exist constants c(N, β) and C(N, β) such that where λ N is sharp, that is, AAT(λ N , β) � ∞.
en, AMT a,b (β) < ∞ if and only if b ≤ N, and λ N is sharp, and we also have in particular,

Finsler Metric and Some Useful Lemmas
Before giving the proof, for the convenience of the readers, we provide some notations and basic facts about the Finsler metric. Let F: R N ↦R be a nonnegative convex function of class C 2 (R N \ 0 { }) which is even and positively homogeneous of degree 1, for any ξ ∈ R N and t ∈ R so that Because of homogeneity of F, there exist two constants If we consider the map which is defined by We can verify that We can see that φ( ≤ r a Wulff ball of radius r with center at 0. Next, according to the assumption of F, we can give some properties of the function F: In the following, we give two lemmas that will be used later.

Lemma 1
and then we have Due to the homogeneity of F(x), we obtain Hence, and the proof is finished.

□
By Lemma 2, when we consider the sharp Trudinger-Moser inequality, we can always assume ‖u‖ L N � 1.

Lemma 2.
e sharp subcritical Trudinger-Moser inequality is the sequence of the sharp critical Trudinger-Moser inequality. More precisely, if AMT a,b (β) is bounded, then AAT(λ, β) is also bounded and in particular, where en, Because ‖F(∇v)‖ a L N + ‖v‖ b L N ≤ 1, we have Mathematical Problems in Engineering 5 □

Equivalence between the Critical and Subcritical Anisotropic Trudinger-Moser Inequalities under the Homogeneous Norm Restriction
In this section, we give the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger-Moser inequalities, show the equivalence between the critical and subcritical anisotropic Trudinger-Moser inequalities under the homogeneous norm restriction, and finish the Proof of eorem 3.
Now, we estimate the volume of Ω u : We rewrite (38) en, By calculation, In the area Ω u , we assume where v ∈ W 1,N 0 (Ω u ), and we can easily have Set ε � λ N /λ − 1, for any a, b, ε > 0 and p > 1.We have that 6 Mathematical Problems in Engineering by using the following elementary inequality: Using the singular Trudinger-Moser inequality under the anisotropic norm in the bounded domain [23], we have that erefore, Next, we show that that AAT(λ N , β) � ∞. Set u k (x) : By calculation, we have Mathematical Problems in Engineering Next, (50) at is, ere exists a quite large constant M 1 and when k ≥ M 1 , we have erefore, Now, we establish the lower bounds of AAT(λ, β):

Mathematical Problems in Engineering
When λ/λ N ≥ (1/2), there exists a very large constant M 2 which is independent of λ; when k ≥ M 2 , we have that en, When λ is close enough to λ N , we can always find a suitable k satisfying 1 ≤ (1 − (λ/λ N ))k ≤ 2 and Since λ is close enough to λ N , we have which is

Critical Anisotropic Trudinger-Moser Inequalities under the Nonhomogenous Norm Restriction and the Relationship with the Subcritical Anisotropic Trudinger-Moser Inequalities
In this section, we prove critical anisotropic Trudinger-Moser inequalities under the nonhomogenous norm restriction and give a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger-Moser inequalities under the nonhomogenous norm restriction.
Proof of where By the simple calculation, By eorem 3, we can obtain Mathematical Problems in Engineering When θ ⟶ 1, we can use L'Hospital's rule to estimate the last but one term and then we can obtain By eorem 3, we can obtain Next, we will prove the constant λ N (1 − (β/N)) is optimal. We can establish the same sequence u k (x) as in (47). We recall that u k (x) satisfies where λ k ∈ (0, 1) satisfies and because λ k ⟶ 1 and so there exists a large enough k, such that λ/λ N λ Now, we will prove when AMT a,b (β) < ∞. By Lemma 2, we have Let u k (x) be the maximizing sequence of AMT a,b (β), i.e., (78) us, we have AAT F ∇u k (x) � � � � � � � �  AAT(λ, β) < ∞. (82) Since AMT a,b (β) < ∞, From eorem 3, we can obtain us, when λ is close enough to λ − N , we have that AAT(λ, β) ∼ (1 − (λ/λ N ) N− 1 ) (β− N)/N . us, which is impossible because b > N. Hence, we complete the proof.

Data Availability
Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.