Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

: Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ω β ( x ) = [ − ln F o ( x )] ( n − 1 ) β . This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when β is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality.


Introduction
It is well-known that important geometric inequalities, for example, the Sobolev inequality, Moser-Trudinger inequality, etc., and the existence of extreme functions play a key role to study partial differential equations.For a bounded domain Ω ⊂ R n with n ≥ 2, we have W 1,p 0 (Ω) ⊂ L q (Ω), 1 ≤ q ≤ np n−p for 1 ≤ p < n by the calssical Sobolev embedding theorem.Particularly, for p = n, W 1,n 0 (Ω) ⊂ L q (Ω), ∀q ≥ 1.But W 1,n 0 (Ω) ⊈ L ∞ (Ω).For the borderline case p = n, the Moser-Trudinger inequality is the perfect replacement.In 1971, Moser [1] proved the sharpening of Trudinger's inequality as follows: for ∀α ≤ α n = nω 1 n−1 n−1 and ω n−1 stands for area of the (n − 1)-sphere.Moreover, α n is sharp, which means that if α > α n , then the inequality (1) can no longer hold.Inequality (1) is the so-called Moser-Trudinger inequality, the extremal of which is related to the existence of solutions of some semi-linear Liouville-type equations.
As far as we know, there have been many important studies related to the Moser-Trudinger inequality, for example, [2][3][4][5][6][7][8], etc.In the references listed above, readers can see the Moser-Trudinger inequality in R n and in hyperbolic spaces, the existence of an extremal function for the Moser-Trudinger inequality, etc.These important geometric inequalities play a key role in geometry analysis, calculus of variations, and PDEs; we refer to [9][10][11][12][13][14] and references therein.And recently, the authors of [15] studied a system of Kirchhoff type driven by the Q-Laplacian in the Heisenberg group H n .They obtained the existence of solutions via variational methods based on a new Moser-Trudinger-type inequality for the Heisenberg group H n .Moreover, in [16], the authors also focus on a Kirchhoff-type problem and establish the existence of a radial solution in the subcritical growth case by the Moser-Trudinger inequality and minimax method.
Recently, many researchers have intended to establish anisotropic Moser-Trudingertype inequalities.Let F ∈ C 2 (R n \{0}) be a nonnegative and convex function, the polar F o (x) of which represents a Finsler metric on R n .By F(x), a Finsler-Laplacian operator ∆ F is defined by where . In Euclidean modulus, ∆ F is nothing but the common Laplacian.The Finsler-Laplacian operator is closely related to the Wulff shape, which was initiated in Wulff' work [21].More details about the properties of F(x) and F o (x) can be seen in Section 2.
For a bounded smooth domain Ω ⊂ R n , Wang and Xia [22] proved that, for where denotes the volume of a unit Wulff ball in R n .
In this paper, we intend to establish the anisotropic Moser-Trudinger-type inequality with logarithmic weight.We believe that these sharp inequalities will be the key tools to study the existence of solutions for some quasi-linear elliptic equations, such as the Finsler-Laplacian equation.For β ∈ [0, 1), we let 1) , which is the weight of logarithmic type defined on a unit Wulff ball Let W 1,n 0,rad (W 1 , ω β ) be the subspace of W 1,n 0 (W 1 , ω β ) of all radial functions with respect to F. In this paper, radial functions with respect to F means that u(x) = ũ(r), where r = F o (x).
In the following, for convenience, we denote We now state our main results.
Next, we prove the existence of an extremal function for the anisotropic Moser-Trudinger-type inequality with logarithmic weight.Theorem 2. There exists Finally, we establish the Lions-type concentration-compactness property, which can be seen as an improvement of the anisotropic Moser-Trudinger-type inequality in Theorem 1 for some situations.

Preliminaries
In this section, we give preliminaries involving the Finsler-Laplacian, co-area formula with respect to F and convex symmetrization u ♯ of u with respect to F.
Let F : R n → R be a function that is C 2 (R n \{0}), convex, and even.And F(x) is a homogenous function, that is, for any t ∈ R, ξ ∈ R n , F(tξ) = |t|F(ξ).
Furthermore, we assume for any ξ ̸ = 0, F(ξ) > 0. By the homogeneity property of F, we can find two positive constants 0 The operator is called Finsler-Laplacian, which was studied by many mathematicians.For some important works involving the Finsler-Laplacian, we refer to [22][23][24][25][26] and the references therein.F o (x) is the support function of F(x), which is defined as F o (x) := sup ξ∈K ⟨x, ξ⟩, where ).And F o (x) is also a convex and homogeneous function.What is more, F o (x) is dual to F(x) in the sense that Denote the unit Wulff ball of center at origin as which is the volume of a unit Wulff ball W 1 .Also, we denote W r as the Wulff ball of center at origin with radius r, i.e., For later use, by the assumptions of F(x), we can obtain some properties of the function F(x); see also [25,27,28].
Lemma 1.We have Now, we give the co-area formula and isoperimetric inequality with respect to F, respectively.For a domain Ω ⊂ R n , G ⊂ Ω, let u ∈ BV(Ω), which we denote as a function of bounded variation.The anisotropic bounded variation of u with respect to F is defined by and the anisotropic perimeter of G with respect to F is defined by where X G is the characteristic function defined on the subset G. Then we have the co-area formula (see [26]) and the isoperimetric inequality Furthermore, ( 9) becomes an equality if and only if G is a Wulff ball.

Anisotropic Moser-Trudinger-Type Inequality with Logarithmic Weight
In this section, we prove Theorem 1. Firstly, we give a useful formula involving the change in functions in a unit Wulff ball Then we have the following lemma.(10); then we have ∥v∥ ω β ≤ 1.
Proof.By the property of F(x) in Lemma 1, we have F n (∇u)ω β dy) Hence, by the co-area Formula (8), we have Next, in this paper, we frequently need to change the variable in the following way.For u ∈ W 1,n 0,rad (W 1 , ω β ), we change the variable as follows: and set Then we have . By Lemma 1 and co-area Formula (8), we can transform the norm as follows: The functional changes as follows: 1 and where λ = λ λ n,β . Now it is easy to prove Theorem 1 by Lemma 2.

Existence of the Extremal Function
In this section, we complete the proof of Theorem 2. Firstly, we give a uniform bound for u ∈ W 1,n 0,rad (W 1 , ω β ).For u ∈ W 1,n 0,rad (W 1 , ω β ), we denote by u(r) the value of u(x) with r = F o (x).By the Hölder inequality and co-area Formula (8), for any 0 < r < s ≤ 1 and u ∈ W 1,n 0,rad (W 1 , ω β ), we have In particular, when s = 1, for any 0 < r ≤ 1 and u ∈ W 1,n 0,rad (W 1 , ω β ), we have The definition of ψ(t) in ( 11) and (12) shows that the anisotropic norm changes as and (13) shows that the functional I β (ψ) and J β (u) changes as For δ ∈ [0, 1), we define Then the existence of an extremal function in Theorem 1 reduces to find ψ 0 ∈ Λ1 such that Let gk (x) be a maximizing sequence of (19), that is, then there exist a subsequence (still denoted by gk ) and a function g0 ∈ W 1,n 0,rad (W 1 , ω β ) such that gk ⇀ g0 , gk → g0 pointwise.
Next, we give an inequality and we will use it several times.For any h ∈ C 1 [0, ∞) and t ≥ A ≥ 0, by the Hölder inequality, we have Now we give a lemma involving concentration-compactness alternative, by which we only need to prove that the maximizing sequence gk (x) in (20) does not concentrate at 0, and then we can pass to the limit in the functional.Firstly, we give a definition.
Proof.We assume that (ii) does not hold; then we only need to show that (i) holds.Since (ii) does not hold, then there exist A > 0 and δ ∈ (0, 1) such that for sufficiently large k, where we use the variable of change By (21), we have Since for any k, we have for t ≥ T, T sufficiently large, We note that in (23), we applied the inequality if a > b > 0, p > 1.Then, for x ∈ R large enough, (1 + ax) p ≤ 1 + b p x p .
We split the integral where Since ṽk converges pointwise to ṽ, then v k also converges pointwise to v.Then, by and the dominated convergence theorem, we have that I 1 (v k ) → I 1 (v).By (23), we have for any small ϵ > 0 and T large enough, which is smaller than ϵ.
The following lemma is proved in [29].For δ, a > 0, let Lemma 4 ( [29]).For each a > 0 and ϕ(t) ∈ Λ a δ , we have where Define f k (t) from fk (x) by the same transformation as in (22).Then, since fk (x) concentrates at 0, we have that fk (x) ⇀ 0 in W 1,n 0,rad (W 1 , ω β ) and converges pointwise to 0. Lemma 5. Let f k (t) be as above.Then one of the following alternatives holds: What is more, if the first alternative (i) holds, we can find a k to be the first point in [1, ∞) satisfying ( 26) and satisfying a k → ∞ as k → ∞.
which implies that we cannot find a k satisfying ( 26) in [0, 1).Now we assume (i) does not hold.Then we have

Define the dominating function as follows:
Then, by the dominated convergence theorem, we obtain that Let (i) hold.We choose the first a k ≥ 1 satisfying (26).We now prove that a k → ∞ as k → ∞.For any large number M, we need to prove that there exist k 0 ∈ N, such that for any k ≥ k 0 , a k ≥ M. Firstly, we choose µ small, such that µt < −2 log t + t, t ∈ [0, M).Now, since fk concentrates, we have for t ∈ [0, M) and any k ≥ k 0 , Then we obtain for any k ≥ k 0 , a k ≥ M. Now we define the concentration level at 0, We can give the estimate for the concentration level.
Lemma 6.For β ∈ [0, 1), we have that Proof.To prove the lemma, it is sufficient to assume the sequences fk satisfy the first alternative in Lemma 5, because if fk satisfy the second alternative, we can obtain the inequality ( 27) by Lemma 5. Firstly, we show that where f k and a k are as in Lemma 5. Since F n (∇ fk )ω β ⇀ 0 and (21), we have that f k → 0 uniformly on compact subsets of R + .Then for any ϵ, A > 0, we obtain On the other hand, lim sup Next, we prove that (1−β) n−1 dt.Then, by (21) with A = 0 and t = a k , we have Define the function Now, applying Lemma 4 with δ = δ * k and a = a k , we obtain where We split G k as follows: We make use of the Maclaurin series expansion.Firstly, for some positive constant C, which depends only on β, n.Thus, we have To estimate I k 2 , we first use the binomial expansion of (1 . Now, using (30) and (33), we obtain Then we have completed the proof of the Lemma.
Proof of Theorem 2. We assume J β ( gk ) does not converge to J β ( g0 ), where gk , g is as in (20).Thus, by Lemma 6, we obtain If we can find some ϕ ∈ Λ1 such that n−1 and thus, we obtain a contradiction.Consider the function h n (t) as follows: where T n = (n − 1)e ( n n−1 ) n − n n−1 + 1.It has been proved in [29] that Now we can choose α = α * sufficiently close to 1 such that Now, by direct calculation, we obtain and where B n = ( n n−1 ) n−1 − 1.Note that by the above estimates, we have Thus, we can choose β = β * , depending only on n, such that I 1 (β * ) + I 2 (β * ) ≤ 1.Thus, we have finished the proof of the Theorem.

Improvement of the Anisotropic Moser-Trudinger Inequality
In this section, we complete the proof of Theorem 3, which can be seen as an improvement of the anisotropic Moser-Trudinger inequality when u k ⇀ u 0 .
Proof of Theorem 3. If u 0 ≡ 0, then we can directly obtain (7) by Theorem 1.Thus, it is left to consider the case u 0 ̸ ≡ 0. By ( 16), we have that u k → u 0 uniformly on W 1 \ W r , ∀r ∈ (0, 1).Then, by (17) and dominated convergence theorem, we have u k → u 0 in L q (W 1 ) for any q < ∞.
For any R > 0, k ∈ N, we define the functions v R,k = min{|u k |, L}sign(u k ) and w R,k = u k − v R,k .
Since v R,k → v R,0 a.e. in W 1 as k → ∞ and v R,k is bounded in W 1,n 0,rad (W 1 , ω β ), up to a subsequence, we can assume that v R,k ⇀ v R,0 weakly in W 1,n 0,rad (W 1 , ω β ).Then we have lim inf Then we can find k 0 ∈ N such that for ∀k ≥ k 0 , we have for any k ≥ k 0 , where C depends only on n, β, ϵ, p and R. Combining (38) with Theorem 1 and the choice of ϵ, we obtain (7).The proof is completed.

Conclusions
In this paper, we mainly study the anisotropic Moser-Trudinger-type inequality for radical Sobolev space with logarithmic weight ω β (x) = [− ln F o (x)] β(n−1) , β ∈ [0, 1).Moreover, we obtain the existence of an extremal function when β is small.The extremal function is densely related to the existence of solutions of Finsler-Liouville-type equations.Finally, we obtain the Lions-type concentration-compactness principle, which is the improvement of an anisotropic Moser-Trudinger-type inequality.However, we note that the singular anisotropic Moser-Trudinger-type inequality with logarithmic weight in a unit Wulff ball W 1 and the anisotropic Moser-Trudinger-type inequality with logarithmic weight in R n are still open questions.