Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian

In this paper, we investigate the Moser-Trudinger inequality when it involves a Finsler-Laplacian operator that is associated with functionals containing \begin{document}$F^2(\nabla u)$\end{document} . Here \begin{document}$F$\end{document} is convex and homogeneous of degree 1, and its polar \begin{document}$F^o$\end{document} represents a Finsler metric on \begin{document}$\mathbb{R}^n$\end{document} . We obtain an existence result on the extremal functions for this sharp geometric inequality.

is optimal in the sense of that if α > nω 1 n−1 n−1 we can find a sequence {u ε } such that Ω e α|u | n n−1 dx diverges. In this paper, we will investigate this Moser-Trudinger type inequality when it involves a Finsler-Laplacian operator Q that is associated with functionals containing F 2 (∇u). Here F is convex and homogeneous of degree 1, and its polar F o represents a Finsler metric on R n . In 2012, Guofang Wang and Chao Xia [18] proved the following Moser-Trudinger type inequality Ω e λu n n−1 dx ≤ C(n)|Ω| (2) for all u ∈ W 1,n 0 (Ω) and Ω F (∇u) n dx ≤ 1. Here λ ≤ λ n = n n n−1 κ 1 n−1 n , κ n is the volume of a unit Wulff ball, i.e. κ n = |{x ∈ R n : F 0 (x) ≤ 1}|. λ n is also optimal in the sense that if λ > λ n we can find a sequence {u k } such that Ω e λu n n−1 k dx diverges.
Another interesting question about Moser-Trudinger inequalities is whether an extremal function exist or not. The first result in this direction is due to Carleson and Chang [2], who proved that supremum in (1) is attained when Ω is a unit ball in R n . Then Flucher [4] proved the same result when Ω is a general bounded smooth domain in R 2 . Later, Lin [8] generalized the existence result to a bounded smooth domain in all dimensions. But it is unknown that whether extremal functions for the Moser-Trudinger functional involving Finsler-Laplacian exist or not. So our goal in this paper is to prove the existence of extremal functions for the inequality (2) in dimension two. Our main Theorem is Theorem 1.1. Let Ω ⊂ R 2 be a Wulff ball with radius R centered at the origin, and κ is the volume of a unit Wulff ball. Then for any 0 < α ≤ 4κ, the supermum can be attained by some function u 0 . Here u 0 ∈ H ∩ C 1 loc (Ω\{0}) ∩ C 0 (Ω) and Ω F 2 (∇u 0 )dx = 1.
When Ω is a general domain in high dimensional space R n for n ≥ 2, for lacking of P. Lions type Lemma (see the Lemma 4.1 below), the situation becomes more subtle. So we will consider it later. For the proof of Theorem 1.1, we use an important tool in geometric analysis, the blow-up analysis. This method is based on two facts. One is that an upper bound of Moser-Trudinger energy in H (see the definition in the section 3) can be obtained from a result of Carleson and Chang [2] under the assumption that certain maximizing sequence blows up. The other is that a sequence of functions f ε can be constructed to show that the Moser-Trudinger energy is larger than the above upper bound. This contradiction implies that the considered maximizing sequence doesn't blow up and consequently this maximizing sequence converges to an extremal function of the considered Moser-Trudinger energy. Though the method we carry out by using the blow-up analysis is routine, we will encounter new difficulties which are caused by Finsler-Laplacian. We should mention that our method is derived from the convex symmetrization method and the level set method which was used in [17] and [18].
We organize this paper as follows. In section 2, we introduce the Finsler-Laplacian, the convex symmetrization of u with respect to F , the corresponding iso-perimetric inequality and the co-area formula. In section 3, we give the existence of maximizers, denote u , in the subcritical case. Some properties of u are also given in section 3. In section 4, we use blow-up analysis to show the blow-up behaviors of u ε near the blow-up point. In section 5, an upper bound is derived if u blows up, and a sequence of functions f ε is constructed to reach a contradiction, which completes the proof of Theorem.
2. preliminaries. In this section, we will give the notations and preliminaries.
Throughout this paper, let F : R 2 → R be a nonnegative convex function of class C 2 (R 2 \{0}) which is even and positively homogenous of degree 1, so that . We further assume that F (ξ) > 0 for any ξ = 0.
If we consider the minimization problem we know that its Euler equation contains an operator of the form Note that these operators are not linear unless F is the Euclidean norm. We call this nonlinear operator as Finsler-Laplacian. This operator Q is closely related to a smooth, convex hypersurface in R 2 , which is called the Wulff shape of F . And this operator Q was studied by many mathematicians, see [16,5,18,1] and the references therein. Usually, we shall assume that the Hess(F 2 ) is positive definite in R 2 \{0}. Then for such a function F , there exist two constants 0 < a ≤ b < ∞ such that a|ξ| ≤ F (ξ) ≤ b|ξ| for any ξ ∈ R 2 and there exist two constants 0 < λ ≤ Λ < ∞ such that in any compact subsets of R 2 \ {ξ = 0}, i.e. this operator Q is uniformly elliptic operator in the compact subsets of R 2 \ {x|∇u(x) = 0}, see [18] and [19]. Consider the map φ : Its image φ(S 1 ) is a smooth, convex hypersurface in R 2 , which is called Wulff shape of F . Let F 0 be the support function of K := {x ∈ R 2 : F (x) ≤ 1}, which is defined by It is easy to verify that F 0 : R 2 → [0, +∞) is also a convex, homogeneous function of class of C 2 (R 2 \{0}). Actually F 0 is dual to F in the sense that x, ξ F 0 (ξ) .

CHANGLIANG ZHOU AND CHUNQIN ZHOU
One can see easily that φ(S 1 ) = {x ∈ R 2 |F 0 (x) = 1}. We denote W F := {x ∈ R 2 |F 0 (x) ≤ 1} and κ := |W F |, the Lebesgue measure of W F . We also use the notion W r (x 0 ) := {x ∈ R 2 |F 0 (x − x 0 ) ≤ r}. we call W r (x 0 ) a Wulff ball of radius r with center at x 0 . For later use, we give some simple properties of the function F , which follows directly from the assumption on F , also see [5,16] Lemma 2.1. We have Next we describe the isoperimetric inequality and co-area formula with respect to F . For a domain Ω ⊂ R 2 , a subset E ⊂ Ω and a function of bounded variation u ∈ BV (Ω), we define the anisotropic bounded variation of u with respect to F is We set anisotropic perimeter of E with respect to F is where X E is the characteristic function of the set E. It is well known (also see [11]) that the co-area formula and the isoperimetric inequality holds. Moreover, the equality in (5) holds if and only if E is a Wulff ball.
In the sequel, we will introduce the convex symmetrization with respect to F . The convex symmetrization generalizes the Schwarz symmetrization. It was defined in [1] and will be an essential tool for this paper. Let us consider a measurable function u in Ω ⊂ R 2 . The one dimensional decreasing rearrangement of u is The convex symmetrization of u with respect to F is defined as Here κF 0 (x) 2 is just the Lebesgue measure of a homothetic Wulff ball with radius F 0 (x) and Ω * is the homothetic Wulff ball centered at the origin having the same measure as Ω. Throughout the following paper, we take Ω = W R (0) and Ω * = Ω.
3. Maximizers for subcritical-Moser-Trudinger functional. In this section, we will show the existence of the maximizers for Moser-Trudinger functional in the subcritical case. In usual, it is easy to show it. For convenience to the readers, we will give the details here. We also will give some properties of the maximizers. We first present a technical Lemma.
Since u 0 is bounded in Ω, we have Ω e 6pu 2 0 dx < +∞. This gives the desired estimate.
For u ∈ C ∞ 0 (Ω) and Ω F (∇u) 2 dx = 1, let u be the convex symmetrization of u with respect to F (x). Since and then we only need to consider non-increase and radially symmetric functions with respect to F 0 (x) for our maximizers. Let , and H be the closure of Σ in W 1,2 0 (Ω). By the Sobolev embedding theorem, we get We begin with the following existence proposition of maximizers for the subcritical Moser-Trudinger functional.
In particular, u satisfies the following equation in the distributional sense, where λ = Ω u 2 e (4κ− )u 2 dx.

CHANGLIANG ZHOU AND CHUNQIN ZHOU
By the weakly lower semi-continuous property of the norm, we have ,j is uniformly bounded in L 1 (Ω), it follows that e (4κ− )u 2 ,j is uniformly bounded in L p (Ω) for some p > 1. Since Thus we have that u attains the supremum. Clearly u ≡ 0. If we suppose that which is a contradiction to the fact that u is a maximizer. Hence ||F (∇u )|| L 2 (Ω) = 1. A straightforward calculation shows that u satisfies the Euler-Lagrange equation (7) in the distributional sense. By an inequality e t 2 ≤ 1 + t 2 e t 2 , This leads to lim inf →0 λ > 0. Moreover, using the Hölder inequality and Lemma 3.1, we can easily know that u λ e (4κ− )u 2 is bounded in L s (B 2t (0)) for some s > 1 and B 2t (0) ⊂ Ω. Hence by Theorem 2 in [6], we have u ∈ L ∞ (B t (0)). Together with u ∈ H, we have u ∈ L ∞ (Ω). It implies u λ e (4κ− )u 2 is bounded in L ∞ (Ω). Then by Theorem 1 in [9], we easily get u ∈ C 1,α (Ω) for some α ∈ (0, 1), which implies that u ∈ C 1 (Ω).
Since u ∈ H is non-increasing and H is continuously embedded in L p (Ω) for any p ≥ 1, then for any bounded sequence {u } ⊂ H, we may assume, by taking a subsequence, that u converges to u weakly in W 1,2 0 (Ω) and a.e in Ω. Therefore to prove our main Theorem, we only need to show that can be attained by some function u 0 ∈ H ∩ C 1 loc (Ω\{0}) with Ω F (∇u 0 ) 2 dx = 1. Thus in the sequel, we always assume that u ∈ H.

4.
Blow-up behaviors of u . In this section, we will develop the blow-up analysis when the sequence u blows up as → 0. Since u is bounded in H from the previous section, we can assume without loss of generality in Ω.
In the following, we can assume M = u (0) → +∞ as → 0. We start to establish a version of P. Lions type Lemma.
Also notice that, for any u ∈ H, u = u(r) = u(F 0 (x)) for r = F 0 (x) and ∇u = u (r)∇F 0 (x). Then by Lemma 2.1 again, we have Thus, for small enough we have p||F (∇u − ∇u 0 )|| 2 L 2 (Ω) < 4κ. Now choosing q > 1 close to 1 and δ > 0 satisfying Moreover, since for small enough. Thus we get the proof of the result.
Next we pay our our attention to the blow up behaviors of u at the blow up point. We set it follows that is uniformly bounded in L ∞ (B r (0)), by Theorem 1 in [13], v is uniformly bounded in C 1,α (B r 2 (0)). By Ascoli-Arzela's theorem, we can find a sequence j → 0 such that v j → v in C 1 loc (R 2 ), where v ∈ C 1 (R 2 ) and satisfies −div(F (∇v)F ξ (∇v)) = 0 in R 2 . Furthermore, we have 0 ≤ v ≤ 1 and v(0) = 1. The Liouville Theorem (see [7]) leads to v ≡ 1.

CHANGLIANG ZHOU AND CHUNQIN ZHOU
On the other hand, we claim Actually we can prove it by using the level-set method. For t ∈ R, let Ω t = {x ∈ Ω|w(x) > t} and µ(t) = |Ω t |. By the divergence theorem, By using the isoperimetric inequality (5) and the co-area formula (4), it follows form the Hölder inequality that Hence which implies the claim. Thus we get that R 2 e 8κw dx = 1, which implies that the equality holds in the above iso-perimetric inequality. Therefore Ω t must be a wulff ball. In other words, w is radial symmetric with respect to F 0 . We can immediately get Proof. We chose (u − M L ) + as a test function of (7) to get For anyR > 0, the estimation of the right hand side of (16) is In virtue of the divergence theorem and Lemma 2.1, the estimation of the left hand side of (16) is Putting (16) (17)(19) together, and takingR → ∞, we obtain Thus the conclusion can be obtained due to the fact Ω F 2 (∇u )dx = 1.
Remark 1. Using inequality (2) to functions L 1 2 u ,L , we get For any L > 1, since e 4κLu 2 ,L is uniformly bounded in L 1 (Ω), then e 4κu 2 ,L is uniformly bounded in L q (Ω) for some q > 1. Noticing that u ,L converges to 0 almost everywhere in Ω, we have e 4κu 2 ,L converges to 1 in L 1 (Ω). Thus and then
Multiplying by < ∇G, x >, and integrating by parts, we have where ν = ∇F 0 |∇F 0 | is the unit outward normal. On ∂W r , we have Inserting (33)-(35) to (32), taking r → 0, we get Here we have used the fact that 1 ∂W 1 |∇F 0 | = 2κ. Recalling the equation (28), we multiply by < ∇g , x > and integrate by parts to get Taking ρ → R and → 0, and using Lemma 4.4, we have The result holds.
Remark 2. For any 0 < ρ < R, we have In fact, Since from (23) we have it follows that (36) holds.
It is easy to see that T is increasing and ≤1 Ω e 4κu 2 dx.
To get a contradiction, we use the similar arguments in [17] and [2]. We first claim a Carleson-Chang type result.
Proof. We can use the Green function G and the solution w(r) to the bubble equation (11)( see also (15)) to construct the following sequence of functions where β ε and γ ε are constants to be chosen later, and R ε = − ln ε. Here and in the sequel we always denote r as F 0 (x).