Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

Abstract

Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ${\omega }_{\beta }(x)={[-ln{F}^o(x)|}^{(n-1)\beta }.$ 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 $\beta$ is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality.

1. 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 $\mathsf{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$, we have $W_0^{1,p}(\mathsf{\Omega })\subset L^q(\mathsf{\Omega })$, $1\le q\le \frac{np}{n-p}$ for $1\le p<n$ by the calssical Sobolev embedding theorem. Particularly, for $p=n$, $W_0^{1,n}(\mathsf{\Omega })\subset L^q(\mathsf{\Omega })$, $\forall q\ge 1$. But $W_0^{1,n}(\mathsf{\Omega })⊈L^{\infty }(\mathsf{\Omega })$. 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:

$$\underset{u\in W_0^{1,n}(\mathsf{\Omega }),{\parallel \nabla u\parallel }_n\le 1}{sup}{\int }_{\mathsf{\Omega }}{e}^{{\alpha |u|}^{\frac{n}{n-1}}}dx\le C,$$

(1)

for $\forall \alpha \le {\alpha }_n=n{\omega }_{n-1}^{\frac{1}{n-1}}$ and ${\omega }_{n-1}$ stands for area of the $(n-1)$-sphere. Moreover, ${\alpha }_n$ is sharp, which means that if $\alpha >{\alpha }_n$, 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 ${\mathbb{R}}^n$ 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 ${\mathbb{H}}^n$. They obtained the existence of solutions via variational methods based on a new Moser–Trudinger-type inequality for the Heisenberg group ${\mathbb{H}}^n$. 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.

Let ${\varrho }_{\beta }(x)={(-ln|x|)}^{\beta (n-1)},$ $0\le \beta <1$. Clalnchi and Ruf [17,18] proved the weighed Moser–Trudinger-type inequality involving the radical functions in unit ball B:

$$\underset{u\in W_{0,rad}^{1,n}(B,{\varrho }_{\beta }),{\parallel u\parallel }_{{\varrho }_{\beta }\le 1}}{sup}{\int }_B{e}^{{\alpha |u|}^{\frac{n}{(n-1)(1-\beta )}}}dx<\infty$$

(2)

for any $\alpha \le {\alpha }_{\beta ,n}=n{[(1-\beta ){\omega }_{n-1}^{\frac{1}{n-1}}]}^{\frac{1}{1-\beta }}$, ${\parallel u\parallel }_{{\varrho }_{\beta }}=({\int }_B{|\nabla u|}^n{\varrho }_{\beta }{(x))}^{\frac{1}{n}}$. Moreover, the constant ${\alpha }_{\beta ,n}$ is sharp, i.e., if $\alpha >{\alpha }_{\beta ,n}$, the supremum in (2) will be infinite. We note that the authors applied Leckband’s inequality [19] to prove the weighed Moser–Trudinger-type inequality (2). Note that when $\beta =0$, by the Pólya-Szegö principle, (2) recovers the classical Moser–Trudinger inequality (1). Furthermore, Roy [20] proved the existence of an extremal function for inequality (2).

Recently, many researchers have intended to establish anisotropic Moser–Trudinger-type inequalities. Let $F\in C^2({\mathbb{R}}^n\setminus \{0\})$ be a nonnegative and convex function, the polar $F^o(x)$ of which represents a Finsler metric on ${\mathbb{R}}^n$. By $F(x)$, a Finsler–Laplacian operator ${\Delta }_F$ is defined by

$${\Delta }_{F}u:=\sum _{i=1}^n\frac{\partial }{\partial x_i}(F(\nabla u)F_{{\xi }_i}(\nabla u)),$$

where $F_{{\xi }_i}=\frac{\partial F}{\partial {\xi }_i}$. In Euclidean modulus, ${\Delta }_F$ 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 $\mathsf{\Omega }\subset {\mathbb{R}}^n$, Wang and Xia [22] proved that, for $\forall \lambda \le {\lambda }_n=n^{\frac{n}{n-1}}{\kappa }_n^{\frac{1}{n-1}}$,

$$\underset{u\in W_0^{1,n}(\mathsf{\Omega }),{\int }_{\mathsf{\Omega }}{F}^n(\nabla u)dx\le 1}{sup}{\int }_{\mathsf{\Omega }}{e}^{{\lambda |u|}^{\frac{n}{n-1}}}dx\le C,$$

(3)

where

$${\kappa }_n=|x\in {\mathbb{R}}^n|F^o(x)\le 1|$$

(4)

denotes the volume of a unit Wulff ball in ${\mathbb{R}}^n$.

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 $\beta \in [0,1)$, we let

$${\omega }_{\beta }(x)={[-ln{F}^o(x)|}^{\beta (n-1)},$$

which is the weight of logarithmic type defined on a unit Wulff ball ${\mathcal{W}}_1=\{x\in {\mathbb{R}}^n:F^o(x)<1\}$. And $W_0^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ represents the functions of completion of $C_0^1({\mathcal{W}}_1)$ with respect to the norm

$${\parallel u\parallel }_{{\omega }_{\beta }}={({\int }_{{\mathcal{W}}_1}{F}^n(\nabla u){\omega }_{\beta }(x)dx)}^{\frac{1}{n}},u\in C_0^1({\mathcal{W}}_1).$$

Let $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ be the subspace of $W_0^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ of all radial functions with respect to F. In this paper, radial functions with respect to F means that $u(x)=\tilde{u}(r)$, where $r=F^o(x)$.

In the following, for convenience, we denote

$$AMT(n,\lambda ,\beta )=\underset{u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta }),{\parallel u\parallel }_{{\omega }_{\beta }\le 1}}{sup}{\int }_{{\mathcal{W}}_1}{e}^{{\lambda \parallel u\parallel }^{\frac{n}{(n-1)(1-\beta )}}}dx.$$

(5)

We now state our main results.

Theorem 1.

For any

$$\lambda \le {\lambda }_{\beta ,n}=n^{1+\frac{1}{(n-1)(1-\beta )}}{[{\kappa }_n^{\frac{1}{n-1}}(1-\beta )]}^{\frac{1}{1-\beta }},$$

(6)

we have $AMT(n,\lambda ,\beta )<\infty$. Moreover, this constant ${\lambda }_{\beta ,n}$ is sharp, i.e., if $\lambda >{\lambda }_{\beta ,n}$, $AMT(n,\lambda ,\beta )$ is infinite.

Next, we prove the existence of an extremal function for the anisotropic Moser–Trudinger-type inequality with logarithmic weight.

Theorem 2.

There exists ${\beta }_0\in [0,1)$ such that, for $\forall \beta \in [0,{\beta }_0)$, $AMT(n,{\lambda }_{\beta ,n},\beta )$ is attained.

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.

Theorem 3.

Let $\{u_k\}$ be a sequence in $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ such that $\parallel u_k{\parallel }_{{\omega }_{\beta }}=1$ and $u_k\rightharpoonup u_0$ in ${\mathcal{W}}_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$. Then we have

$$\underset{k\to \infty }{lim\; sup}{\int }_{{\mathcal{W}}_1}{e}^{p{\lambda }_{\beta ,n}{|u_k|}^{\frac{n}{(n-1)(1-\beta )}}}dx<\infty ,$$

(7)

for any $p<p(u_0):=(1-\parallel u_0{{\parallel }_{{\omega }_{\beta }}^n)}^{-\frac{1}{(n-1)(1-\beta )}}$.

2. 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 $F:{\mathbb{R}}^n\mapsto \mathbb{R}$ be a function that is $C^2({\mathbb{R}}^n\setminus \{0\})$, convex, and even. And $F(x)$ is a homogenous function, that is, for any $t\in \mathbb{R},\xi \in {\mathbb{R}}^n$,

$$F(t\xi )=|t|F(\xi ).$$

Furthermore, we assume for any $\xi \ne 0$, $F(\xi )>0.$

By the homogeneity property of F, we can find two positive constants $0<c_1\le c_2<\infty$ such that

$$c_1|\xi |\le F(\xi )\le c_2|\xi |,\forall \xi \in {\mathbb{R}}^n.$$

The operator

$${\Delta }_{F}u:=\sum _{i=1}^n\frac{\partial }{\partial x_i}(F(\nabla u)F_{{\xi }_i}(\nabla u))$$

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 $F^o(x):=\underset{\xi \in K}{sup}\langle x,\xi \rangle$, where $K=\{x\in {\mathbb{R}}^n:F(x)\le 1\}$. Then we can check that $F^o(x)$ is also a function that is $C^2({\mathbb{R}}^n\setminus \{0\})$. 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

$$F^o(x)=\underset{\xi \ne 0}{sup}\frac{\langle x,\xi \rangle }{F(\xi )},F(x)=\underset{\xi \ne 0}{sup}\frac{\langle x,\xi \rangle }{F^o(\xi )}.$$

Denote the unit Wulff ball of center at origin as

$${\mathcal{W}}_1:=\{x\in {\mathbb{R}}^n|F^o(x)\le 1\}$$

and

$${\kappa }_n:=|{\mathcal{W}}_1|,$$

which is the volume of a unit Wulff ball ${\mathcal{W}}_1$. Also, we denote ${\mathcal{W}}_r$ as the Wulff ball of center at origin with radius r, i.e.,

$${\mathcal{W}}_r:=\{x\in {\mathbb{R}}^n|F^o(x)\le r\}.$$

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

(i)

$|F(m)-F(n)|\le F(m+n)\le F(m)+F(n)$;

(ii)

$\frac{1}{C}\le |\nabla F(m)|\le C$, and $\frac{1}{C}\le |\nabla F^o(m)|\le C$ for some $C>0$ and $m\ne 0$;

(iii)

$\langle m,\nabla F(m)\rangle =F(m),\langle m,\nabla F^o(m)\rangle =F^o(m)$ for $m\ne 0$;

(iv)

$F(\nabla F^o(m))=1$, $F^o(\nabla F(m))=1$ for $m\ne 0$;

(v)

$F^o(m)F_{\xi }(\nabla F^o(m))=m$ for $m\ne 0$.

Now, we give the co-area formula and isoperimetric inequality with respect to F, respectively. For a domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$, $G\subset \mathsf{\Omega }$, let $u\in BV(\mathsf{\Omega })$, which we denote as a function of bounded variation. The anisotropic bounded variation of u with respect to F is defined by

$${\int }_{\mathsf{\Omega }}{|\nabla u|}_F=sup\{{\int }_{\mathsf{\Omega }}u\mathrm{div}\tau dx,\tau \in C_0^1(\mathsf{\Omega };{\mathbb{R}}^n),F^o(\tau )\le 1\},$$

and the anisotropic perimeter of G with respect to F is defined by

$$H_F(G):={\int }_{\mathsf{\Omega }}{|\nabla {\mathcal{X}}_G|}_{F}dx,$$

where ${\mathcal{X}}_G$ is the characteristic function defined on the subset G. Then we have the co-area formula (see [26])

$${\int }_{\mathsf{\Omega }}{|\nabla u|}_F={\int }_0^{\infty }{H}_F(|u|>t)dt$$

(8)

and the isoperimetric inequality

$$H_F(G)\ge n{\kappa }_n^{\frac{1}{n}}{|G|}^{1-\frac{1}{n}}.$$

(9)

Furthermore, (9) becomes an equality if and only if G is a Wulff ball.

3. 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 ${\mathcal{W}}_1$. For $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ and any $0\le \tilde{\beta }<\beta$, we let

$$v(x)={(\frac{{\lambda }_{\beta ,n}}{{\lambda }_{\tilde{\beta },n}})}^{\frac{(n-1)(1-\tilde{\beta })}{n}}u(x){|u(x)|}^{\frac{\beta -\tilde{\beta }}{1-\beta }}.$$

(10)

Then we have the following lemma.

Lemma 2.

Let $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ with ${\parallel u\parallel }_{{\omega }_{\beta }}\le 1$. Define v by (10); then we have ${\parallel v\parallel }_{{\omega }_{\tilde{\beta }}}\le 1$.

Proof. 

By the property of $F(x)$ in Lemma 1, we have

$$\begin{array}{ccc}\hfill F^n(\nabla v)& =& {(\frac{{\lambda }_{\beta ,n}}{{\lambda }_{\tilde{\beta },n}})}^{(n-1)(1-\tilde{\beta })}\frac{{(1-\tilde{\beta })}^n}{{(1-\beta )}^n}{F}^n(\nabla u){|u(x)|}^{\frac{n(\beta -\tilde{\beta })}{1-\beta }}\hfill \\ & \le & \frac{(1-\tilde{\beta })}{(1-\beta )}{F}^n(\nabla u)\frac{{\omega }_{\beta }(x)}{{\omega }_{\tilde{\beta }}(x)}{({\int }_{{\mathcal{W}}_1\setminus {\mathcal{W}}_{F^o(x)}}{F}^n(\nabla u){\omega }_{\beta }dy)}^{\frac{\beta -\tilde{\beta }}{1-\beta }}.\hfill \end{array}$$

Hence, by the co-area Formula (8), we have

$$\begin{array}{ccc}& & {\parallel v\parallel }_{{\omega }_{\tilde{\beta }}}^n={\int }_{{\mathcal{W}}_1}{F}^n(\nabla v){\omega }_{\tilde{\beta }}(x)dx\hfill \\ & \le & \frac{(1-\tilde{\beta })}{(1-\beta )}{\int }_{{\mathcal{W}}_1}{F}^n(\nabla u){\omega }_{\beta }(x)({\int }_r^{1}n{\kappa }_n|u^{\prime }(s){{|}^n{s}^{n-1}{\omega }_{\beta }(s)ds)}^{\frac{\beta -\tilde{\beta }}{1-\beta }}dx\hfill \\ & =& \frac{(1-\tilde{\beta })}{(1-\beta )}{(n{\kappa }_n)}^{\frac{(1-\tilde{\beta })}{(1-\beta )}}{\int }_0^1|u^{\prime }(r){|}^n{\omega }_{\beta }(r)r^{n-1}({\int }_r^1|u^{\prime }(s){{|}^n{\omega }_{\beta }(s)s^{n-1}ds)}^{\frac{\beta -\tilde{\beta }}{1-\beta }}dr\hfill \\ & =& -{(n{\kappa }_n)}^{\frac{(1-\tilde{\beta })}{(1-\beta )}}{\int }_0^1\frac{d}{dr}[({\int }_r^1|u^{\prime }(s){|}^n{\omega }_{\beta }(s)s^{n-1}ds{)}^{\frac{(1-\tilde{\beta })}{(1-\beta )}}]dr\hfill \\ & =& {(n{\kappa }_n)}^{\frac{(1-\tilde{\beta })}{(1-\beta )}}({\int }_0^1|u^{\prime }(r){{|}^n{\omega }_{\beta }(r)r^{n-1}dr)}^{\frac{(1-\tilde{\beta })}{(1-\beta )}}\hfill \\ & =& {({\int }_{{\mathcal{W}}_1}{F}^n(\nabla u){\omega }_{\beta }dx)}^{\frac{1-\tilde{\beta }}{1-\beta }}\hfill \\ & \le & 1.\hfill \end{array}$$

□

Next, in this paper, we frequently need to change the variable in the following way.

For $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, we change the variable as follows:

$$F^o(x)=e^{-\frac{t}{n}}$$

and set

$$\begin{array}{c}\hfill \psi (t)={\kappa }_n^{\frac{1}{n}}{n}^{\frac{1+(n-1)(1-\beta )}{n}}{(1-\beta )}^{\frac{n-1}{n}}u(x).\end{array}$$

(11)

Then we have ${\psi }^{\prime }(t)=-n^{-\beta \frac{n-1}{n}}{\kappa }_n^{\frac{1}{n}}{(1-\beta )}^{\frac{n-1}{n}}{\tilde{u}}^{\prime }(e^{-\frac{t}{n}})e^{-\frac{t}{n}}$. By Lemma 1 and co-area Formula (8), we can transform the norm as follows:

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{W}}_1}{F}^n(\nabla u){|log{F}^o(x)|}^{\beta (n-1)}dx\hfill \\ \hfill & ={\int }_{{\mathcal{W}}_1}{F}^n({\tilde{u}}^{\prime }(F^o(x))\nabla F^o(x)){|log{F}^o(x)|}^{\beta (n-1)}dx\hfill \\ \hfill & ={\int }_{{\mathcal{W}}_1}{[{\tilde{u}}^{\prime }(F^o(x))|}^n{|log{F}^o(x)|}^{\beta (n-1)}dx\hfill \\ \hfill & ={\int }_0^{1}n{\kappa }_n{[{\tilde{u}}^{\prime }(F^o(x))|}^n{|log{F}^o(x)|}^{\beta (n-1)}{(F^o(x))}^{n-1}d{F}^o(x)\hfill \\ \hfill & ={\int }_0^{+\infty }{\kappa }_n{[{\tilde{u}}^{\prime }(e^{-\frac{t}{n}})|}^n{\left|\frac{t}{n}\right|}^{\beta (n-1)}{e}^{-t}dt\hfill \\ \hfill & ={\int }_0^{+\infty }\frac{|{\psi }^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt.\hfill \end{array}$$

(12)

The functional changes as follows:

$$\begin{array}{c}\hfill \frac{1}{{\kappa }_n}{\int }_{{\mathcal{W}}_1}{e}^{{\lambda }_{n,\beta }{|u|}^{\frac{n}{(n-1)(1-\beta )}}}dx={\int }_0^{+\infty }{e}^{{|\psi |}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \frac{1}{{\kappa }_n}{\int }_{{\mathcal{W}}_1}{e}^{{\lambda |u|}^{\frac{n}{(n-1)(1-\beta )}}}dx={\int }_0^{+\infty }{e}^{\overline{\lambda }{|\psi |}^{\frac{n}{(n-1)(1-\beta )}}-t}dt,\end{array}$$

(14)

where $\overline{\lambda }=\frac{\lambda }{{\lambda }_{n,\beta }}$.

Now it is easy to prove Theorem 1 by Lemma 2.

Proof of Theorem 1. 

Let $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ with ${\parallel u\parallel }_{{\omega }_{\beta }}\le 1$. Define v by (10). By Lemma 2, we have ${\parallel v\parallel }_{{\omega }_{\tilde{\beta }}}\le 1$ for $\forall \tilde{\beta }\le \beta$. By the definition of $AMT(n,{\lambda }_{\beta ,n},\beta )$, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathcal{W}}_1}{e}^{{\lambda }_{\beta ,n}{|u|}^{\frac{n}{(n-1)(1-\beta )}}}dx={\int }_{{\mathcal{W}}_1}{e}^{{\lambda }_{\tilde{\beta },n}{|v|}^{\frac{n}{(n-1)(1-\tilde{\beta })}}}dx\le AMT(n,{\lambda }_{\tilde{\beta },n},\tilde{\beta }).\end{array}$$

(15)

Since (15) holds for $\forall u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ with ${\parallel u\parallel }_{{\omega }_{\beta }}\le 1$, then we have

$$AMT(n,{\lambda }_{\beta ,n},\beta )\le AMT(n,{\lambda }_{\tilde{\beta },n},\tilde{\beta }),$$

for any $0\le \tilde{\beta }\le \beta <1$. Hence, we obtain that the function $\beta \mapsto AMT(n,{\lambda }_{\beta ,n},\beta )$ is decreasing on $[0,1)$. Thus, by the anisotropic Moser–Trudinger-type inequality (3), we obtain $AMT(n,{\lambda }_{\beta ,n},\beta )<\infty$.

Now we prove the constant ${\lambda }_{\beta ,n}$ is sharp. We need to show that, if $\lambda >{\lambda }_{n,\beta }$, $AMT(n,\lambda ,\beta )$ is infinite. By (14), we only need to test

$${\int }_0^{+\infty }{e}^{\overline{\lambda }{|\psi |}^{\frac{n}{(n-1)(1-\beta )}}-t}dt,$$

where $\overline{\lambda }>1$.

Consider the family of functions of Moser’s type

$${\eta }_m(t)=\left\{\begin{array}{cc}\frac{t^{1-\beta }}{m^{\frac{1-\beta }{n}}},\hfill & t\le m,\hfill \\ m^{\frac{(1-\beta )(n-1)}{n}},\hfill & t\ge m.\hfill \end{array}\right.$$

By direct computation, we have ${\int }_0^{+\infty }\frac{|{\eta }_m^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt=1$. However, as $m\to +\infty$,

$${\int }_0^{+\infty }{e}^{\overline{\lambda }{|{\eta }_m|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\ge {\int }_m^{+\infty }{e}^{\overline{\lambda }m-t}dt\to +\infty ,\mathrm{if}\overline{\lambda }>1.$$

The proof of Theorem 1 is completed. □

4. Existence of the Extremal Function

In this section, we complete the proof of Theorem 2. Firstly, we give a uniform bound for $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$. For $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, 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 $0<r<s\le 1$ and $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, we have

$$\begin{array}{ccc}\hfill & \hfill & |u(r)-u(s)|\hfill \\ \hfill & =\hfill & |{\int }_s^r{u}^{\prime }(t)dt|\hfill \\ \hfill & \le \hfill & {\int }_s^r|{\tilde{u}}^{\prime }(t)|t^{\frac{n-1}{n}}{|logt|}^{\frac{(n-1)\beta }{n}}{t}^{-\frac{n-1}{n}}{|logt|}^{-\frac{(n-1)\beta }{n}}dt\hfill \\ \hfill & \le \hfill & {(n{\kappa }_n)}^{-\frac{1}{n}}{(\frac{1}{1-\beta })}^{\frac{n-1}{n}}{({\int }_{{\mathcal{W}}_s\setminus {\mathcal{W}}_r}{F}^n(\nabla u){\omega }_{\beta }dx)}^{\frac{1}{n}}{(-ln\frac{r}{s})}^{\frac{(n-1)(1-\beta )}{n}}\hfill \\ \hfill & =\hfill & {(\frac{n}{{\lambda }_{\beta ,n}})}^{\frac{(n-1)(1-\beta )}{n}}{({\int }_{{\mathcal{W}}_s\setminus {\mathcal{W}}_r}{F}^n(\nabla u){\omega }_{\beta }dx)}^{\frac{1}{n}}{(-ln\frac{r}{s})}^{\frac{(n-1)(1-\beta )}{n}}.\hfill \end{array}$$

(16)

In particular, when $s=1$, for any $0<r\le 1$ and $u\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, we have

$$\begin{array}{c}\hfill |u(r)|\le {(\frac{n}{{\lambda }_{\beta ,n}})}^{\frac{(n-1)(1-\beta )}{n}}{({\int }_{{\mathcal{W}}_1\setminus {\mathcal{W}}_r}{F}^n(\nabla u){\omega }_{\beta }dx)}^{\frac{1}{n}}{(-lnr)}^{\frac{(n-1)(1-\beta )}{n}}.\end{array}$$

(17)

The definition of $\psi (t)$ in (11) and (12) shows that the anisotropic norm changes as

$$\Gamma (\psi ):={\int }_0^{+\infty }\frac{|{\psi }^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt={\int }_{{\mathcal{W}}_1}{F}^n(\nabla u){|log{F}^o(x)|}^{\beta (n-1)}dx.$$

and (13) shows that the functional $I_{\beta }(\psi )$ and $J_{\beta }(u)$ changes as

$$\begin{array}{c}\hfill I_{\beta }(\psi ):={\int }_0^{+\infty }{e}^{{|\psi |}^{\frac{n}{(n-1)(1-\beta )}}-t}dt=\frac{1}{{\kappa }_n}{\int }_{{\mathcal{W}}_1}{e}^{{\lambda }_{n,\beta }{|u|}^{\frac{n}{(n-1)(1-\beta )}}}dx:=J_{\beta }(u).\end{array}$$

(18)

For $\delta \in [0,1)$, we define

$${\tilde{\mathsf{\Lambda }}}_{\delta }=\{\psi \in C^1[0,\infty )|\psi (0)=0,\Gamma (\psi )\le \delta \}.$$

Then the existence of an extremal function in Theorem 1 reduces to find ${\psi }_0\in {\tilde{\mathsf{\Lambda }}}_1$ such that

$$Q_{\beta }:=I_{\beta }({\psi }_0)=\underset{\psi \in {\tilde{\mathsf{\Lambda }}}_1}{sup}{I}_{\beta }(\psi ).$$

(19)

Let ${\tilde{g}}_k(x)$ be a maximizing sequence of (19), that is, $J_{\beta }({\tilde{g}}_k)\to Q_{\beta }$. Since

$${\int }_{{\mathcal{W}}_1}{F}^n(\nabla {\tilde{g}}_k){|log{F}^o(x)|}^{\beta (n-1)}dx\le 1,$$

then there exist a subsequence (still denoted by ${\tilde{g}}_k$) and a function ${\tilde{g}}_0\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ such that

$${\tilde{g}}_k\rightharpoonup {\tilde{g}}_0,{\tilde{g}}_k\to {\tilde{g}}_0\mathrm{pointwise}.$$

(20)

Next, we give an inequality and we will use it several times. For any $h\in C^1[0,\infty )$ and $t\ge A\ge 0$, by the Hölder inequality, we have

$$\begin{array}{ccc}h(t)\hfill & =\hfill & h(A)+{\int }_A^t{h}^{\prime }(s)ds\hfill \\ \hfill & =\hfill & h(A)+{\int }_A^t{h}^{\prime }(s)s^{\frac{\beta (n-1)}{n}}{s}^{-\frac{(n-1)\beta }{n}}ds\hfill \\ \hfill & \le \hfill & h(A)+({\int }_A^t|h^{\prime }(s){{|}^n{s}^{\beta (n-1)}ds)}^{\frac{1}{n}}{({\int }_A^t{s}^{-\beta }ds)}^{\frac{n-1}{n}}\hfill \\ \hfill & =\hfill & h(A)+({\int }_A^t|h^{\prime }(s){{|}^n{s}^{\beta (n-1)}ds)}^{\frac{1}{n}}{(t^{1-\beta }-A^{1-\beta })}^{\frac{n-1}{n}}.\hfill \end{array}$$

(21)

Now we give a lemma involving concentration-compactness alternative, by which we only need to prove that the maximizing sequence ${\tilde{g}}_k(x)$ in (20) does not concentrate at 0, and then we can pass to the limit in the functional. Firstly, we give a definition.

We say a sequence of functions $u_k\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ concentrates at $x=0$, denoted by

$$F^n(\nabla u_k){\omega }_{\beta }dx\rightharpoonup {\delta }_0,$$

if $\parallel u_k{\parallel }_{{\omega }_{\beta }}\le 1$ and any $1>r>0$, ${\int }_{{\mathcal{W}}_1\setminus {\mathcal{W}}_r}{F}^n(\nabla u_k){\omega }_{\beta }dx\to 0$.

Lemma 3.

[Concentration-compactness alternative] For any sequence ${\tilde{v}}_k$, $\tilde{v}\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, such that ${\tilde{v}}_k\rightharpoonup \tilde{v}$ in $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, then up to a subsequence (still denoted by ${\tilde{v}}_k$), either (i) $J_{\beta }({\tilde{v}}_k)\to J_{\beta }(\tilde{v})$, or (ii) ${\tilde{v}}_k$ concentrates at $x=0$.

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 $\delta \in (0,1)$ such that for sufficiently large k,

$${\int }_{{\mathcal{W}}_1\setminus {\mathcal{W}}_{e^{-\frac{A}{n}}}}{F}^n(\nabla {\tilde{v}}_k){|log{F}^o(x)|}^{\beta (n-1)}dx={\int }_0^A\frac{|v_k^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt\ge \delta ,$$

where we use the variable of change

$$F^o(x)=e^{-\frac{t}{n}}\mathrm{and}{\lambda }_{n,\beta }^{\frac{(n-1)(1-\beta )}{n}}{\tilde{v}}_k(x)=v_k(t).$$

(22)

By (21), we have

$$|v_k(t)-v_k{(A)|\le (1-\delta )}^{\frac{1}{n}}{(t^{1-\beta }-A^{1-\beta })}^{\frac{n-1}{n}}\le {(1-\delta )}^{\frac{1}{n}}{t}^{\frac{(n-1)(1-\beta )}{n}}.$$

Since for any k,

$$|v_k(A)|\le A^{\frac{(n-1)(1-\beta )}{n}},$$

we have for $t\ge T$, T sufficiently large,

$$\begin{array}{cc}{v}_k{(t)}^{\frac{n}{(n-1)(1-\beta )}}\hfill & \le {[A^{\frac{(n-1)(1-\beta )}{n}}+{(1-\delta )}^{\frac{1}{n}}{t}^{\frac{(n-1)(1-\beta )}{n}}]}^{\frac{n}{(n-1)(1-\beta )}}\hfill \\ \hfill & \le A+{(1-\frac{\delta }{2})}^{\frac{1}{(n-1)(1-\beta )}}t.\hfill \end{array}$$

(23)

We note that in (23), we applied the inequality if $a>b>0$, $p>1$. Then, for $x\in \mathbb{R}$ large enough, ${(1+ax)}^p\le 1+b^p{x}^p$.

We split the integral $I_{\beta }(v_k)=I_1(v_k)+I_2(v_k)$, where

$$I_1(v_k)={\int }_0^T{e}^{|v_k{(t)|}^{\frac{(n-1)(1-\beta )}{n}}-t}dt,$$

and

$$I_2(v_k)={\int }_T^{\infty }{e}^{|v_k{|}^{\frac{(n-1)(1-\beta )}{n}}-t}dt.$$

Since ${\tilde{v}}_k$ converges pointwise to $\tilde{v}$, then $v_k$ also converges pointwise to v. Then, by $|v_k(t)|\le t^{\frac{(n-1)(1-\beta )}{n}}$ and the dominated convergence theorem, we have that $I_1(v_k)\to I_1(v)$.

By (23), we have for any small $ϵ>0$ and T large enough,

$$\begin{array}{cc}{I}_2(v_k)\hfill & ={\int }_T^{\infty }{e}^{|v_k{|}^{\frac{(n-1)(1-\beta )}{n}}-t}dt\hfill \\ \hfill & \le e^A{\int }_T^{\infty }{e}^{[{(1-\frac{\delta }{2})}^{\frac{1}{(1-\beta )(n-1)}}-1|t}dt,\hfill \end{array}$$

(24)

which is smaller than $ϵ$. Then $I_{\beta }(v_k)\to I_{\beta }(v)$, that is, $J_{\beta }({\tilde{v}}_k)\to J_{\beta }(\tilde{v})$. □

The following lemma is proved in [29]. For $\delta$, $a>0$, let

$${\mathsf{\Lambda }}_{\delta }^a=\{\varphi \in C^1[0,\infty )|\varphi (0)=0,{\int }_a^{\infty }|{\varphi }^{\prime }{|}^{n}dt\le \delta \}.$$

Lemma 4

([29]). For each $a>0$ and $\varphi (t)\in {\mathsf{\Lambda }}_{\delta }^a$, we have

$$\begin{array}{c}\hfill {\int }_a^{\infty }{e}^{{\varphi }^{\frac{n}{n-1}}(t)-t}dt\le \frac{e^{{\varphi }^{\frac{n}{n-1}}(a)-a}}{1-{\delta }^{\frac{1}{n-1}}}{e}^{\frac{c^n}{n}{(\frac{n-1}{n})}^{n-1}{\beta }_n}{e}^{1+\frac{1}{2}+···+\frac{1}{n-1}},\end{array}$$

(25)

where ${\beta }_n=\delta {(1-{\delta }^{\frac{1}{n-1}})}^{-n+1}$ and $c=\frac{n}{n-1}{\varphi }^{\frac{1}{n-1}}(a)$. The inequality tends to an equality if $c^n{\beta }_n\to \infty$, $a\to \infty$ and $\delta \to 0$.

Let ${\tilde{f}}_k(x)\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ such that ${\tilde{f}}_k(x)$ concentrates at 0, that is, $\parallel {\tilde{f}}_k{\parallel }_{{\omega }_{\beta }}\le 1$, $|F^n(\nabla {\tilde{f}}_k)|{\omega }_{\beta }\rightharpoonup {\delta }_0$. Define $f_k(t)$ from ${\tilde{f}}_k(x)$ by the same transformation as in (22). Then, since ${\tilde{f}}_k(x)$ concentrates at 0, we have that ${\tilde{f}}_k(x)\rightharpoonup 0$ in $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$ and converges pointwise to 0.

Lemma 5.

Let $f_k(t)$ be as above. Then one of the following alternatives holds:

(i) We can find points $a_k\in [1,\infty )$ such that

$$|f_k(a_k){|}^{\frac{n}{(n-1)(1-\beta )}}-a_k=-2log{a}_k;$$

(26)

(ii) If such $a_k$ does not exist, then

$$\underset{k\to \infty }{lim\; sup}{\int }_0^{\infty }{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt=1.$$

What is more, if the first alternative (i) holds, we can find $a_k$ to be the first point in $[1,\infty )$ satisfying (26) and satisfying $a_k\to \infty$ as $k\to \infty$.

Proof. 

Since $|f_k(t)|\le t^{\frac{(n-1)(1-\beta )}{n}}$, then if $t\in [0,1)$, $|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t\le 0$. However, if $t\in [0,1)$, $-2logt>0$, then $|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t<-2logt$, which implies that we cannot find $a_k$ satisfying (26) in $[0,1)$.

Now we assume $(i)$ does not hold. Then we have $|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t<-2logt$, $t\in [1,\infty )$. Furthermore, we have

$$e^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}\le t^{-2},\mathrm{if}t\in [1,\infty ).$$

Define the dominating function as follows:

$$h(t)=\left\{\begin{array}{cc}1,\hfill & t\in (0,1),\hfill \\ \frac{1}{t^2},\hfill & t\in [1,\infty ).\hfill \end{array}\right.$$

Then, by the dominated convergence theorem, we obtain that $I_{\beta }(f_k)\to 1$.

Let $(i)$ hold. We choose the first $a_k\ge 1$ satisfying (26). We now prove that $a_k\to \infty$ as $k\to \infty$. For any large number M, we need to prove that there exist $k_0\in \mathbb{N}$, such that for any $k\ge k_0$, $a_k\ge M$. Firstly, we choose $\mu$ small, such that

$$\mu t<-2logt+t,t\in [0,M).$$

Now, since ${\tilde{f}}_k$ concentrates, we have for $t\in [0,M)$ and any $k\ge k_0$,

$$|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}\le {({\int }_0^M\frac{|f_k^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt)}^{\frac{1}{(n-1)(1-\beta )}}t<\mu t\le t-2logt.$$

Then we obtain for any $k\ge k_0$, $a_k\ge M$. □

Now we define the concentration level at 0,

$$J_{\beta ,{\omega }_{\beta }}^{\delta }(0)=\underset{{\tilde{f}}_k\in W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })}{sup}\{\underset{k\to \infty }{lim\; sup}{J}_{\beta }({\tilde{f}}_k)|F^n(\nabla {\tilde{f}}_k){\omega }_{\beta }\rightharpoonup {\delta }_0\}.$$

We can give the estimate for the concentration level.

Lemma 6.

For $\beta \in [0,1)$, we have that

$$J_{\beta ,{\omega }_{\beta }}^{\delta }(0)\le 1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}.$$

(27)

Proof. 

To prove the lemma, it is sufficient to assume the sequences ${\tilde{f}}_k$ satisfy the first alternative in Lemma 5, because if ${\tilde{f}}_k$ satisfy the second alternative, we can obtain the inequality (27) by Lemma 5.

Firstly, we show that

$$\underset{k\to \infty }{lim}{\int }_0^{a_k}{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}=1,$$

where $f_k$ and $a_k$ are as in Lemma 5. Since $F^n(\nabla {\tilde{f}}_k){\omega }_{\beta }\rightharpoonup 0$ and (21), we have that $f_k\to 0$ uniformly on compact subsets of ${\mathbb{R}}^{+}$. Then for any $ϵ$, $A>0$, we obtain $|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}\le ϵ$ for $t\le A$ and k large enough. By the property of $a_k$, that is, for $t\le a_k$, $|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}\le t-2logt$, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{a_k}{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt& =& {\int }_0^A{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt+{\int }_{a_k}^A{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\hfill \\ & \le & e^{ϵ}{\int }_0^A{e}^{-t}dt+{\int }_A^{a_k}{e}^{-2logt}dt\hfill \\ & =& e^{ϵ}(1-e^{-A})+(\frac{1}{A}-\frac{1}{a_k}).\hfill \end{array}$$

Therefore,

$$\underset{k\to \infty }{lim\; sup}{\int }_0^{a_k}{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\le e^{ϵ}(1-e^{-A})+\frac{1}{A}.$$

Now, as $ϵ\to 0$ and $A\to \infty$, we have

$$\underset{k\to \infty }{lim\; sup}{\int }_0^{a_k}{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\le 1.$$

On the other hand,

$$\underset{k\to \infty }{lim\; sup}{\int }_0^{a_k}{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\ge {\int }_0^{a_k}{e}^{-t}dt=1-e^{-a_k}\to 1.$$

Next, we prove that

$$\underset{k\to \infty }{lim}{\int }_{a_k}^{\infty }{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}\le e^{1+\frac{1}{2}+···+\frac{1}{n-1}}.$$

Set ${\delta }_k={\int }_{a_k}^{\infty }\frac{|f_k^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt$. Then, by (21) with $A=0$ and $t=a_k$, we have

$$\begin{array}{ccc}\hfill {\delta }_k=1-{\int }_0^{a_k}\frac{|f_k^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt& \le & 1-{(\frac{|f_k(a_k){|}^{\frac{n}{(n-1)(1-\beta )}}}{a_k})}^{(1-\beta )(n-1)}\hfill \\ & =& 1-{(1-\frac{2log{a}_k}{a_k})}^{(1-\beta )(n-1)}.\hfill \end{array}$$

(28)

Define the function $g_k(t)={|f_k(t)|}^{\frac{1}{1-\beta }}$. Then

$${\int }_{a_k}^{\infty }{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt={\int }_{a_k}^{\infty }{e}^{g_k{(t)}^{\frac{n}{n-1}}-t}dt.$$

(29)

By $|f_k^{\frac{n}{(n-1)(1-\beta )}}(t)|\le t$, we have

$$\begin{array}{ccc}{\int }_{a_k}^{\infty }{|g_k^{\prime }|}^{n}dt\hfill & =\hfill & \frac{1}{{(1-\beta )}^n}{\int }_{a_k}^{\infty }{f}_k^{\frac{n\beta }{1-\beta }}{|f_k^{\prime }|}^{n}dt\hfill \\ \hfill & \le \hfill & \frac{1}{{(1-\beta )}^n}{\int }_{a_k}^{\infty }{t}^{(n-1)\beta }{|f_k^{\prime }|}^{n}dt\hfill \\ \hfill & \le \hfill & \frac{{\delta }_k}{1-\beta }:={\delta }_k^{\ast }\to 0.\hfill \end{array}$$

(30)

Now, applying Lemma 4 with $\delta ={\delta }_k^{\ast }$ and $a=a_k$, we obtain

$$\begin{array}{cc}\hfill & {\int }_{a_k}^{\infty }{e}^{|f_k{(t)|}^{\frac{n}{(n-1)(1-\beta )}}-t}dt\hfill \\ \le \hfill & \frac{e^{g_k{(a_k)}^{\frac{n}{n-1}}}-a_k}{1-|{\delta }_k^{\ast }{|}^{\frac{1}{n-1}}}{e}^{\frac{c^n}{n}{(\frac{n-1}{n})}^{n-1}{\beta }_n+1+\frac{1}{2}+···+\frac{1}{n-1}},\hfill \end{array}$$

(31)

where ${\beta }_n={\delta }_k^{\ast }(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{-n+1}$ and $c=\frac{n}{n-1}{g}_k{(a_k)}^{\frac{1}{n-1}}$. Therefore, it is left to show that

$$\underset{k\to \infty }{lim\; sup}{G}_k:=\underset{k\to \infty }{lim\; sup}[g_k{(a_k)}^{\frac{n}{n-1}}-a_k+\frac{g_k{(a_k)}^{\frac{n}{n-1}}{\delta }_k^{\ast }}{(n-1)(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}|\le 0.$$

We split $G_k$ as follows:

$$\begin{array}{ccc}{G}_k\hfill & =\hfill & -2log{a}_k+\frac{(a_k-2log{a}_k){\delta }_k}{(n-1)(1-\beta )(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}\hfill \\ \hfill & =\hfill & -2log{a}_k+\frac{a_k{\delta }_k}{(n-1)(1-\beta )(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}-\frac{2(log{a}_k){\delta }_k}{(n-1)(1-\beta )(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}\hfill \\ \hfill & =\hfill & \{-2log{a}_k+\frac{a_k{\delta }_k}{(n-1)(1-\beta )}\}+\frac{a_k{\delta }_k(1-(1-|{\delta }_k^{\ast }{|}^{\frac{1}{n-1}}{)}^{n-1})}{(n-1)(1-\beta )(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}\hfill \\ \hfill & \hfill & -\frac{2(log{a}_k){\delta }_k}{(n-1)(1-\beta )(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}}\hfill \\ \hfill & =:\hfill & I_1^k+I_2^k-I_3^k.\hfill \end{array}$$

(32)

We make use of the Maclaurin series expansion. Firstly,

$$\begin{array}{cc}{\delta }_k\hfill & =1-{(-\frac{2log{a}_k}{a_k}+1)}^{(1-\beta )(n-1)}\hfill \\ \hfill & =(1-\beta )(n-1)\frac{2log{a}_k}{a_k}+C{(\frac{2log{a}_k}{a_k})}^2+o({(\frac{log{a}_k}{a_k})}^2),\hfill \end{array}$$

(33)

for some positive constant C, which depends only on $\beta ,n$. Thus, we have

$$|I_1^k|=4C\frac{{(log{a}_k)}^2}{a_k}+a_{k}o({(\frac{log{a}_k}{a_k})}^2)\to 0,\mathrm{as}k\to \infty .$$

Also,

$$|I_3^k|\le C_1\frac{{(log{a}_k)}^2}{a_k}+C_2\frac{{(log{a}_k)}^3}{a_k^2}+(log{a}_k)o({(\frac{log{a}_k}{a_k})}^2)\to 0,\mathrm{as}k\to \infty .$$

To estimate $I_2^k$, we first use the binomial expansion of $(1-|{\delta }_k^{\ast }{{|}^{\frac{1}{n-1}})}^{n-1}$ to obtain $|I_2^k|\le C{a}_k{\delta }_k{|{\delta }_k^{\ast }|}^{\frac{1}{n-1}}$. Now, using (30) and (33), we obtain

$$|I_2^k|\le C\frac{{(log{a}_k)}^{\frac{n}{n-1}}}{a_k^{\frac{1}{n-1}}}\to 0,\mathrm{as}k\to \infty .$$

Then we have completed the proof of the Lemma. □

Proof of Theorem 2. 

We assume $J_{\beta }({\tilde{g}}_k)$ does not converge to $J_{\beta }({\tilde{g}}_0)$, where ${\tilde{g}}_k$, $\tilde{g}$ is as in (20). Thus, by Lemma 6, we obtain

$$M_{\beta }=\underset{k\to \infty }{lim}{J}_{\beta }({\tilde{g}}_k)\le 1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}.$$

If we can find some $\varphi \in {\tilde{\mathsf{\Lambda }}}_1$ such that

$$I_{\beta }(\varphi )>1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}},$$

then clearly $Q_{\beta }>1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}$ and thus, we obtain a contradiction.

Consider the function $h_n(t)$ as follows:

$$h_n(t)=\left\{\begin{array}{cc}(1-\frac{1}{n}){(n-1)}^{-\frac{1}{n}}t,\hfill & 0\le t\le n,\hfill \\ {(t-1)}^{1-\frac{1}{n}},\hfill & n\le t\le T_n,\hfill \\ {(T_n-1)}^{1-\frac{1}{n}},\hfill & t\ge T_n,\hfill \end{array}\right.$$

where $T_n=(n-1)e^{{(\frac{n}{n-1})}^n-\frac{n}{n-1}}+1$. It has been proved in [29] that ${\int }_0^{\infty }{|h_n^{\prime }|}^{n}dt\le 1$ and

$$\begin{array}{cc}{\int }_0^{\infty }{e}^{h_n{(t)}^{\frac{n}{n-1}}-t}dt\hfill & =1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}+{\gamma }^{\ast }(n)\hfill \\ \hfill & >1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}.\hfill \end{array}$$

(34)

Set ${\varphi }_n^{\alpha }(t)={[\alpha h_n(t)|}^{1-\beta }$ for $\alpha \in (0,1)$. Then

$$\begin{array}{ccc}\hfill I_{\beta }({\varphi }_n^{\alpha })& =& {\int }_0^{\infty }{e}^{{[\alpha h_n(t)|}^{\frac{n}{n-1}}-t}dt\hfill \\ & =& {\int }_0^{\infty }{e}^{({\alpha }^{\frac{n}{n-1}}-1+1)h_n{(t)}^{\frac{n}{n-1}}-t}dt\hfill \\ & \ge & e^{({\alpha }^{\frac{n}{n-1}}-1){\parallel h_n\parallel }_{\infty }}(1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}+{\gamma }^{\ast }(n)).\hfill \end{array}$$

Now we can choose $\alpha ={\alpha }_{\ast }$ sufficiently close to 1 such that $I_{\beta }({\varphi }_n^{\alpha })>1+e^{1+\frac{1}{2}+···+\frac{1}{n-1}}$. Let us estimate the term $\Gamma ({\varphi }_n^{{\alpha }_{\ast }})$. Since ${({\varphi }_n^{{\alpha }_{\ast }})}^{\prime }=0$ for $t\ge T_n$, we have

$$\begin{array}{ccc}\Gamma ({\varphi }_n^{{\alpha }_{\ast }})\hfill & =\hfill & {\int }_0^{+\infty }\frac{|{({\varphi }_n^{{\alpha }_{\ast }})}^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt\hfill \\ \hfill & =\hfill & {\int }_0^n\frac{|{({\varphi }_n^{{\alpha }_{\ast }})}^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt+{\int }_n^{T_n}\frac{|{({\varphi }_n^{{\alpha }_{\ast }})}^{\prime }{|}^n{t}^{\beta (n-1)}}{{(1-\beta )}^{n-1}}dt\hfill \\ \hfill & =\hfill & I_1(\beta )+I_2(\beta ).\hfill \end{array}$$

(35)

Now, by direct calculation, we obtain

$$\begin{array}{ccc}\hfill I_1(\beta )& =& (1-\beta ){\alpha }_{\ast }^{(1-\beta )n}{\int }_0^n{h}_n^{-n\beta }{|h_n^{\prime }|}^n{t}^{\beta (n-1)}dt\hfill \\ & =& {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){\int }_0^n{(1-\frac{1}{n})}^{n(1-\beta )}{(n-1)}^{\beta -1}{t}^{-\beta }\hfill \\ & =& {\alpha }_{\ast }^{(1-\beta )n}{(1-\frac{1}{n})}^{n(1-\beta )}{(n-1)}^{\beta -1}{n}^{1-\beta }\hfill \\ & =& {\alpha }_{\ast }^{(1-\beta )n}{(1-\frac{1}{n})}^{(n-1)(1-\beta )}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill I_2(\beta )& =& {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){\int }_n^{T_n}{h}_n^{-n\beta }{|h_n^{\prime }|}^n{t}^{\beta (n-1)}dt\hfill \\ & =& {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){(1-\frac{1}{n})}^n{\int }_{n-1}^{T_n-1}\frac{1}{s}{(1+\frac{1}{s})}^{\beta (n-1)}ds\hfill \\ & \le & {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){(1-\frac{1}{n})}^n{(1+\frac{1}{n-1})}^{\beta (n-1)}{\int }_{n-1}^{T_n-1}\frac{1}{s}ds\hfill \\ & \le & {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){(1-\frac{1}{n})}^n{(\frac{n}{n-1})}^{\beta (n-1)}[{(\frac{n}{n-1})}^n-\frac{n}{n-1}]\hfill \\ & =& {\alpha }_{\ast }^{(1-\beta )n}(1-\beta ){(1-\frac{1}{n})}^{n-1}{(\frac{n}{n-1})}^{\beta (n-1)}{B}_n,\hfill \end{array}$$

where $B_n={(\frac{n}{n-1})}^{n-1}-1$. Note that by the above estimates, we have

$$I_1(0)+I_2(0)\le {\alpha }_{\ast }^n<1.$$

Thus, we can choose $\beta ={\beta }_{\ast }$, depending only on n, such that $I_1({\beta }_{\ast })+I_2({\beta }_{\ast })\le 1$. Thus, we have finished the proof of the Theorem. □

5. 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 $u_k\rightharpoonup u_0$.

Proof of Theorem 3. 

If $u_0\equiv 0$, then we can directly obtain (7) by Theorem 1. Thus, it is left to consider the case $u_0\not\equiv 0$. By (16), we have that $u_k\to u_0$ uniformly on ${\mathcal{W}}_1\setminus {\mathcal{W}}_r$, $\forall r\in (0,1)$. Then, by (17) and dominated convergence theorem, we have $u_k\to u_0$ in $L^q({\mathcal{W}}_1)$ for any $q<\infty$.

For any $R>0$, $k\in \mathbb{N}$, we define the functions

$$v_{R,k}=min\{|u_k|,L\}sign(u_k)\mathrm{and}{w}_{R,k}=u_k-v_{R,k}.$$

Since $\underset{R\to \infty }{lim}\parallel v_{R,0}{\parallel }_{{\omega }_{\beta }}^n={\parallel u_0\parallel }_{{\omega }_{\beta }}^n$, for $\forall p<p(u_0)$, then there exist R large enough such that

$$p_0:=p(1-\parallel v_{R,0}{{\parallel }_{{\omega }_{\beta }}^n)}^{\frac{1}{(n-1)(1-\beta )}}<1.$$

Since $v_{R,k}\to v_{R,0}$ a.e. in ${\mathcal{W}}_1$ as $k\to \infty$ and $v_{R,k}$ is bounded in $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$, up to a subsequence, we can assume that $v_{R,k}\rightharpoonup v_{R,0}$ weakly in $W_{0,rad}^{1,n}({\mathcal{W}}_1,{\omega }_{\beta })$. Then we have

$$\underset{k\to \infty }{lim\; inf}\parallel v_{R,k}{\parallel }_{{\omega }_{\beta }}^n\ge {\parallel v_{R,0}\parallel }_{{\omega }_{\beta }}^n$$

and

$$\underset{k\to \infty }{lim\; sup}\parallel w_{R,k}{\parallel }_{{\omega }_{\beta }}^n=1-\underset{k\to \infty }{lim\; inf}\parallel v_{R,k}{\parallel }_{{\omega }_{\beta }}^n\le 1-{\parallel v_{R,0}\parallel }_{{\omega }_{\beta }}^n.$$

Then we can find $k_0\in \mathbb{N}$ such that for $\forall k\ge k_0$, we have

$$p\parallel w_{R,k}{\parallel }_{{\omega }_{\beta }}^{\frac{n}{(n-1)(1-\beta )}}\le \frac{p_0+1}{2}<1.$$

(36)

Using $u_k=w_{R,k}+v_{R,k}$ and $|v_{R,k}|\le R$, we obtain

$$|u_k{|}^{\frac{n}{(n-1)(1-\beta )}}\le (1+ϵ){|w_{R,k}|}^{\frac{n}{(n-1)(1-\beta )}}+C(n,\beta ,ϵ)R^{\frac{n}{(n-1)(1-\beta )}},$$

where

$$C(n,\beta ,ϵ)={(1-{(1+ϵ)}^{-\frac{(n-1)(1-\beta )}{n\beta +1-\beta }})}^{-\frac{n\beta +1-\beta }{(n-1)(1-\beta )}}.$$

(37)

Now we choose $ϵ>0$ such that $\frac{(1+ϵ)(1+p_0)}{2}<1$. By (36), we have

$$\begin{array}{ccc}\hfill & & {\int }_{{\mathcal{W}}_1}{e}^{p{\lambda }_{\beta ,n}{|u_k|}^{\frac{n}{(n-1)(1-\beta )}}}dx\hfill \\ & \le & {\int }_{{\mathcal{W}}_1}{e}^{p{\lambda }_{\beta ,n}(1+ϵ){|w_{R,k}|}^{\frac{n}{(n-1)(1-\beta )}}+p{\lambda }_{\beta ,n}C(n,\beta ,ϵ)R^{\frac{n}{(n-1)(1-\beta )}}}dx\hfill \\ & \le & C{\int }_{{\mathcal{W}}_1}{e}^{p{\lambda }_{\beta ,n}(1+ϵ)\parallel w_{R,k}{\parallel }_{{\omega }_{\beta }}^{\frac{n}{(n-1)(1-\beta )}}{|\frac{w_{R,k}}{\parallel w_{R,k}{\parallel }_{{\omega }_{\beta }}}|}^{\frac{n}{(n-1)(1-\beta )}}}dx\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& \le & C{\int }_{{\mathcal{W}}_1}{e}^{{\lambda }_{\beta ,n}(1+ϵ)\frac{(1+ϵ)(1+p_0)}{2}{|\frac{w_{R,k}}{\parallel w_{R,k}{\parallel }_{{\omega }_{\beta }}}|}^{\frac{n}{(n-1)(1-\beta )}}}dx\hfill \end{array}$$

(39)

for any $k\ge k_0$, where C depends only on $n,\beta ,ϵ,p$ and R. Combining (38) with Theorem 1 and the choice of $ϵ$, we obtain (7). The proof is completed. □

6. Conclusions

In this paper, we mainly study the anisotropic Moser–Trudinger-type inequality for radical Sobolev space with logarithmic weight ${\omega }_{\beta }(x)={[-ln{F}^o(x)|}^{\beta (n-1)}$, $\beta \in [0,1)$. Moreover, we obtain the existence of an extremal function when $\beta$ 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 ${\mathcal{W}}_1$ and the anisotropic Moser–Trudinger-type inequality with logarithmic weight in ${\mathbb{R}}^n$ are still open questions.