The sharp Hardy--Moser--Trudinger inequality in dimension n
aa r X i v : . [ m a t h . F A ] S e p The sharp Hardy–Moser–Trudinger inequality in dimension n Van Hoang NguyenSeptember 30, 2019
Abstract
In this paper, we prove a Hardy–Moser–Trudinger inequality in the unit ball B n in R n which improves both the classical singular Moser–Trudinger inequality and the classical Hardyinequality at the same time. More precisely, we show that for any β ∈ [0 , n ) there exists aconstant C > n and β such thatsup u ∈ W ,n ( B n ) , H ( u ) ≤ Z B n e (1 − βn ) α n | u | nn − | x | − β dx ≤ C where α n = nω n − n − with ω n − being the surface area of the unit sphere S n − = ∂B , and H ( u ) = Z B n |∇ u | n dx − (cid:18) n − n (cid:19) n Z B n | u | n (1 − | x | ) n dx. This extends an inequality of Wang and Ye in dimension two to higher dimensions and tothe singular case as well. The proof is based on the method of transplantation of Green’sfunctions and without using the blow-up analysis method. As a consequence, we obtain asingular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmativelya conjecture by Mancini, Sandeep and Tintarev [27, Conjecture 5 . R n which is analogue of the conjecture of Wang and Ye in higher dimensions. It is well-known that the Sobolev embedding is a basic and important tool in many aspects ofMathematics such as Analysis, Geometry, Partial of Differential Equations, Calculus of Variations,etc. Let Ω be a bounded domain in R n with n ≥ p ∈ (1 , ∞ ). We denote by W ,p (Ω) the usualfirst order Sobolev space on Ω which is the completion of the space C ∞ (Ω) under the Dirichletnorm k∇ u k L p (Ω) := (cid:0)R Ω |∇ u | p dx (cid:1) p , u ∈ C ∞ (Ω). For 1 < p < n , we have the following well-knownSobolev inequality C (cid:18)Z Ω | u | q dx (cid:19) q ≤ k∇ u k L p (Ω) , u ∈ W ,p (Ω) (1.1)for any 1 ≤ q < p ∗ = np/ ( n − p ) where C > n, p, q and Ω. Inother words, we have the embedding W ,p (Ω) ֒ → L q (Ω) for q ∈ [1 , p ∗ ]. However, in the limit case p = n (thus, p ∗ = ∞ ) the embedding W ,n (Ω) ֒ → L ∞ (Ω) fails. In this situation, it was proved Email: [email protected]
Mathematics Subject Classification : 26D10, 35A23, 46E35
Key words and phrases : Moser–Trudinger inequality, Hardy inequality, Hardy–Moser–Trudinger inequality,sharp constant, Green functions W ,n (Ω) can be embeddedinto an Orlicz space L ϕ n (Ω) generated by the Young function ϕ n ( t ) = e c | t | nn − − c > c . More precisely, wehave the following Moser–Trudinger inequalitysup u ∈ W ,n (Ω) , k∇ u k Ln (Ω) ≤ Z Ω e α | u | nn − dx < ∞ , (1.2)if and only if α ≤ α n := nω n − n − where ω n − denotes the surface area of the unit sphere in R n .The Moser–Trudinger inequality plays the role of the Sobolev inequality in the limit case withmany applications in many branches of Mathematics such as Analysis, Geometry and PartialDifferential Equations, especially in studying the quasi-linear equations with exponential growthnonlinearity. It has been become an interesting subject to study. In fact, there have been manygeneralizations of the Moser–Trudinger inequality in many directions (e.g., to higher order (orfractional order) Sobolev spaces [2, 29], to unbounded domain in R n [1, 18, 21, 41], to singularweighted case [4, 6, 37] or to Riemannian manifolds [5, 26, 27, 33, 36, 51, 52]). In 2004, Adimurthiand Druet improved the Moser–Trudinger inequality (1.2) in dimension 2 by replacing the integral R Ω e α | u | nn − dx ≤ R Ω e α (1+ γ k u k nLn (Ω) ) n − | u | nn − dx with 0 ≤ γ < λ (Ω) := inf {k∇ u k L (Ω) : u ∈ H (Ω); k u k L (Ω) = 1 } . A higher dimension version of the Moser–Trudinger inequality is spiritof Adimurthi and Druet was established by Yang [50]. Tintarev [44] improve the inequality ofAdimurthi and Druet (but still in dimension 2) by replacing the condition k∇ u k L (Ω) ≤ k∇ u k L (Ω) − γ k u k L (Ω) ≤ ≤ γ < λ (Ω). In [35, 36], the author extendsthe result of Tintarev to the higher dimension as well as to the case of the singular–Moser–Trudinger inequality, respectively. Among these generalization of the Moser–Trudinger inequality(1.2), let us quote the singular Moser–Trudinger inequlity due to Adimurthi and Sandeep [4]: forany bounded domain Ω ⊂ R n containing the origin in its interior and β ∈ [0 , n ), it holdssup u ∈ W ,n (Ω) , k∇ u k Ln (Ω) ≤ Z Ω e α (1 − βn ) | u | nn − dx < ∞ , (1.3)if and only if α ≤ α n := nω n − n − .Another important inequality in the unit ball is the Hardy inequality which asserts that Z B n |∇ u | n dx ≥ (cid:18) n − n (cid:19) n Z B n | u | n (1 − | x | ) n dx, u ∈ C ∞ ( B n ) . (1.4)The constant ( n − n ) n is sharp and never attained. Furthermore, it was proved by Mancini,Sandeep and Tintarev (see [27, Lemma 2 . p ∈ ( n, ∞ ) there exists a constant S n,p > n and p such that Z B n |∇ u | n dx − (cid:18) n − n (cid:19) n Z B n | u | n (1 − | x | ) n dx ≥ S n,p (cid:18)Z B n | u | p (1 − | x | ) n dx (cid:19) np , u ∈ C ∞ ( B n ) . (1.5)Denote H ( u ) = Z B n |∇ u | n dx − (cid:18) n − n (cid:19) n Z B n | u | n (1 − | x | ) n dx, u ∈ C ∞ ( B n ) . By (1.5), the functional u → p H ( u ) defines a norm on C ∞ ( B ). Let H ( B ) denote the completionof C ∞ ( B ) under this norm. We have that H ( B ) is a proper subspace of H ( B ). Notice thatwhen n ≥ u → n p H ( u ) defines a norm on C ∞ ( B n ).However, by the density, H is well-defined on W ,n ( B n ).In [47], Wang and Ye obtained another improvement of the Moser–Trudinger inequality (1.2)in the unit disc B ⊂ R which combines both the Moser–Trudinger inequality (1.2) and the Hardy2nequality (2.11). Their inequality states thatsup u ∈H ( B ) , H ( u ) ≤ Z B e πu dx < ∞ . (1.6)The proof of (1.6) given in [47] is based on the blow-up analysis method which is now a standardmethod to study the problems of this type. We refer the readers to [3, 19–21, 35, 37, 41, 47, 49, 53]and references therein for more details on this method. The Hardy–Moser–Trudinger inequality(1.6) is a special case of the inequality of Tintarev [44] aforementioned in which H is replaced bythe functional H V ( u ) = R B |∇ u | dx − R B V u dx for some potential V so that H V satisfies a weakcoercive condition. There have been a lot of generalizations of (1.6) (see [17,22,25,48,49,53]). It isvery remarkable that the inequality (1.6) can be seen as the analogue of the Hardy–Sobolev–Maz’yainequality in dimension two. Recall that the Hardy–Sobolev –Maz’ya inequality (see [31, Section2 . .
6, Corollary 3]) says that there exists a constant
C > u ∈ W , ( B n ) with n >
2, it holds Z B n |∇ u | dx − Z B n | u | (1 − | x | ) dx ≥ C (cid:18)Z B n | u | nn − dx (cid:19) n − n . Moreover, let C n denote the best constant so that the above inequality holds. It is well knownthat C n < S n and is attained if n > C = S and is not attained (see [8]) where S n is the best constant in the Sobolev inequality (1.1) with p = 2 and q = 2 ∗ . The L p version ofthe above inequality in the hyperbolic space was considered in [34] by the author.A new proof of (1.6) without using the blow-up analysis method was recently given by theauthor [38]. This new proof is based on the transplantation of Green functions. This methodwas previously used by Flucher [15] to prove the existence of maximizer for the Moser–Trudingerinequality in dimension two, and then was used by Lin [23] in any dimension. It also was suc-cessfully applied to prove the existence of maximizers for the singular Moser–Trudinger inequal-ity [11–13]. Let us explain briefly on this method. We know that G B ( x ) = − π ln | x | is the Greenfunction of − ∆ in B with pole at 0 and Dirichlet boundary condition. It was proved in [47]that the equation − ∆ u − −| x | ) u = δ in the distribution sense has a unique radial solution G ∈ H ( B ) + W ,p ( B / ) where B r = { x ∈ R : | x | < r } . The function G is strictly decreasingand has the decomposition G = − π ln | x | + C G + ψ ( x ) for some constant C G , where ψ ∈ C ,αloc ( B )and ψ ( x ) = O ( | x | α ) as x → α ∈ (0 , | x | . By abusing notation, we write u ( r ) for the value of u ( x ) with | x | = r and a radial function u . For a radial function u ∈ H ( B ), we define the new radial function v on B such that u ( x ) = v ( e − πG ( x ) ). The main computations in [38] implies that v ∈ W , ( B )and k∇ v k L ( B ) ≤ H ( u ), and Z B e πu dx ≤ e πC G Z B e πv dx where C G appears in the decomposition of the Green function G above. Then the inequality (1.6)follows from the classical Moser–Trudinger (1.2) in B .The Moser–Trudinger inequality in the hyperbolic spaces was established by Mancini andSandeep [26] (see also [5] by Adimurthi and Tintarev). In [27], by using the inequality (1.6),Mancini, Sandeep and Titarev have established the following Moser–Trudinger inequality in thehyperbolic spaces H sup u ∈H ( B ) , H ( u ) ≤ Z B e πu − − πu (1 − | x | ) dx < ∞ . (1.7)In fact, it was show in in [24] that the inequalities (1.6) and (1.7) are equivalent as well. Thehigher dimension version of (1.7) was conjectured in [27] (see the Conjecture 5 .
2) as followssup u ∈ C ∞ ( B n ) , H ( u ) ≤ Z B n e α n | u | nn − − P n − ( α n | u | nn − )(1 − | x | ) n dx < ∞ (1.8)3here P k ( t ) = e t − P ki =0 t k k ! , t ≥ , k ≥
0. It also was shown in [27] thatsup u ∈ C ∞ ( B n ) , H ( u ) ≤ Z B n e α n | u | nn − − P n − ( α n | u | nn − )(1 − | x | ) n dx = ∞ . The original motivation of this paper is to prove the conjectured inequality (1.8). In fact, weshall establish a singular Moser–Trudinger inequality in hyperbolic spaces which is more generalthan (1.8) (see Theorem 1.2 below). In order to prove the conjectured inequality (1.8), we willprove the following singular Hardy–Moser–Trudinger inequality in the unit ball B n which is thefirst main result in this paper. Theorem 1.1.
Let n ≥ and ≤ β < n , then there exists a constant C ( n, β ) depending only on n and β such that sup u ∈ W ,n ( B n ) , H ( u ) ≤ Z B n e (1 − βn ) α n | u | nn − | x | − β dx ≤ C ( n, β ) . (1.9)Obviously, the inequality (1.9) is stronger than the singular Moser–Trudinger inequality (1.3)in B n . Furthermore, it combines both the singular Moser–Trudinger inequality (1.3) and the Hardyinequality (1.4). In the dimension two, the inequality (1.9) was recently proved by Wang [49] byusing the blow-up analysis method following the lines in the proof of Wang and Ye [47]. Our proofof (1.9) is completely different with their proofs. In fact, we follow the arguments in [38] in whichthe new proof of (1.6) is provided. The main feature in the proof is the existence of a Greenfunction G which is the weak solution of the equation − ∆ n G − (cid:18) n − n (cid:19) n G n − (1 − | x | ) n = δ in B n in the distribution sense, where ∆ n G = div( |∇ G | n − ∇ G ) is the n − Laplace operator.The existence of G follows from the deep results of Pinchover and Tintarev concerning to the p − Laplacian problems [39]. Some important properties of G are given in Lemma 2.1 below.It should be notice here that our approach can be applied to prove a more general class ofthe improvements of the singular Moser–Trudinger inequality (1.3) in B n by replacing H ( u ) by H V ( u ) = k∇ u k nL n ( B n ) − R B n V | u | n dx with the potential V satisfying some suitable condition. Thedetails of this fact will be mentioned in the remark at the end of this paper.As a consequence of Theorem 1.1, we obtain the following singular Moser–Trudinger inequalityin the hyperbolic spaces H n which confirms affirmatively the inequality (1.8) of Mancini, Sandeepand Tintarev. Theorem 1.2.
Let n ≥ and ≤ β < n , then there exists a constant ˜ C ( n, β ) depending only on n and β such that sup u ∈ W ,n ( B n ) , H ( u ) ≤ Z B n e (1 − βn ) α n | u | nn − − P n − (cid:16)(cid:16) − βn (cid:17) α n | u | nn − (cid:17) (1 − | x | ) n | x | − β dx < ˜ C ( n, β ) . (1.10)Evidently, when β = 0, the inequality (1.10) is exactly the inequality (1.8). Theorem 1.2 hencenot only confirms affirmatively the inequality (1.8) but also extends this inequality to the singularcase 0 < β < n .It is an interesting question on the extremal functions for the Moser–Trudinger inequality.The existence of extremals for the Moser–Trudinger inequality was first proved by Carleson andChang [10] when Ω = B n (Another proof of this result was given in [14]). Later, this existence resultwas proved for any domain in R by Flucher [15] and for any domain in R n by Lin [23]. Notice thatthe method used in [15, 23] is based on the transplantation of Green functions. This method wassuccessfully applied in [11–13] to prove the existence of extremals for the singular Moser–Trudinger4nequality. For the improved Moser–Trudinger inequality, the existence of extremals was provedin [35, 37, 54] and the references therein. In [19, 20], Li developed a blow-up analysis method toestablish the existence of extremals for the Moser–Trudinger inequality on Riemannian manifolds.Concerning to the Hardy–Moser–Trudinger inequality, it was proved by Wang and Ye [47] (byusing the blow-up analysis method) that the extremals for the inequality (1.6) exists in H ( B ) butnot in W , ( B ). Similarly, again by the the blow-up analysis method, Yang and Zhu proved theexistence of extremals for the improvement version of (1.6) and Wang proved the existence of thesingular Hardy–Moser–Trudinger inequality (1.9) in B . It remains an open question in this paperwhich is whether or not the extremals for the singular Hardy–Moser–Trudinger inequality (1.9)exists when n ≥
3. The main difficult is to determine the suitable space for which the extremals(if exist) belong to. Let us recall that when n ≥
3, we do not know the functional u → H ( u ) n is anorm on C ∞ ( B n ) or not. So we can’t talk about the completion of C ∞ ( B n ) under this functionalalso the weak convergence with respect to this functional. This is the crucial different with thecase n = 2. We will come back this question in the future research.As a final remark, it is well known that for a convex domain domain Ω ⊂ R n the followingHardy’s inequality holds (see, e.g., [28, 30]) Z Ω |∇ u | n dx ≥ (cid:18) n − n (cid:19) n Z Ω | u | n d ( x, ∂ Ω) n dx, u ∈ C ∞ (Ω) , where d ( x, ∂ Ω) = inf {| x − y | : y ∈ ∂ Ω. The constant ( n − n /n n is sharp and never attained.Hence, H Ω ( u ) = Z Ω |∇ u | n dx − (cid:18) n − n (cid:19) n Z Ω | u | n d ( x, ∂ Ω) n dx > , u ∈ C ∞ (Ω) \ { } . We wonder if the inequality (1.9) can be extended to any convex domain Ω in R n . In this direction,we propose the following inequalitysup u ∈ C ∞ (Ω) ,H Ω ( u ) ≤ Z Ω e α n (1 − βn ) | u | nn − | x | − β dx < ∞ . (1.11)Since d ( x, ∂ B n ) = 1 − | x | ≥ −| x | , then the inequality (1.11) holds when Ω = B n by (1.9).In dimension two, the inequality (1.11) for β = 0 was conjectured by Wang and Ye (see [47,Conjecture, page 4]) and was recently settled by Lu and Yang [24].The rest of this paper is organized as follows. In the section § B n . We also prove the existence of the Green function G and its propertiesin this section. Finally, we define a transformation of functions (based on the transplantationof Green functions) and make some useful computations which is useful in the proof of(1.9) insubsection § §
3. We also make somefurther comments on the application of our method to obtain the other improvements of thesingular Moser–Trudinger inequality in B n concerning to the potential V . In this section, we recall some useful facts and make some crucial estimates which will be used inthe proof of Theorem 1.1. We first recall the rearrangement argument applied to the hyperbolicspaces.
In this subsection, we consider the hyperbolic space H n as the unit ball B n equipped with theRiemannian metric g ( x ) = (cid:18) − | x | (cid:19) ( dx + dx + · · · + dx n ) . g is given by dv H n = (cid:16) −| x | (cid:17) n dx and ∇ g = ( −| x | ) ∇ . This model of hyperbolic space is especially useful for questions involvingrotational symmetry. The geodesic distance between x and 0 is given by ρ ( x ) = ln | x | −| x | and wedenote by B H n (0 , r ) the geodesic ball in H n with center at 0 and radius r , i.e., B H n (0 , r ) = { x ∈ H n : ρ ( x ) < r } . For a measurable subset A ∈ H n , we use the notation v H n ( A ) = R A dv H n . Let u be a measur-able function in H n such that v H n ( { x : | u ( x ) | > t } ) < ∞ for any t >
0. The non-increasingrearrangement function of u , denoted by u ∗ , is defined as u ∗ ( x ) = inf { s > v H n ( { x : | u ( x ) | > s } ) ≤ v H n ( B H n (0 , ρ ( x ))) } . From the definition, we have R B n ( u ∗ ) n dv H n = R B n | u | n dv H n . The well-known P´olya–Sz¨ego principlein hyperbolic spaces [7] says that if u ∈ W ,n ( B n ) then u ∗ ∈ W ,n ( B n ) and Z B n |∇ u ∗ | n dx = Z B n |∇ g u ∗ | ng dv H n ≤ Z B n |∇ g u | ng dv H n = Z B n |∇ u | n dx. Thus, H ( u ∗ ) ≤ H ( u ).Furthermore, by the Hardy–Littlewood inequality (see [9]), we have Z B n e (1 − βn ) α n | u | nn − | x | − β dx = 2 − n Z B n e (1 − βn ) α n | u | nn − | x | − β (1 − | x | ) n dv H n ≤ − n Z B n e (1 − βn ) α n | u ∗ | nn − | x | − β (1 − | x | ) n dv H n = Z B n e (1 − βn ) α n | u ∗ | nn − | x | − β dx by noticing that the rearrangement of | x | − β (1 − | x | ) n is just itself. Therefore we only need toconsider nonincreasing, radially symmetric functions in proving (1.9). Let us defineΣ = { u ∈ C ∞ ( B n ) : u ( x ) = u ( r ) with | x | = r ; u ′ ≤ } , and H be the closure of Σ in W ,n ( B n ). So, to prove Theorem 1.1, we need only to show thatthere exists some constant C ( n, β ) depending only on n and β such thatsup u ∈ Σ , H ( u ) ≤ Z B n e (1 − βn ) α n | u | nn − | x | − β dx ≤ C ( n, β ) . Throughout this subsection, we denote by V ( x ) = (cid:18) n − n (cid:19) n − | x | ) n , and Q V ( u ) = H ( u ) , u ∈ C ∞ ( B n ), i.e., Q V ( u ) = Z B n |∇ u | n dx − Z B n V ( x ) | u ( x ) | n dx, u ∈ C ∞ ( B n ) . We have Q V ≥ C ∞ ( B n ) by the Hardy inequality (1.4). By [39, Theorem 5 . Q ′ V ( u ) = 0 has (up to a multiple constant) a unique positive solution v in B n \ { } of minimalgrowth in a neighborhood of infinity in B n (see [39, Definition 5 .
3] for the definition of positive6olution of minimal growth in a neighborhood of infinity). Furthermore, v is either a globalminimal solution of the equation Q ′ V ( u ) = 0 in B n , or v has a nonremovable singularity at 0.By the Hardy–Sobolev inequality (1.5), there exists a positive constant C > Q V ( u ) ≥ C Z B n | u | n dx, u ∈ C ∞ ( B n ) . In terminology of [39, Definition 1 . Q V has a weighted spectral gap in B n (or Q V is strictly positive in B n ). This fact together with [39, Theorem 5 .
5] implies that the solution v of the equation Q ′ V ( u ) = 0 in B n \ { } above has a nonremovable singularity at 0. By Lemma 5 . x → v ( x ) − ln | x | = C for some C >
0. By normalizing, we assume this solution satisfieslim x → v ( x ) − ln | x | = ω − n − n − . (2.1)Let G ( x ) denote such a solution v , and we call it the Green function of the equation Q ′ V ( u ) = 0in B n with a pole at 0. It is not hard to see that G is the weak solution of the equation − ∆ n G − (cid:18) n − n (cid:19) n G n − (1 − | x | ) n = δ in the distribution sense in B n . We have the following results on G . Lemma 2.1. G is radially symmetric and strictly decreasing in | x | . There exists C > such that G ( x ) ≤ C (1 − | x | ) n − n , ≤ | x | < . (2.2) Furthermore, we have the following decomposition of GG ( x ) = − ω − n − n − ln | x | + C G + H ( x ) , (2.3) with H ∈ C ,αloc ( B ) and H ( r ) = O ( r α ) as r → for any α ∈ (0 , .Proof. Since V ∈ C ∞ ( B n ) and G ∈ W ,nloc ( B n \ { } ), then by the standard regularity [43, 45]we have G ∈ C ( B n \ { } ). For any R ∈ O ( n ) the group of the n × n orthogonal matrices.Denote G R ( x ) = G ( Rx ) , x ∈ B n \ { } . It is easy to check that G R is a solution of the equation Q ′ V ( u ) = 0 in B n \ { } and satisfies (2.1). Hence G R ≡ G by the uniqueness. In other word, wehave G ( Rx ) = G ( x ) for any R ∈ O ( n ). This implies that G is radially symmetric in | x | .By (2.1), we have G ∈ L ploc ( B n , dv H n ) for any p < ∞ . For any 0 < a < b <
1, we chose k > k a ≥ k (1 − b ) >
1. For any k ≥ k , we define the function ψ k ( x ) = ≤ | x | < a − k or b + k ≤ | x | < − k ( a − | x | ) if a − k ≤ | x | < a a ≤ | x | < b − k ( | x | − b ) if b ≤ | x | < b + k .Testing the equation Q ′ V ( G ) = 0 by ψ k and using the radially symmetric of G , we have ω n − k Z aa − k | G ′ ( r ) | n − G ′ ( r ) r n − dr − k Z b + k b | G ′ ( r ) | n − G ′ ( r ) r n − dr ! = Z B n V G N − ψ k dx. k → ∞ and using the facts G ∈ C ( B n \ { } ) and G ∈ L ploc ( B n , dv H n ) for any p < ∞ andusing the Lebesgue dominated convergence theorem, we get ω n − (cid:0) | G ′ ( a ) | N − G ′ ( a ) a N − − | G ′ ( b ) | N − G ′ ( b ) b N − (cid:1) = Z { a ≤| x |
1) such that a i → i → ∞ and G ′ ( a i ) <
0. Thisfact together with (2.4) implies G ′ ( r ) < < r <
1. Hence G is strictly decreasing in | x | .We next prove (2.2). Let B r = { x : | x | < r } for 0 < r <
1. From the proof of Theorem 5 . G is locally uniform limit in B n \ { } of the sequence G N , N ≥ Q ′ V ( G N ) = 0 in B − N \ { } and satisfies the condition G N = 0 on ∂B − N andlim x → G N ( x ) − ln | x | = ω − n − n − . Fix a δ ∈ (0 , G N ( x ) ≤ C δ on ∂B δ for any N ≥ C δ > δ . Let ψ ( x ) = ( − ln | x | ) n − n . By a direct computation, we have − ∆ n ψ ( x ) − V ( x ) ψ ( x ) n − n = (cid:18) n − n (cid:19) n ψ ( x ) n − | x | n (cid:18) − ln | x | ) n − (cid:18) | x | − | x | (cid:19) n (cid:19) . Using the elementary inequality − r ln r ≤ − r , r ∈ (0 , , we obtain that − ∆ n ψ ( x ) − V ( x ) ψ ( x ) n − n > , x ∈ B n \ { } . Notice that ψ > ∂B − N . Furthermore, multiplying ψ by a large constant C , we see that Cψ ≥ C / ≥ G N on ∂B / for any N . Applying the comparison principle (see [39, Theorem 2 . G N ( x ) ≤ Cψ ( x ) for any N and ≤ | x | <
1. Letting N → ∞ wehave G ( x ) ≤ C ( − ln | x | ) n − n ≤ ˜ C (1 − | x | ) n − n , ≤ | x | < . as wanted.From (2.4), we see that there existslim a → | G ′ ( a ) | N − G ′ ( a ) a n − = | G ′ ( b ) | n − G ′ ( b ) b n − + ω − n − Z B b V G n − dx. (2.5)Notice that G ′ <
0, hence there exists the limitlim r → − G ′ ( r ) r = γ ≥ . This limit together with (2.1) and L’Hˆopital theorem implies γ = ω − n − n − . Furthermore, we havefrom (2.5) | G ′ ( b ) | n − G ′ ( b ) b n − + ω − n − Z B b V G n − dx = − γ n − , ∀ < b < , or equivalently, − ω n − n − G ′ ( b ) b = (cid:18) Z B b V G n − dx (cid:19) n − , ∀ < b < . (2.6)Again, from (2.4), we get − G ′ ( r ) r = (cid:18) γ n − + ω − n − n − Z B r V G n − dx (cid:19) n − = γ + ψ ( r ) , ψ ( r ) = (cid:18) γ n − + ω − n − n − Z B r V G n − dx (cid:19) n − − γ. From (2.1), we have ψ ( r ) = O (( − ln r ) n − r n ) as r →
0. Furthermore, we have ψ ∈ C ,αloc ( B ) forany α ∈ (0 , − G ′ ( r ) − γr = ψ ( r ) r = O (( − ln r ) n − r n − ) (2.7)as r → < s < r | − G ( r ) − γ ln( r ) − ( − G ( s ) − γ ln s ) | = Z rs ψ ( t ) t dt → r, s →
0. Hence, there exits the limits lim r → ( − G ( r ) − γ ln( r )) = − C G . Hence, we get from(2.7) that − G ( r ) − γ ln r + C G = Z r ψ ( s ) s ds. Let H ( r ) = − R r ψ ( s ) s ds , we obtain G ( r ) = − ω − n − n − ln r + C G + H ( r )by noticing that γ = ω − n − n − . From the definition of H , we have H ( r ) = O (( − ln r ) n − r n ) as r → H ∈ C ,αloc ( B ) for any α ∈ (0 , Let us recall that the n -Green function with pole at 0 of the operator − ∆ n in B is given G B n ( x ) = − ω − n − n − ln | x | , i.e., G B n is the weak solution of the equation − ∆ n G B n = δ in B n and G B n = 0 on ∂B .Let u ∈ Σ be a given function and we define a new function v in B n by v ( r ) = u ( G − ◦ G B n ( r )) (2.8)or equivalently u ( r ) = v ( e − ω n − n − G ( r ) ) . A simple computation shows u ′ ( r ) = − v ′ ( e − ω n − n − G ( r ) )) e − ω n − n − G ( r ) ω n − n − G ′ ( r ) . Thus, we have Z B n |∇ u | n dx = ω n − Z | u ′ ( r ) | n r n − dr = ω n − Z | v ′ ( e − ω n − n − G ( r ) )) e − ω n − n − G ( r ) ω n − n − G ′ ( r ) | n r n − dr. Making the change of variable t = e − ω n − n − G ( r ) and define a ( t ) = G − ( − ω − n − n − ln t ) . G is strictly decreasing, then a is strictly increasing, a (0) = 0 and a (1) = 1. Fur-thermore, a ∈ C ([0 , r = a ( t ) and dr = − ( G − ) ′ ( − ω − n − n − ln t ) ω − n − n − t − dt and Z B n |∇ u | n dx = − ω n − Z | v ′ ( t ) | n t n ω nn − n − | G ′ ( a ( t )) | n a ( t ) n − ( G − ) ′ ( − ω − n − n − ln t ) ω − n − n − t − dt = ω n − Z | v ′ ( t ) | n t n − ω n − | G ′ ( a ( t )) | n − a ( t ) n − dt = ω n − Z | v ′ ( t ) | n t n − dt + ω n − Z | v ′ ( t ) | n t n − Φ( t ) dt, (2.9)with Φ( t ) = ω n − | G ′ ( a ( t )) | n − a ( t ) n − − > , ∀ t ∈ (0 , G ′ ( G − ( a ))( G − ) ′ ( a ) = 1 for the second equality. Note thatΦ ′ ( t ) = − ω n − ∆ n G ( a ( t )) a ( t ) n − a ′ ( t )= (cid:18) n − n (cid:19) n ( − ln t ) n − a ( t ) n − (1 − a ( t ) ) n − ω n − n − G ′ ( a ( t )) t In the other hand (cid:18) n − n (cid:19) n Z B n | u | n (1 − | x | ) n dx = (cid:18) n − n (cid:19) n ω n − Z | v ( e − ω n − n − G ( r ) ) | n (1 − r ) n r n − dr = (cid:18) n − n (cid:19) n ω n − Z v ( t ) n a ( t ) n − (1 − a ( t ) ) n dt − ω n − n − G ′ ( a ( t )) t = ω n − Z v ( t ) n Φ ′ ( t )( − ln t ) n − dt. (2.10)To continue, we need a Hardy type inequality as follows Lemma 2.2.
For any v ∈ C ([0 , which is non-increasing, it holds Z | v ′ ( t ) | n t n − Φ( t ) dt ≥ Z v ( t ) n Φ ′ ( t )( − ln t ) n − dt. (2.11) Proof.
Let v ( t ) = w ( t )( − ln t ). We have v ′ ( t ) = w ′ ( t )( − ln t ) − w ( t ) t . Notice that v ′ ( t ) ≤
0. Usingthe simple inequality | a − b | n ≥ | b | n − nb n − a + | a | n , for any b ≥ a − b ≤
0, we get | v ′ ( t ) | n ≥ w ( t ) n t n + n w ( t ) n − w ′ ( t ) t n − ln t + | w ′ ( t ) | n ( − ln t ) n = ( w ( t ) n ln t ) ′ t − n + | w ′ ( t ) | n ( − ln t ) n . Using integration by parts, we get Z | v ′ ( t ) | n t n − Φ( t ) dt ≥ Z ( w ( t ) n ln t ) ′ Φ( t ) dt + Z | w ′ ( t ) | n ( − ln t ) n t n − Φ( t ) dt ≥ Z w ( t ) n ( − ln t )Φ ′ ( t ) dt = Z v ( t ) n Φ ′ ( t )( − ln t ) n − dt, as desired. 10ombining (2.9), (2.10) and (2.11), we arrive H ( u ) n ≥ Z B n |∇ v | n dx. (2.12)The inequality (2.12) is the key in our proof of Theorem 1.1. With the estimate (2.12) at hand, we are ready to prove the inequality (1.9) in Theorem 1.1.
Proof of Theorem 1.1.
As mentioned in subsection § u ∈ Σ with H ( u ) ≤
1, we define the new function v by (2.8). Notice that v ∈ W ,n ( B n ) and by (2.12) we have R B n |∇ v | n dx ≤ Z B n e (1 − βn ) α n | u | nn − | x | − β dx = ω n − Z e (1 − βn ) α n v ( e − ω n − n − G ( r ) )) nn − r n − β − dr = ω n − Z e (1 − βn ) α n v ( t ) nn − t n − β − (cid:18) a ( t ) t (cid:19) n − β − ω n − n − G ′ ( a ( t )) a ( t ) dt = ω n − Z e α n v ( t ) nn − t n − β − Ψ( t ) dt, with Ψ( t ) = 1 − ω n − n − G ′ ( a ( t )) a ( t ) a ( t ) n − β t n − β . By (2.6), we have − ω n − n − G ′ ( a ( t )) a ( t ) > , ∀ t ∈ (0 , . From the definition of a ( t ), we have G ′ ( a ( t )) a ′ ( t ) = − ω − n − n − t − , hence (cid:18) a ( t ) t (cid:19) ′ = a ′ ( t ) t − a ( t ) t = − − ω n − n − G ′ ( a ( t )) a ( t ) ω n − n − t G ′ ( a ( t ) < , ∀ t ∈ (0 , G ′ <
0. Then the function a ( t ) /t is strictly decreasing. Furthermore, from (2.3) we have G ( r ) = − ω − n − n − ln r + C G + H ( r )which implies a ( t ) t = e ω n − n − ( C G + H ( a ( t ))) . So, we have a ( t ) t < lim t → a ( t ) t = e ω n − n − C G , ∀ t ∈ (0 , t → a ( t ) = 0. Therefore, it holds Z B n e (1 − βn ) α n | u | nn − | x | − β dx ≤ e (1 − βn ) α n C G Z B n e (1 − βn ) | v | nn − | x | − β dx.
11n the light of the classical singular Moser–Trudinger inequality (1.3) in B n , it holds Z B n e (1 − βn ) α n | u | nn − | x | − β dx ≤ e (1 − βn ) α n C G sup w ∈ W ,n ( B ) , R B n |∇ w | n dx ≤ Z B n e (1 − βn ) α n | w | nn − | x | − β dx =: C ( n, β ) < ∞ , for any u ∈ Σ with H ( u ) ≤
1. This finishes the proof of Theorem 1.1.We next prove Theorem 1.2.
Proof of Theorem 1.2.
Again, by the standard rearrangement argument in the hyperbolic spacesfrom subsection § u ∈ Σ with H ( u ) ≤ u , we have u ( r ) ≤ C n,p (1 − r ) n − p , ∀ r ∈ (1 / , , here p > n is any number and C n,p depends only on n and p (see [27, Lemma 5 . e (1 − βn ) α n | u | nn − − P n − (cid:16)(cid:16) − βn (cid:17) α n | u | nn − (cid:17) ≤ ˜ C n,p,β (1 − r ) n p , ∀ r ∈ (1 / , , here p > n is any number and ˜ C n,p,β depends only on n, p and β . Choosing p such that n < p < n n − hence n p − n + 1 >
0. By splitting the integral, we have Z B n e (1 − βn ) α n | u | nn − − P n − (cid:16)(cid:16) − βn (cid:17) α n | u | nn − (cid:17) (1 − | x | ) n | x | − β dx = Z {| x |≤ } e (1 − βn ) α n | u | nn − − P n − (cid:16)(cid:16) − βn (cid:17) α n | u | nn − (cid:17) (1 − | x | ) n | x | − β dx + Z { < | x | < } e (1 − βn ) α n | u | nn − − P n − (cid:16)(cid:16) − βn (cid:17) α n | u | nn − (cid:17) (1 − | x | ) n | x | − β dx ≤ (cid:16) (cid:17) n Z {| x |≤ } e (1 − βn ) α n | u | nn − | x | − β dx + ˜ C n,p,β Z { < | x | < } (1 − | x | ) n p − n dx ≤ (cid:16) (cid:17) n Z B n e (1 − βn ) α n | u | nn − | x | − β dx + ˜ C n,p,β Z { < | x | < } (1 − | x | ) n p − n dx =: ˜ C ( n, β ) < ∞ , here we use the inequality (1.9) in Theorem 1.1 and the fact n p − n + 1 >
0. This completes theproof of Theorem 1.2.Finally, we make some further comments on our approach in this paper to the other improve-ment of the singular Moser–Trudinger inequality (1.3) in B n . Let V : B n → (0 , ∞ ) be a radiallysymmetric, continuous potential such that (1 −| x | ) n V ( x ) is non-increasing in | x | (this assumption12nables us to apply the rearrangement argument in the hyperbolic spaces). We further assumethat the functional Q V ( u ) = Z B n |∇ u | n dx − Z B n V | u | n dx has a spectral gap (or strictly positive) in B n in the sense of [39, Definition 1 . § − ∆ n u − V u n − = δ , in the distribution sense in B n has a unique radially symmetric, strictly decreasing (in | x | ), positivesolution G . Hence, there exists a = lim r → G ( r ) ≥
0. We show that a = 0. Indeed, from theproof of Theorem 5 . G is locally uniform limit in B n \ { } of the sequence G N which solves the equation Q ′ V ( G N ) = 0 in B − N \ { } and satisfies the condition G N = 0 on ∂B − N and lim x → G N ( x ) − ln | x | = ω − n − n − . If a >
0, denote H ( x ) = G ( x ) − a . We have − ∆ n H − V H n − = − ∆ n G − V H n − = V ( G n − − H n − ) ≥ { x : < | x | < } . Notice that H > B n and G N is uniformly bounded in ∂B sowe can apply the comparison principle (see [39, Theorem 2 .
2] or [16, Theorem 5]) to get that G N ( x ) ≤ CH ( x ) for any ≤ | x | <
1, and any N ≥ C ≥
1. Letting N → ∞ we get G ( x ) ≤ CH ( x ) for some C ≥
1. Letting | x | → a = 0. Therefore,the function G : (0 , → (0 , ∞ ) is a bijection. Furthermore, by the same arguments in subsection § G has the form G ( x ) = − ω − n − n − ln | x | + C G + ψ ( x )with ψ ∈ C ,αloc ( B n ) and ψ ( r ) = O ( r α ) as r → α ∈ (0 , u ∈ W ( B n ) ,Q V ( u ) ≤ Z B n e (1 − βn ) α n | u | nn − | x | − β dx < ∞ , (3.1)for any 0 ≤ β < n . In dimension two, the inequality (3.1) was considered by Tintarev [44] when β = 0. A special example of the potential V which satisfies our assumptions in V ( x ) = α ∈ [0 , λ ,n ( B n )) where λ ,n ( B n ) = inf (cid:26)Z B n |∇ u | n dx : u ∈ W ,n ( B n ); Z B n | u | n dx = 1 (cid:27) . In this case, we obtain the results in [35, 37] from (3.1)sup u ∈ W ,n ( B n ) , k∇ u k nLn ( B n ) − α k u k nLn ( B n ) ≤ Z B n e α n (1 − βn ) | u | nn − | x | − β dx < ∞ , β ∈ [0 , n ) . Another example is the improvement of the singular Hardy–Moser–Trudinger inequality (1.9). Let λ = inf (cid:26) H ( u ) : u ∈ C ∞ ( B n ); Z B n | u | n dx = 1 (cid:27) . The Poincar´e–Sobolev inequality (1.5) implies that λ >
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