On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem
aa r X i v : . [ m a t h . A P ] M a y ON A WEIGHTED TRUDINGER-MOSER TYPE INEQUALITY ONTHE WHOLE SPACE AND RELATED MAXIMIZING PROBLEM
VAN HOANG NGUYEN AND FUTOSHI TAKAHASHI
Abstract.
In this paper, we establish a weighted Trudinger-Moser type inequality withthe full Sobolev norm constraint on the whole Euclidean space. Main tool is the singularTrudinger-Moser inequality on the whole space recently established by Adimurthi andYang, and a transformation of functions. We also discuss the existence and non-existenceof maximizers for the associated variational problem. Introduction
Let Ω ⊂ R N , N ≥ W ,N (Ω) ֒ → L q (Ω) for any q ∈ [1 , + ∞ ), however, as the functionlog (log( e/ | x | )) ∈ W ,N ( B ), B the unit ball in R N , shows, the embedding W ,N (Ω) ֒ → L ∞ (Ω) does not hold. Instead, functions in W ,N (Ω) enjoy the exponential summability: W ,N (Ω) ֒ → { u ∈ L N (Ω) : Z Ω exp (cid:16) α | u | NN − (cid:17) dx < ∞ for any α > } , see Yudovich [31], Pohozaev [26], and Trudinger [30]. Moser [22] improved the aboveembedding as follows, now known as the Trudinger-Moser inequality: Define T M ( N, Ω , α ) = sup u ∈ W ,N k∇ u k LN (Ω) ≤ | Ω | Z Ω exp( α | u | NN − ) dx. Then we have
T M ( N, Ω , α ) ( < ∞ , α ≤ α N , = ∞ , α > α N , here and henceforth α N = N ω N − N − and ω N − denotes the area of the unit sphere S N − in R N . On the attainability of the supremum, Carleson-Chang [6], Flucher [13], and Lin [17]proved that T M ( N, Ω , α ) is attained on any bounded domain for all 0 < α ≤ α N . Date : August 29, 2018.2010
Mathematics Subject Classification.
Primary 35A23; Secondary 26D10.
Key words and phrases.
Trudinger-Moser inequality, weighted Sobolev spaces, maximizing problem.
Later, Adimurthi-Sandeep [2] established a weighted (singular) Trudinger-Moser inequal-ity as follows: Let 0 ≤ β < N and put α N,β = (cid:0) N − βN (cid:1) α N . Define g T M ( N, Ω , α, β ) = sup u ∈ W ,N k∇ u k LN (Ω) ≤ | Ω | Z Ω exp( α | u | NN − ) dx | x | β . Then it is proved that g T M ( N, Ω , α, β ) ( < ∞ , α ≤ α N,β , = ∞ , α > α N,β . On the attainability of the supremum, recently Csat´o-Roy [10], [11] proved that g T M (2 , Ω , α, β )is attained for 0 < α ≤ α ,β = 2 π (2 − β ) for any bounded domain Ω ⊂ R . For other typesof weighted Trudinger-Moser inequalities, see for example, [7], [8], [9], [14], [18], [28], [29],[32], to name a few.On domains with infinite volume, for example on the whole space R N , the Trudinger-Moser inequality does not hold as it is. However, several variants are known on the wholespace. In the following, let Φ N ( t ) = e t − N − X j =0 t j j !denote the truncated exponential function.First, Ogawa [23], Ogawa-Ozawa [24], Cao [5], Ozawa [25], and Adachi-Tanaka [1] provedthat the following inequality holds true, which we call Adachi-Tanaka type Trudinger-Moserinequality: Define A ( N, α ) = sup u ∈ W ,N ( R N ) \{ }k∇ u k LN ( R N ) ≤ k u k NL N ( R N ) Z R N Φ N ( α | u | NN − ) dx. (1.1)Then(1.2) A ( N, α ) ( < ∞ , α < α N , = ∞ , α ≥ α N . The functional in (1.1) F ( u ) = 1 k u k NL N ( R N ) Z R N Φ N ( α | u | NN − ) dx enjoys the scale invariance under the scaling u ( x ) u λ ( x ) = u ( λx ) for λ >
0, i.e., F ( u λ ) = F ( u ) for any u ∈ W ,N ( R N ) \ { } . Note that the critical exponent α = α N isnot allowed for the finiteness of the supremum. On the attainability of the supremum,Ishiwata-Nakamura-Wadade [16] proved that A ( N, α ) is attained for any α ∈ (0 , α N ). Inthis sense, Adachi-Tanaka type Trudinger-Moser inequality has a subcritical nature of theproblem. On the other hand, Ruf [27] and Li-Ruf [20] proved that the following inequality holdstrue: Define B ( N, α ) = sup u ∈ W ,N ( R N ) k u k W ,N ( R N ) ≤ Z R N Φ N ( α | u | NN − ) dx. (1.3)Then(1.4) B ( N, α ) ( < ∞ , α ≤ α N , = ∞ , α > α N . Here k u k W ,N ( R N ) = (cid:16) k∇ u k NL N ( R N ) + k u k NL N ( R N ) (cid:17) /N is the full Sobolev norm. Note thatthe scale invariance ( u u λ ) does not hold for this inequality. Also the critical exponent α = α N is permitted to the finiteness of (1.3). Concerning the attainability of B ( N, α ),it is known that B ( N, α ) is attained for 0 < α ≤ α N if N ≥ N = 2, there exists an explicit constant α ∗ > R such that B (2 , α ) is attained for α ∗ < α ≤ α (= 4 π ) [27], [15]. However,if α > B (2 , α ) is not attained [15]. The non-attainabilityof B (2 , α ) for α sufficiently small is attributed to the non-compactness of “vanishing”maximizing sequences, as described in [15].In the following, we are interested in the weighted version of the Trudinger-Moser in-equalities on the whole space. Let N ≥ −∞ < γ < N and define the weighted Sobolevspace X ,Nγ ( R N ) as X ,Nγ ( R N ) = ˙ W ,N ( R N ) ∩ L N ( R N , | x | − γ dx )= { u ∈ L loc ( R N ) : k u k X ,Nγ ( R N ) = (cid:0) k∇ u k NN + k u k NN,γ (cid:1) /N < ∞} , where we use the notation k u k N,γ for (cid:16)R R N | u | N | x | γ dx (cid:17) /N . We also denote by X ,Nγ,rad ( R N ) thesubspace of X ,Nγ ( R N ) consisting of radial functions. We note that a special form of theCaffarelli-Kohn-Nirenberg inequality in [4]:(1.5) k u k N,β ≤ C k u k N − βN − γ N,γ k∇ u k − N − βN − γ N implies that X ,Nγ ( R N ) ⊂ X ,Nβ ( R N ) when γ ≤ β . From now on, we assume(1.6) N ≥ , −∞ < γ ≤ β < N and put α N,β = (cid:0) N − βN (cid:1) α N .Recently, Ishiwata-Nakamura-Wadade [16] proved that the following weighted Adachi-Tanaka type Trudinger-Moser inequality holds true: Define(1.7) ˜ A rad ( N, α, β, γ ) = sup u ∈ X ,Nγ,rad ( R N ) \{ }k∇ u k LN ( R N ) ≤ k u k N ( N − βN − γ ) N,γ Z R N Φ N ( α | u | NN − ) dx | x | β . VAN HOANG NGUYEN AND FUTOSHI TAKAHASHI
Then for
N, β, γ satisfying (1.6), we have˜ A rad ( N, α, β, γ ) ( < ∞ , α < α N,β , = ∞ , α ≥ α N,β . (1.8)Later, Dong-Lu [12] extends the result in the non-radial setting. Let(1.9) ˜ A ( N, α, β, γ ) = sup u ∈ X ,Nγ ( R N ) \{ }k∇ u k LN ( R N ) ≤ k u k N ( N − βN − γ ) N,γ Z R N Φ N ( α | u | NN − ) dx | x | β . Then the corresponding result holds true also for ˜ A ( N, α, β, γ ). Attainability of the bestconstant (1.7), (1.9) is also considered in [16] and [12]: ˜ A rad ( N, α, β, γ ) and ˜ A ( N, α, β, γ )are attained for any 0 < α < α
N,β .First purpose of this note is to establish the weighted Li-Ruf type Trudinger-Moserinequality on the weighted Sobolev space X ,Nγ ( R N ) with N, β, γ satisfying (1.6). Define˜ B rad ( N, α, β, γ ) = sup u ∈ X ,Nγ,rad ( R N ) k u k X ,Nγ ( R N ) ≤ Z R N Φ N ( α | u | NN − ) dx | x | β , (1.10) ˜ B ( N, α, β, γ ) = sup u ∈ X ,Nγ ( R N ) k u k X ,Nγ ( R N ) ≤ Z R N Φ N ( α | u | NN − ) dx | x | β . (1.11) Theorem 1. (Weighted Li-Ruf type inequality) Assume (1.6) and put α N,β = (cid:0) N − βN (cid:1) α N .Then we have ˜ B rad ( N, α, β, γ ) ( < ∞ , α ≤ α N,β , = ∞ , α > α N,β . (1.12) Furthermore if ≤ γ ≤ β < N , we have ˜ B ( N, α, β, γ ) ( < ∞ , α ≤ α N,β , = ∞ , α > α N,β . (1.13)We also study the existence and non-existence of maximizers for the weighted Trudinger-Moser inequalities (1.12) and (1.13). Theorem 2.
Assume (1.6). Then the following statements hold. (i) If N ≥ then ˜ B rad ( N, α, β, γ ) is attained for any < α ≤ α N,β . (ii) If N = 2 then ˜ B rad (2 , α, β, γ ) is attained for any < α ≤ α ,β if β > γ , while thereexists α ∗ > such that ˜ B rad (2 , α, β, β ) is attained for any α ∗ < α < α ,β . (iii) ˜ B rad (2 , α, β, β ) is not attained for sufficiently small α > . Theorem 3.
Let N ≥ , ≤ γ ≤ β < N . Then the following statements hold. (i) If N ≥ then ˜ B ( N, α, β, γ ) is attained for any < α ≤ α N,β . (ii) If N = 2 then ˜ B (2 , α, β, γ ) is attained for any < α ≤ α ,β if β > γ , while thereexists α ∗ > such that ˜ B (2 , α, β, β ) is attained for any α ∗ < α < α ,β . (iii) ˜ B (2 , α, β, β ) is not attained for sufficiently small α > . Next, we study the relation between the suprema of Adachi-Tanaka type and Li-Ruftype weighted Trudinger-Moser inequalities, along the line of Lam-Lu-Zhang [19]. Set˜ B rad ( N, β, γ ) = ˜ B rad ( N, α
N,β , β, γ ) in (1.10), and ˜ B ( N, β, γ ) = ˜ B ( N, α
N,β , β, γ ) in (1.11),i.e., ˜ B rad ( N, β, γ ) = sup u ∈ X ,Nγ,rad ( R N ) k u k X ,Nγ ≤ Z R N Φ N ( α N,β | u | NN − ) dx | x | β , (1.14) ˜ B ( N, β, γ ) = sup u ∈ X ,Nγ ( R N ) k u k X ,Nγ ≤ Z R N Φ N ( α N,β | u | NN − ) dx | x | β , (1.15)for N, β, γ satisfying (1.6). Then ˜ B rad ( N, β, γ ) < ∞ , and ˜ B ( N, β, γ ) < ∞ if γ ≥
0, byTheorem 1.
Theorem 4. (Relation) Assume (1.6). Then we have ˜ B rad ( N, β, γ ) = sup α ∈ (0 ,α N,β ) − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ ˜ A rad ( N, α, β, γ ) . Furthermore if γ ≥ , we have ˜ B ( N, β, γ ) = sup α ∈ (0 ,α N,β ) − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ ˜ A ( N, α, β, γ ) . Note that this implies ˜ A rad ( N, α, β, γ ) < ∞ for N, β, γ satisfying (1.6), and ˜ A ( N, α, β, γ ) < ∞ if 0 ≤ γ ≤ β < N , by Theorem 1.Furthermore, we prove how ˜ A rad ( N, α, β, γ ) and ˜ A ( N, α, β, γ ) behaves as α approachesto α N,β from the below:
Theorem 5. (Asymptotic behavior of Adachi-Tanaka supremum) Assume (1.6). Thenthere exist positive constants C , C (depending on N , β , and γ ) such that for α closeenough to α N,β , the estimate C − (cid:16) αα N,β (cid:17) N − N − βN − γ ≤ ˜ A rad ( N, α, β, γ ) ≤ C − (cid:16) αα N,β (cid:17) N − N − βN − γ holds. Corresponding estimates hold true for ˜ A ( N, α, β, γ ) if γ ≥ . VAN HOANG NGUYEN AND FUTOSHI TAKAHASHI
Note that the estimate from the above follows from Theorem 4. On the other hand,we will see that the estimate from the below follows from a computation using the Mosersequence.The organization of the paper is as follows: In section 2, we prove Theorem 1. Main toolsare a transformation which relates a function in X ,Nγ ( R N ) to a function in W ,N ( R N ), andthe singular Trudinger-Moser type inequality recently proved by Adimurthi and Yang [3],see also de Souza and de O [29]. In section 3, we prove the existence part of Theorems 2, 3(i) (ii). In section 4, we prove the nonexistence part of Theorem 2, 3 (iii). Finally in section5, we prove Theorem 4 and Theorem 5. The letter C will denote various positive constantwhich varies from line to line, but is independent of functions under consideration.2. Proof of Theorem 1.
In this section, we prove Theorem 1. We will use the following singular Trudinger-Moserinequality on the whole space R N : For any β ∈ [0 , N ), define(2.1) ˜ B ( N, α, β,
0) = sup u ∈ W ,N ( R N ) , k u k W ,N ≤ Z R N Φ N ( α | u | NN − ) dx | x | β . Then it holds(2.2) ˜ B ( N, α, β, ( < ∞ , α ≤ α N,β , = ∞ , α > α N,β . Here k u k W ,N = (cid:0) k∇ u k NN + k u k NN (cid:1) /N denotes the full norm of the Sobolev space W ,N ( R N ).Note that the inequality (2.2) was first established by Ruf [27] for the case N = 2 and β = 0. It then was extended to the case N ≥ β = 0 by Li and Ruf [20]. The case N ≥ β ∈ (0 , N ) was proved by Adimurthi and Yang [3], see also de Souza and de O[29]. Proof of Theorem 1 : We define the vector-valued function F by F ( x ) = (cid:18) N − γN (cid:19) NN − γ | x | γN − γ x. Its Jacobian matrix is DF ( x ) = (cid:18) N − γN (cid:19) NN − γ | x | γN − γ (cid:18) Id N + γN − γ x | x | ⊗ x | x | (cid:19) = N − γN | F ( x ) | γN (cid:18) Id N + γN − γ x | x | ⊗ x | x | (cid:19) . where Id N denotes the N × N identity matrix and v ⊗ v = ( v i v j ) ≤ i,j ≤ N denotes the matrixcorresponding to the orthogonal projection onto the line generated by the unit vector v = ( v , · · · , v N ) ∈ R N , i.e., the map x ( x · v ) v . Since a matrix of the form I + αv ⊗ v , α ∈ R , has eigenvalues 1 (with multiplicity N −
1) and 1 + α (with multiplicity 1), we seethat(2.3) det ( DF ( x )) = (cid:18) N − γN (cid:19) N − | F ( x ) | γ . Let u ∈ X ,Nγ ( R N ) be such that k u k X ,Nγ ≤
1. We introduce a change of functions asfollows.(2.4) v ( x ) = (cid:18) N − γN (cid:19) N − N u ( F ( x )) . A simple calculation shows that ∇ v ( x ) = (cid:18) N − γN (cid:19) N − N DF ( x )( ∇ u ( F ( x )))= (cid:18) N − γN (cid:19) N − N | F ( x ) | γN (cid:18) ∇ u ( F ( x )) + γN − γ (cid:18) ∇ u ( F ( x )) · x | x | (cid:19) x | x | (cid:19) , and hence |∇ v ( x ) | = (cid:18) N − γN (cid:19) N − N | F ( x ) | γN |∇ u ( F ( x )) | + γ (2 N − γ )( N − γ ) (cid:18) ∇ u ( F ( x )) · x | x | (cid:19) ! . Since (cid:16) ∇ u ( F ( x )) · x | x | (cid:17) ≤ |∇ u ( F ( x )) | , we then have(2.5) |∇ v ( x ) | ≤ (cid:18) N − γN (cid:19) N − N | F ( x ) | γN |∇ u ( F ( x )) | = (det( DF ( x ))) N |∇ u ( F ( x )) | if γ ≥
0, with equality if and only if (cid:16) ∇ u ( F ( x )) · x | x | (cid:17) = |∇ u ( F ( x )) | when γ >
0. If γ = 0the inequality (2.5) is an equality. Note that the inequality (2.5) does not hold if γ < u is not radial function. In fact, a reversed inequality occurs in this case. Moreover,(2.5) becomes an equality if u is a radial function for any −∞ < γ < N . Integrating bothsides of (2.5) on R N , we obtain(2.6) k∇ v k N ≤ k∇ u k N . Moreover, for any function G on [0 , ∞ ), using the change of variables, we get(2.7) Z R N G (cid:16) | u ( x ) | NN − (cid:17) | x | − δ dx = (cid:18) N − γN (cid:19) N − N ( γ − δ ) N − γ Z R N G (cid:18) NN − γ | v ( y ) | NN − (cid:19) | y | N ( γ − δ ) N − γ dy. Consequently, by choosing G ( t ) = t N − and δ = γ , we get k u k N,γ = k v k N and hence(2.8) k u k NX ,Nγ = k∇ u k NN + Z R N | u ( x ) | N | x | − γ dx ≥ k∇ v k NN + k v k NN = k v k NW ,N . VAN HOANG NGUYEN AND FUTOSHI TAKAHASHI
We remark again that (2.6) and (2.8) become equalities if u is radial function for any γ < N . Thus k v k W ,N ≤ k u k X ,Nγ ≤
1. By choosing G ( t ) = Φ N ( αt ) and δ = β ≥ γ , weget(2.9) Z R N Φ N (cid:16) α | u ( x ) | NN − (cid:17) | x | − β dx = (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ Z R N Φ N (cid:18) NN − γ α | v ( y ) | NN − (cid:19) | y | − N ( β − γ ) N − γ dy. Denote ˜ β = N ( β − γ ) N − γ ∈ [0 , N ) . By using (2.8) and (2.9) and applying the singular Trudinger-Moser inequality (2.2), weget sup u ∈ X ,Nγ ( R N ) , k u k X ,Nγ ≤ Z R N Φ N (cid:16) α | u ( x ) | NN − (cid:17) | x | − β dx ≤ (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ sup v ∈ W ,N ( R N ) , k v k W ,N ≤ Z R N Φ N (cid:18) NN − γ α | v ( y ) | NN − (cid:19) | y | − ˜ β dy = (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ ˜ B (cid:18) N, NN − γ α, ˜ β, (cid:19) < ∞ , since NN − γ α ≤ NN − γ α N,β = N − βN − γ α N = (cid:16) N − ˜ βN (cid:17) α N = α N, ˜ β .If u is radial then so is v . In this case, (2.5), (2.6) become equalities, and hence so does(2.8). Then the conclusion follows again from the singular Trudinger-Moser inequality(2.2).We finish the proof of Theorem 1 by showing that ˜ B ( N, α, β, γ ) = ∞ and ˜ B rad ( N, α, β, γ ) = ∞ when α > α N,β . Since ˜ B rad ( N, α, β, γ ) ≤ ˜ B ( N, α, β, γ ), it is enough to prove that˜ B rad ( N, α, β, γ ) = ∞ . Suppose the contrary that ˜ B rad ( N, α, β, γ ) < ∞ for some α > α N,β .Using again the transformation of functions (2.4) for radial functions u ∈ X ,Nγ , we thenhave equalities in (2.5), (2.6), and hence in (2.8). Evidently, the transformation of func-tions (2.4) is a bijection between X ,Nγ,rad and W ,Nrad and preserves the equality in (2.8).Consequently, we have˜ B rad ( N, α, β, γ ) = (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ ˜ B rad (cid:18) N, NN − γ α, ˜ β, (cid:19) , with ˜ β = N ( β − γ ) N − γ ∈ [0 , N ). Hence ˜ B rad (cid:16) N, NN − γ α, ˜ β, (cid:17) < ∞ . By rearrangement argument,we have ˜ B (cid:18) N, NN − γ α, ˜ β, (cid:19) = ˜ B rad (cid:18) N, NN − γ α, ˜ β, (cid:19) < ∞ which violates the result of Adimurthi and Yang since NN − γ α > α N, ˜ β .For the later purpose, we also prove here directly ˜ B rad ( N, α, β, γ ) = ∞ when α > α N,β by using the weighted Moser sequence as in [16], [19]: Let −∞ < γ ≤ β < N and for n ∈ N set A n = (cid:18) ω N − (cid:19) /N (cid:18) nN − β (cid:19) − /N , b n = nN − β , so that ( A n b n ) NN − = n/α N,β . Put u n = A n b n , if | x | < e − b n ,A n log(1 / | x | ) , if e − b n < | x | < , , if 1 ≤ | x | . (2.10)Then direct calculation shows that k∇ u n k L N ( R N ) = 1 , (2.11) k u n k NN,γ = N − β ( N − γ ) N +1 Γ( N + 1)(1 /n ) + o (1 /n )(2.12)as n → ∞ . Thus u n ∈ X ,Nγ,rad ( R N ). In fact for (2.12), we compute k u n k NN,γ = ω N − Z e − bn ( A n b n ) N r N − − γ dr + ω N − Z e − bn A Nn (log(1 /r )) N r N − − γ dr = I + II.
We see I = ω N − ( A n b n ) N (cid:20) r N − γ N − γ (cid:21) r = e − bn r =0 = ω N − (cid:18) nα N,β (cid:19) N − e − ( N − γN − β ) n N − γ = o (1 /n )as n → ∞ . Also II = (cid:18) N − βn (cid:19) Z e − bn (log(1 /r )) N r N − − γ dr = (cid:18) N − βn (cid:19) Z b n ρ N e − ( N − γ ) ρ dρ = N − β ( N − γ ) N +1 (1 /n ) Z ( N − γ ) b n ρ N e − ρ dρ = N − β ( N − γ ) N +1 (1 /n )Γ( N + 1) + o (1 /n ) . Thus we obtain (2.12).
Now, put v n ( x ) = λ n u n ( x ) where u n is the weighted Moser sequence in (2.10) and λ n > λ Nn + λ Nn k u n k NN,γ = 1. Thus we have k∇ v n k NL N + k v n k NN,γ = 1 for any n ∈ N .By (2.12) with β = γ , we see that λ Nn = 1 − O (1 /n ) as n → ∞ . For α > α N,β , we calculate Z R N Φ N ( α | v n | NN − ) dx | x | β ≥ Z { ≤| x |≤ e − bn } Φ N ( α | v n | NN − ) dx | x | β = Z { ≤| x |≤ e − bn } e α | v n | NN − − N − X j =0 α j j ! | v n | NjN − ! dx | x | β ≥ (cid:26) exp (cid:18) nαα N,β λ NN − n (cid:19) − O ( n N − ) (cid:27) Z { ≤| x |≤ e − bn } dx | x | β ≥ (cid:26) exp (cid:18) nαα N,β (cid:18) − O (cid:18) n N − (cid:19)(cid:19)(cid:19) − O ( n N − ) (cid:27) (cid:18) ω N − N − β (cid:19) e − n → + ∞ as n → ∞ . Here we have used that for 0 ≤ | x | ≤ e − b n , α | v n | NN − = αλ NN − n ( A n b n ) NN − = nαα N,β λ NN − n by definition of A n and b n . Also we used that for 0 ≤ | x | ≤ e − b n , | v n | NjN − = λ NjN − n ( A n b n ) NjN − ≤ Cn j ≤ Cn N − for 0 ≤ j ≤ N − n is large. This proves Theorem 1 completely. (cid:3) Existence of maximizers for the weighted Trudinger-Moser inequality
As explained in the Introduction, the existence and non-existence of maximizers for (2.1)is well known. Now, let us recall it here.
Proposition 1.
The following statements hold, (i) If N ≥ then ˜ B ( N, α, , is attained for any < α ≤ α N (see [15, 20] ). (ii) If N = 2 , there exists < α ∗ < α = 4 π such that ˜ B (2 , α, , is attained for any α ∗ < α ≤ α (see [15, 27] ). (iii) If β ∈ (0 , N ) and N ≥ then ˜ B ( N, α, β, is attained for any < α ≤ α N,β (see [21] ). (iv) ˜ B (2 , α, , is not attained for any sufficiently small α > (see [15] ). The existence part (iii) of Proposition 1 is recently proved by X. Li, and Y. Yang [21]by a blow-up analysis.
Remark . By a rearrangement argument, the maximizers for (2.1), if exist, must be adecreasing spherical symmetric function if β ∈ (0 , N ) and up to a translation if β = 0.The proofs of the existence part (i) (ii) of Theorem 2 and 3 are completely similar byusing the formula of change of functions (2.4) and the results on the existence of maximizers for (2.1). So we prove Theorem 3 only here. As we have seen from the proof of Theorem1 that ˜ B ( N, α, β, γ ) ≤ (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ ˜ B (cid:18) N, NN − γ α, ˜ β, (cid:19) if 0 ≤ γ ≤ β < N , where ˜ β = N ( β − γ ) / ( N − γ ) ∈ [0 , N ). If N, α, β and γ satisfy thecondition (i) and (ii) of Theorem 3, then N , N α/ ( N − γ ) and ˜ β satisfy the condition (i)–(iii) of Proposition 1, hence there exists a maximizer v ∈ W ,N ( R N ) for ˜ B (cid:16) N, NN − γ α, ˜ β, (cid:17) with k v k NN + k∇ v k NN = 1 and Z R N Φ N (cid:18) NN − γ α | v ( y ) | NN − (cid:19) | y | − ˜ β dy = ˜ B (cid:18) N, NN − γ α, ˜ β, (cid:19) . As mentioned in Remark 1, we can assume that v is a radial function. Let u ∈ X ,Nγ bea function defined by (2.4). Note that u is also a radial function, hence (2.5) becomes anequality. So do (2.6) and (2.8). Hence, we get k u k NX ,Nγ = k∇ v k NN + k v k NN = 1 , and by (2.9) Z R N Φ N (cid:16) α | u ( x ) | NN − (cid:17) | x | − β dx = (cid:18) N − γN (cid:19) N − N ( γ − β ) N − γ ˜ B (cid:18) N, NN − γ α, ˜ β, (cid:19) . This shows that u is a maximizer for ˜ B ( N, α, β, γ ). (cid:3) Non-existence of maximizers for the weighted Trudinger-Moserinequality
In this section, we prove the non-existence part (iii) of Theorem 3. The proof of (iii) ofTheorem 2 is completely similar. We follow Ishiwata’s argument in [15].Assume 0 ≤ β <
2, 0 < α ≤ α ,β = 2 π (2 − β ) and recall˜ B (2 , α, β, β ) = sup u ∈ X , β ( R k u k X , β ( R ≤ Z R (cid:16) e αu − (cid:17) dx | x | β . We will show that ˜ B (2 , α, β, β ) is not attained if α > M = n u ∈ X , β ( R ) : k u k X , β = (cid:0) k∇ u k + k u k ,β (cid:1) / = 1 o be the unit sphere in the Hilbert space X , β ( R ) and J α : M → R , J α ( u ) = Z R (cid:16) e αu − (cid:17) dx | x | β be the corresponding functional defined on M . Actually, we will prove the stronger claimthat J α has no critical point on M when α > Assume the contrary that there existed v ∈ M such that v is a critical point of J α on M . Define an orbit on M through v as v τ ( x ) = √ τ v ( √ τ x ) τ ∈ (0 , ∞ ) , w τ = v τ k v τ k X , β ∈ M. Since w τ | τ =1 = v , we must have(4.1) ddτ (cid:12)(cid:12)(cid:12) τ =1 J α ( w τ ) = 0 . Note that k∇ v τ k L ( R ) = τ k∇ v k L ( R ) , k v τ k pp,β = τ p + β − k v k pp,β for p >
1. Thus, J α ( w τ ) = Z R (cid:16) e αw τ − (cid:17) dx | x | β = Z R ∞ X j =1 α j j ! v jτ ( x ) k v τ k jX , β dx | x | β = ∞ X j =1 α j j ! k v τ k j j,β (cid:0) k∇ v τ k + k v τ k ,β (cid:1) j = ∞ X j =1 α j j ! τ j − β k v k j j,β (cid:16) τ k∇ v k + τ β k v k ,β (cid:17) j . By using an elementary computation f ( τ ) = τ j − β c ( τ a + τ β b ) j , a = k∇ v k , b = k v k ,β , c = k v k j j,β ,f ′ ( τ ) = (1 − β τ j − β c ( τ a + τ β b ) j +1 {− τ a + ( j − b } , we estimate ddτ (cid:12)(cid:12)(cid:12) τ =1 J α ( w τ ): ddτ (cid:12)(cid:12)(cid:12) τ =1 J α ( w τ )= ∞ X j =1 " α j j ! (1 − β τ j − β/ k v k j j,β (cid:0) τ k∇ v k + τ β/ k v k ,β (cid:1) j +1 (cid:8) − τ k∇ v k + ( j − k v k ,β (cid:9) τ =1 = − α (1 − β k∇ v k k v k ,β + ∞ X j =2 α j j ! (1 − β k v k j j,β (cid:8) −k∇ v k + ( j − k v k ,β (cid:9) ≤ α (1 − β k∇ v k k v k ,β ( − ∞ X j =2 α j − ( j − k v k j j,β k∇ v k k v k ,β ) , (4.2)since −k∇ v k + ( j − k v k ,β ≤ j .Now, we state a lemma. Unweighted version of the next lemma is proved in [15]:Lemma3.1, and the proof of the next is a simple modification of the one given there using the weighted Adachi-Tanaka type Trudinger-Moser inequality:˜ A (2 , α, β, β ) = sup u ∈ X , β ( R \{ }k∇ u k L R ≤ k u k ,β Z R (cid:16) e αu − (cid:17) dx | x | β < ∞ for α ∈ (0 , α ,β ) if β ≥
0, and the expansion of the exponential function.
Lemma 1.
For any α ∈ (0 , α ,β ) , there exists C α > such that k u k j j,β ≤ C α j ! α j k∇ u k j − k u k ,β holds for any u ∈ X , β ( R ) and j ∈ N , j ≥ . By this lemma, if we take α < ˜ α < α ,β and put C = C ˜ α , we see k v k j j,β k∇ v k k v k ,β ≤ C j !˜ α j k∇ v k j − j ≤ C j !˜ α j for j ≥ v ∈ M . Thus we have ∞ X j =2 α j − ( j − k v k j j,β k∇ v k k v k ,β ≤ ∞ X j =2 Cα j − ( j − j !˜ α j = ( Cα ˜ α ) ∞ X j =2 (cid:16) α ˜ α (cid:17) j − j ≤ αC ′ for some C ′ >
0. Inserting this into the former estimate (4.2), we obtain ddτ (cid:12)(cid:12)(cid:12) τ =1 J α ( w τ ) ≤ (1 − β α k∇ v k k v k ,β ( − C ′ α ) < α > (cid:3) Proof of Theorem 4 and 5.
In this section, we prove Theorem 4 and Theorem 5. As stated in the Introduction, wefollow the argument by Lam-Lu-Zhang [19]. First, we prepare several lemmata.
Lemma 2.
Assume (1.6) and set (5.1) b A ( N, α, β, γ ) = sup u ∈ X ,Nγ ( R N ) \{ }k∇ u k LN ( R N ) ≤ k u k N,γ =1 Z R N Φ N ( α | u | NN − ) dx | x | β . Let ˜ A ( N, α, β, γ ) be defined as in (1.9). Then ˜ A ( N, α, β, γ ) = b A ( N, α, β, γ ) for any α > .Similarly, ˜ A rad ( N, α, β, γ ) = b A rad ( N, α, β, γ ) for any α > , where b A rad ( N, α, β, γ ) isdefined similar to (5.1) and b A rad ( N, α, β, γ ) is defined in (1.7).Proof. For any u ∈ X ,Nγ ( R N ) \ { } and λ >
0, we put u λ ( x ) = u ( λx ) for x ∈ R N . Then itis easy to see that(5.2) ( k∇ u λ k NL N ( R N ) = k∇ u k NL N ( R N ) , k u λ k NN,γ = λ − ( N − γ ) k u k NN,γ . Thus for any u ∈ X ,Nγ ( R N ) \ { } with k∇ u k L N ( R N ) ≤
1, if we choose λ = k u k N/ ( N − γ ) N,γ , then u λ ∈ X ,Nγ ( R N ) satisfies k∇ u λ k L N ( R N ) ≤ k u λ k NN,γ = 1 . Thus b A ( N, α, β, γ ) ≥ Z R N Φ N ( α | u λ | NN − ) dx | x | β = 1 k u k N ( N − β ) N − γ N,γ Z R N Φ N ( α | u | NN − ) dx | x | β which implies b A ( N, α, β, γ ) ≥ ˜ A ( N, α, β, γ ). The opposite inequality is trivial. (cid:3)
Lemma 3.
Assume (1.6) and set ˜ B ( N, β, γ ) as in (1.15). Then we have ˜ A ( N, α, β, γ ) ≤ (cid:16) αα N,β (cid:17) N − − (cid:16) αα N,β (cid:17) N − N − βN − γ ˜ B ( N, β, γ ) for any < α < α N,β . The same relation holds for ˜ A rad ( N, α, β, γ ) in (1.7) and ˜ B rad ( N, β, γ ) in (1.14).Proof. Choose any u ∈ X ,Nγ with k∇ u k L N ( R N ) ≤ k u k N,γ = 1. Put v ( x ) = Cu ( λx )where C ∈ (0 ,
1) and λ > C = (cid:18) αα N,β (cid:19) N − N and λ = (cid:18) C N − C N (cid:19) / ( N − γ ) . Then by scaling rules (5.2), we see k v k NX ,Nγ = k∇ v k NN + k v k NN,γ = C N k∇ u k NN + λ − ( N − γ ) C N k u k NN,γ ≤ C N + λ − ( N − γ ) C N = 1 . Also we have Z R N Φ N ( α N,β | v | NN − ) dx | x | β = λ − ( N − β ) Z R N Φ N (cid:16) α N,β C NN − | u | NN − (cid:17) dx | x | β = λ − ( N − β ) Z R N Φ N (cid:16) α | u | NN − (cid:17) dx | x | β . Thus testing ˜ B ( N, β, γ ) by v , we see˜ B ( N, β, γ ) ≥ (cid:18) − C N C N (cid:19) N − βN − γ Z R N Φ N (cid:16) α | u | NN − (cid:17) dx | x | β . By taking the supremum for u ∈ X ,Nγ with k∇ u k L N ( R N ) ≤ k u k N,γ = 1, we have˜ B ( N, β, γ ) ≥ − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ b A ( N, α, β, γ ) . Finally, Lemma 2 implies the result. The proof of˜ B rad ( N, β, γ ) ≥ − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ b A rad ( N, α, β, γ )is similar. (cid:3)
Proof of Theorem 4 : We prove the relation between ˜ B ( N, β, γ ) and ˜ A ( N, α, β, γ ) only. Theassertion that ˜ B ( N, β, γ ) ≥ sup α ∈ (0 ,α N,β ) − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ ˜ A ( N, α, β, γ )follows from Lemma 3. Note that ˜ B ( N, β, γ ) < ∞ when 0 ≤ γ ≤ β < N by Theorem 1.Let us prove the opposite inequality. Let { u n } ⊂ X ,Nγ ( R N ), u n = 0, k∇ u n k NL N + k u n k NN,γ ≤
1, be a maximizing sequence of ˜ B ( N, β, γ ): Z R N Φ N ( α N,β | u n | NN − ) dx | x | β = ˜ B ( N, β, γ ) + o (1)as n → ∞ . We may assume k∇ u n k NL N ( R N ) < n ∈ N . Define v n ( x ) = u n ( λ n x ) k∇ u n k N , ( x ∈ R N ) λ n = (cid:16) −k∇ u n k NN k∇ u n k NN (cid:17) / ( N − γ ) > . Thus by (5.2), we see k∇ v n k NL N ( R N ) = 1 , k v n k N ( N − β ) N − γ N,γ = λ − ( N − γ ) n k∇ u n k NN k u n k NN,γ ! N − βN − γ = k u n k NN,γ − k∇ u n k NN ! N − βN − γ ≤ , since k∇ u n k NN + k u n k NN,γ ≤
1. Thus, setting α n = α N,β k∇ u n k NN − N < α N,β for any n ∈ N , we may test ˜ A ( N, α n , β, γ ) by { v n } , which results in˜ B ( N, β, γ ) + o (1) = Z R N Φ N ( α N,β | u n ( y ) | NN − ) dy | y | β = λ N − βn Z R N Φ N ( α N,β k∇ u n k NN − N | v n ( x ) | NN − ) dx | x | β = λ N − βn Z R N Φ N ( α n | v n ( x ) | NN − ) dx | x | β ≤ λ N − βn k v n k NN,β ! N − βN − γ Z R N Φ N ( α n | v n ( x ) | NN − ) dx | x | β ≤ λ N − βn ˜ A ( N, α n , β, γ ) = (cid:18) − k∇ u n k NN k∇ u n k NN (cid:19) N − βN − γ ˜ A ( N, α n , β, γ )= − (cid:16) α n α N,β (cid:17) N − (cid:16) α n α N,β (cid:17) N − N − βN − γ ˜ A ( N, α n , β, γ ) ≤ sup α ∈ (0 ,α N,β ) − (cid:16) αα N,β (cid:17) N − (cid:16) αα N,β (cid:17) N − N − βN − γ ˜ A ( N, α, β, γ ) . Here we have used a change of variables y = λ n x for the second equality, and k v n k N ( N − β ) N − γ N,γ ≤ n → ∞ , we have the desired result. (cid:3) Proof of Theorem 5 : Again, we prove theorem for ˜ A ( N, α, β, γ ) only. The assertion that˜ A ( N, α, β, γ ) ≤ C − (cid:16) αα N,β (cid:17) N − N − βN − γ follows form Theorem 4 and the fact that ˜ B ( N, β, γ ) < ∞ when 0 ≤ γ ≤ β < N .For the rest, we need to prove that there exists C > α < α
N,β sufficiently close to α N,β , it holds that(5.3) C − (cid:16) αα N,β (cid:17) N − N − βN − γ ≤ ˜ A ( N, α, β, γ ) . For that purpose, we use the weighted Moser sequence (2.10) again. By (2.12), we have N ∈ N such that if n ∈ N satisfies n ≥ N , then it holds(5.4) k u n k NN,γ ≤ N − γ )Γ( N + 1)( N − β ) N +1 (1 /n ) . On the other hand, Z R N Φ N ( α | u n | N/ ( N − ) dx | x | β ≥ ω N − Z e − bn Φ N (cid:0) α ( A n b n ) N/ ( N − (cid:1) r N − − β dr = ω N − N − β Φ N (( α/α N,β ) n ) (cid:2) r N − β (cid:3) r = e − bn r =0 = ω N − N − β Φ N (( α/α N,β ) n ) e − n . Note that there exists N ∈ N such that if n ≥ N then Φ N (( α/α N,β ) n ) ≥ e ( α/α N,β ) n .Thus we have(5.5) Z R N Φ N ( α | u n | N/ ( N − ) dx | x | β ≥ (cid:18) ω N − N − β (cid:19) e − (1 − ααN,β ) n . Combining (5.4) and (5.5), we have C ( N, β, γ ) > k u n k N ( N − β ) N − γ N,γ Z R N Φ N ( α | u n | N/ ( N − ) dx | x | β ≥ C ( N, β, γ ) n N − βN − γ e − (1 − ααN,β ) n holds when n ≥ max { N , N } .Note that lim x → (cid:16) − x N − − x (cid:17) = N −
1, thus1 − ( α/α N,β ) N − − ( α/α N,β ) ≥ N − α/α N,β < α > α N,β so that max { N , N } < (cid:16) − α/α N,β (cid:17) , − ( α/α N,β ) N − − ( α/α N,β ) ≥ N − , (5.7)we can find n ∈ N such that max { N , N } ≤ n ≤ (cid:16) − α/α N,β (cid:17) , (cid:16) − α/α N,β (cid:17) ≤ n. (5.8) We fix n ∈ N satisfying (5.8). Then by 1 ≤ n (1 − α/α N,β ) ≤
2, (5.6) and (5.7), we have1 k u n k NN,β Z R N Φ N ( α | u n | N/ ( N − ) dx | x | β ≥ C ( N, β, γ ) n N − βN − γ e − ≥ C ( N, β, γ ) (cid:18) − ( α/α N,β ) (cid:19) N − βN − γ ≥ N − C ( N, β, γ ) (cid:18) − ( α/α N,β ) N − (cid:19) N − βN − γ = C ( N, β, γ ) (cid:18) − ( α/α N,β ) N − (cid:19) N − βN − γ , where C ( N, β, γ ) = e − C ( N, β, γ ) and C ( N, β, γ ) = N − C ( N, β, γ ). Thus we have (5.3)for some
C > α which is sufficiently close to α N,β . (cid:3) Acknowledgments.
The first author (V. H. N.) was supported by CIMI’s postdoctoral research fellowship.The second author (F.T.) was supported by JSPS Grant-in-Aid for Scientific Research (B),No.15H03631, and JSPS Grant-in-Aid for Challenging Exploratory Research, No.26610030.
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E-mail address : [email protected] Department of Mathematics, Osaka City University & OCAMI, Sumiyoshi-ku, Osaka,558-8585, Japan
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