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Trudinger–Moser type inequalities with logarithmic weights in dimension N
Nonlinear Analysis (
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Trudinger–Moser type inequalities with logarithmic weights in dimension N Marta Calanchi, Bernhard Ruf ∗ Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy
article
abstract
info
We consider borderline embeddings of Trudinger–Moser type for weighted Sobolev spaces in bounded domains in RN . The embeddings go into Orlicz spaces with exponential growth functions. It turns out that the most interesting weights are powers of the logarithm, for which an explicit dependence of the maximal growth functions can be established. Corresponding Moser type results are also proved, with explicit sharp exponents. In the particular case of a logarithmic weight with the limiting exponent N − 1, a maximal growth of double exponential type is obtained, while for any larger exponent the embedding goes into L∞ . © 2015 Elsevier Ltd. All rights reserved.
Communicated by S. Carl Dedicated to Enzo Mitidieri on the occasion of his 60th birthday Keywords: Trudinger–Moser inequality Limiting Sobolev embedding Weighted Sobolev space Orlicz space
1. Introduction 1,N
(Ω ) the standard first order Sobolev space given by N1 |∇ u|N dx .
Let Ω ⊂ RN denote a bounded domain in RN , and denote with W0
1 ,N W0 (Ω ) = cl u ∈ C0∞ |∇ u|N dx < ∞ ,
Ω
∥u∥ :=
Ω
1 ,N
This space is a limiting case for the Sobolev embedding theorem, which yields W0 easy examples show that
1,N W0
⊂ Lp (Ω ), for all 1 ≤ p < ∞, but
̸⊂ L (Ω ). One knows from the works by Yudovich [30], Peetre [25], Pohozaev [26] and ∞
′
1 ,N W0
N Trudinger [29] that (Ω ) embeds into the Orlicz space Lϕ (Ω ), with N-function ϕ(t ) = e|t | − 1, where N ′ = embedding was made more precise by J. Moser [23] by the following sharp inequality
sup ∥∇ u∥N ≤1 Ω
N′
N . N −1
This
1
−1 eα|u| dx < ∞ ⇐⇒ α ≤ N ωNN− 1,
(1)
where ωN −1 denotes the area of the unit sphere in RN . This result has lead to a very rich literature on related results; we just mention a few: – – – – –
Extension of Moser’s inequality to unbounded domains, see Ruf [27], Li–Ruf [19], and Ishiwata [15] Attainability of the supremum in (1): positive answers have been given by Carleson–Chang [6], Flucher [12], and Lin [20] Trudinger–Moser inequalities on manifolds, see L. Fontana [13] Improvements and sharpening of Moser’s inequality: Adimurthi–Druet [2], and Masmoudi–Sani [22] Moser inequalities with other boundary conditions, see Cianchi [9]
∗
Corresponding author. E-mail addresses: [email protected] (M. Calanchi), [email protected] (B. Ruf).
http://dx.doi.org/10.1016/j.na.2015.02.001 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
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– Extension of the Trudinger–Moser inequality to higher order Sobolev spaces: Adams [1], and Ruf–Sani [28] – TM-inequalities with weights, see Calanchi–Terraneo [5], and Adimurthi–Sandeep [3], see also [10,11,16] In the papers by Calanchi–Terraneo [5] and Adimurthi–Sandeep [3] the influence of weights in the integral term on the maximal growth was studied. In the present paper, we take a different interest, and study the influence of weights in the Sobolev norm; that is, we change the function space in which we work by adding some weights. More precisely, let w ∈ C (Ω ) be a non-negative function, and consider the weighted Sobolev space 1 ,N
W0
|∇ u|N w(x)dx < ∞ . (Ω , w) = cl u ∈ C0∞ (Ω ); Ω
For a general embedding theory for weighted Sobolev spaces, see Kufner–Persson [17]. For Trudinger–Moser type embeddings of weighted Sobolev spaces, the following observations are in order: (a) it turns out that the most interesting weights are of logarithmic type; (b) an effect on the maximal growth is only present if we restrict attention to radially symmetric functions. We are therefore lead to consider problems of the following type: let B ⊂ RN denote the unit ball in RN , and consider the weighted Sobolev space of radial functions
1,N
N W0,rad (B, w) = cl u ∈ C0∞ ,rad (B); ∥u∥w :=
|∇ u|N w(x)dx < ∞
B
where
1 β(N −1) e β(N −1) w(x) = log or w(x) = log , |x| | x|
β ≥ 0.
(2)
We will prove the following. Theorem 1. Let β ∈ [0, 1) and let w(x) be given by (2). Then
(a)
γ
B(0)
e|u| dx < +∞,
1,N
for all u ∈ W0,rad (B, w),
if and only if
γ ≤ γN ,β :=
N
(N − 1)(1 − β)
=
N′ 1−β
,
and
(b)
sup
∥u∥w ≤1,rad B(0)
γN ,β
eα|u|
dx < +∞
if and only if 1 1−β 1/N −1 α ≤ αN ,β = N ωN −1 (1 − β)
(critical growth),
(3)
where ωN −1 is the area of the unit sphere SN −1 in RN . Remark 2. (a) gives the generalization of Trudinger’s theorem to the weighted case (for radial functions), while (b) extends the Trudinger–Moser inequality; indeed, for β = 0 we recover the classical Trudinger–Moser inequality.
β(N −1) 1,N Next, we consider the case β > 1: then, considering the weight w1 (x) = log |xe| , the functions in W0,rad (B, w1 ) are bounded. Theorem 3. Let β > 1. Then we have the following embedding 1 ,N W0,rad (B, w1 ) ↩→ L∞ (B).
For the borderline case β = 1, one has a critical growth of double exponential type, as described in the following. Theorem 4. Let w1 (x) = | log |xe| |N −1 (i.e. β = 1). Then, setting N ′ =
(a)
B(0)
|u|N
ee
′
dx < +∞,
N ∀ u ∈ W01,,rad (B, w1 ),
N N −1
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while 1
(b)
sup
∥u∥w1 ≤1,rad B(0)
e
ω N −1 |u|N a e N −1
′
dx < +∞ ⇐⇒ a ≤ N .
We point out that the special case N = 2, i.e. the corresponding inequality for W 1,2 (B, w), B ⊂ R2 was studied in [4]. The proof is based on ideas by Leckband [18]. Indeed, in [18] a very general optimal integral inequality is proved. The proof in [4] specializes (and simplifies) the proof of Leckband to the particular situation of W 1,2 (B, w). In this note we show that after some suitable manipulations, the results in Theorems 1 and 4 can be directly deduced from Leckband’s inequality. A simplified direct proof in the spirit of [4] seems also possible. 2. Trudinger–Moser embeddings with weights In order to prove Theorem 1 we start with a Radial Lemma which is of crucial importance in the sequel. Lemma 5. Let u be a radially symmetric C01 function on the ball B = B(0) ⊂ RN . Then one has
1−β log |x| N ′
∥u∥w , ∀ β < 1, (ωN −1 )1/N (1 − β)1/N ′ β(N −1) for w = w0 (x) = log |x| ; while for w = w1 (x) = | log( |xe| )|β(N −1) e 1−β 1/N ′ log − 1 | x| (ii) |u(x)| ≤ ∥u∥w , β ̸= 1 (ωN −1 )1/N |1 − β|1/N ′ 1 ′ (iii) |u(x)| ≤ 1/N log1/N log(e/|x|) ∥u∥w , β = 1. ωN −1 (i) |u(x)| ≤
Proof. (i) Let v(|x|) = u(x). Then, by Hölder’s inequality
|u(x)| = v(|x|) − v(1) = 1
|v ′ (t )| t
=
N −1 N
| x| 1
v ′ (t )dt
| log t |β/N t − ′
N −1 N
′
| log t |−β/N dt
| x| 1
|v (t )| t ′
≤
N
N −1
β(N −1)
| log t |
dt
1/N
|x|
|x|
1
1
1
β(N −1)
1
dt β
NN−1
t | log t |
1/N
| log t |1−β |x| 1 1−β
ωN −1 |v (t )| t | log t | dt 1/N ωN −1 | x| 1−β log |x| N ′ ∥ u∥ w . ≤ (ωN −1 )1/N (1 − β)1/N ′ By a similar procedure, substituting | log t | with | log(e/t )|, one obtains (ii). =
′
N
N −1
For completeness we prove (iii): as above we get
|x| |u(x)| = v ′ (t )dt 1 1 1/N ′ − N −1 −1/N ′ N −1 ≤ |v ′ (t )| t N log(e/t ) t N log(e/t ) dt | x|
1
|v (t )| t ′
≤
N
N −1
(N −1) log(e/t ) dt
1/N
|x|
=
ω N −1
1
(N −1) |v ′ (t )|N t N −1 log(e/t ) dt
| x|
log1/N log(e/|x|) ′
≤
1/N
ω N −1
∥ u∥ w .
NN−1 dt |x| t log(e/t ) 1/N 1/N ′ log log(e/|x|) 1
1
1/N
ωN −1
NN−1
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β(N −1) Proof of Theorem 1. For the sufficiency part it is enough to consider only the case w(x) = w0 (x) = log |x| . Indeed,
β(N −1) observe that for β ∈ [0, 1) one has, with w1 (x) = log |xe|
H01 (B, w1 ) ↩→ H01 (B, w0 ). We prove the theorem in the critical case, i.e. for γ = γN ,β , the subcritical case is then a direct consequence. Since C0∞ (Ω ) is dense in H01 (B, w0 ), we may assume that u ∈ C0∞ (Ω ) and u ≥ 0 (otherwise replace u by |u|). Since the problem is radially symmetric we introduce the variable t by
|x| = e−t /N and set 1/N
ψ(t ) = ωN −1 N
N −1 (1−β) N
(1 − β)
N −1 N
u(x).
(4)
Then the weighted Sobolev norm becomes
β(N −1) |∇ u| log |x| dx = N
B(0)
+∞
0
|ψ ′ |N t β(N −1) dt (1 − β)N −1
(5)
and the exponential integral
N
ω N −1
α|u|γ
e
+∞
γ −t ¯ eα|ψ| dt
dx =
B
(6)
0
where
γ = γN ,β =
N
(N − 1)(1 − β)
,
and
α , αN ,β
α¯ =
where αN ,β is the critical growth given in (3). We first prove (a), that is we show that the integral in (6), namely
+∞ 0
|ψ ′ |N t β(N −1) (1−β)N −1
+∞ 0
γ −t ¯ eα|ψ| dt, is finite for any ψ satisfying
dt < +∞.
Arguing as in the paper of Moser [23], for all ε > 0 there exists T = T (ε) such that +∞
T
|ψ ′ |N t β(N −1) dt < ε N . (1 − β)N −1
Hence, by Hölder’s inequality
ψ(t ) = ψ(T ) +
t
ψ ′ (s) ds = ψ(T ) +
T
≤ ψ(T ) +
t
|ψ ′ (s)| sβ/N s−β/N ds ′
′
T t
1/N
t
|ψ ′ (s)|N sβ(N −1) ds
T
NN−1
s−β ds T
1/N NN−1 |ψ ′ (s)|N sβ(N −1) 1−β 1−β = ψ(T ) + ds t − T (1 − β)N −1 T NN−1 ≤ ψ(T ) + ε t 1−β − T 1−β for all t ≥ T . t
This implies that there exists T¯ such that N
α¯ ψ (1−β)(N −1) (t ) ≤
1 2
t,
for all t ≥ T¯ .
This is sufficient to guarantee the existence of the integral in (6). In order to prove the uniform estimate Theorem 1(b), remark that by the radial lemma +∞
if 0
|ψ ′ |N t β(N −1) dt ≤ 1, (1 − β)N −1
then ψ(t ) ≤ t
Hence, if α¯ < 1, it follows easily by (7) that +∞
0
γ −t ¯ eα|ψ| dt ≤
+∞
0
eα¯ t −t dt < +∞.
N −1 (1−β) N
, ∀ t > 0.
(7)
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Sharpness: Now we prove that the theorem is sharp in the sense that if α > αN ,β Theorem 1(b) is infinite.
5
⇐⇒ α¯ > 1, then the supremum in
β(N −1) Again we consider first the case w = w0 (x) = log |x| . It is sufficient to test
+∞
γ −t ¯ eα|ψ| dt
0
on the family of functions of Moser type
ηk (t ) =
1−β t
t ≤ k, 1−β k(1N−β)(N −1) N
t ≥ k.
k
One easily sees that Then, with γ = +∞
|ηk′ |N t β(N −1)
+∞
0 (1−β)N −1 N (N −1)(1−β)
¯ k | γ −t eα|η dt ≥
0
+∞
dt = 1.
eα¯ k−t dt → +∞,
if α¯ > 1.
k
β(N −1) For the case w(x) = log |xe| the family of functions becomes (with the change of variables t = N − N log |x|)
t 1−β − N 1−β 1/N , ξk (t ) = (k + N )1−β − N 1−β N −1 (k + N )1−β − N 1−β N ,
N ≤t ≤k+N t ≥ k + N.
Then ξk has norm 1, and +∞
¯ k | γ −t eα|ξ dt ≥
N
+∞
1 1−β −N 1−β ] 1−β
¯ k+N ) eα[(
−t
dt
k+N 1 1−β −N 1−β ] 1−β
¯ k+N ) = eα[(
−(k+N )
→ +∞,
as k → ∞, if α¯ > 1.
The critical case α¯ = 1. This case is more delicate. We will complete the proof in Section 4, applying a result by Leckband [18], which we will also use for the double exponential case.
β(N −1) Proof of Theorem 3. In the next result we show that in the case of w1 (x) = log |xe| and β > 1 the functions in 1 ,N W0,rad (B) are bounded. This follows easily by the radial lemma. In fact by (ii) in Lemma 5 we have |u(x)| ≤
1/N ′ e 1−β log − 1 |x|
∥ u∥ w ≤
(ωN −1 )1/N |1 − β|1/N ′
2∥u∥w . (ωN −1 )1/N |1 − β|1/N ′
3. A limiting case with double exponential growth In this section we consider the limiting case β = 1, i.e. weights of the form (log |xe| )N −1 . As stated in Theorem 4 we obtain a critical growth of double exponential type. We remark that embeddings into Orlicz spaces with double exponential growth have recently been obtained by Cianchi [8] and belonging to Sobolev–Zygmund spaces WLn Log α L(Ω ), Ω ⊂ RN , i.e. for functions α Hencl [14] for functions u such that Ω |∇ u|N log |∇ u| dx is bounded. See also the extensions by Černý–Maškowá [7] to the case of multiple exponential growths. These results are of different type than the ones in Theorem 4: they involve a logarithmic Sobolev norm, while our results are based on standard Sobolev norms with logarithmic weights. Proof of Theorem 4 (Subcritical and Supercritical Cases). (a) can be proved with the same argument used in the first part of Theorem 1. 1/N In order to prove (b), set ψ(t ) = ωN −1 u(x) with |x| = e−t . Then
e N −1 |∇ u(x)|N log dx = |x| B
+∞
|ψ ′ (t )|N (1 + t )N −1 dt 0
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and 1
e
ω N −1 |u|N ae N −1
′
dx =
B
ωN −1
+∞
N
eae
′ |ψ|N (t ) −Nt
dt .
0
Thus, we need to prove that
ω N −1
N
+∞
′ |ψ|N (t ) −Nt
eae
for all ψ with
dt ≤ C
+∞
0
|ψ ′ (t )|N (1 + t )N −1 dt = 1.
(8)
0
By (iii) of the Radial Lemma one has +∞
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1 ⇒ |ψ(t )| ≤ log
N −1 N
(1 + t ),
∀ t > 0.
0
Therefore +∞
ae|ψ|
e
N ′ (t )
−Nt
e
+∞
dt ≤
0
ea(1+t )−Nt dt < +∞,
if a < N .
0
Sharpness. The necessary condition If a > N it is sufficient to test the functional on the following sequence
log(1 + t ) 0≤t ≤k ψk (t ) = log1/N (1 + k) N −1 log N (1 + k) t ≥ k. We have +∞
|ψk′ (t )|N (1 + t )N −1 dt = 1 0
and +∞
eae
′ ψkN (t )
e−Nt dt ≥
+∞
eae
′ ψkN (t )
e−Nt dt
k
0
+∞
ea+ak−Nt dt → +∞,
=
as k → +∞.
k
Critical exponent: If a = N the estimate is more delicate. We have to prove that there exists C > 0 such that +∞
eNe
′ |ψ|N (t ) −Nt
dt < C ,
∀ ψ with
0
+∞
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1. 0
In order to prove this, we make use of a result due to Leckband, which we explain in the following section. 4. The Leckband–Neugebauer result and the proofs of the critical cases We begin with the definition of a special class of functions, the so-called C ∗ -functions, see [24,18]. Definition 6. A continuous function ρ : [0, +∞) → [0, +∞) is called a C ∗ -function if there is a constant Cρ > 0 such that for every 0 < d < +∞, there exists a constant D > 0 with
ρ((l + d)s) ≤ Cρ ρ (l − 1)s ,
for all l ≥ D and 0 < s < +∞.
One easily notes that Example 7. Any concave function is a C ∗ -function. The product of C ∗ -functions is a C ∗ -function. Next, we introduce the notion of a C ∗ -convex function by the following implicit definition. Definition 8. A C 1 -function M : [0, +∞) → [0, +∞) is called a C ∗ -convex function if M is convex, and the function ρ defined by the differential equation ρ(M (t )) = M ′ (t ) is a C ∗ -function.
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Some typical examples of C ∗ -convex functions: p
Example 9. M1 (t ) = t p and M2 (t ) = et − 1 with p > 1 are C ∗ -convex functions. Indeed their respective C ∗ -functions are
ρ1 (s) = ps
p−1 p
and ρ2 (s) = p(s + 1) log
p−1 p
(s + 1).
We now state a general functional inequality by Leckband [18]. This quite unimpressive looking inequality is actually quite magic, and contains many of the known borderline inequalities. It is a generalization of a previous result by Neugebauer [24], which in turn is a generalization of Moser’s integral inequality [23], see also [21]. Theorem 10 (Leckband’s Inequality [18]). Let f ∈ LN ([0, +∞)) such that ∥f ∥N = 1, ϕ : R+ → R+ with ϕ ≥ 0 is locally integrable, and set G(x) =
x
ϕ
N N −1
NN−1
(y)dy
F (x) =
and
x
f (y)ϕ(y)dy. 0
0
Let Φ ≥ 0 be a nonincreasing function on [0, +∞), and M (t ) a C ∗ -convex function. Then there exists a constant C > 0 +∞
Φ M G(t ) − M F (t ) d M G(t ) ≤ C ∥Φ ∥1 .
(9)
0
Proof of Theorem 1 (Critical Case). We now complete the proof of Theorem 1 in the critical case α¯ = 1, i.e. α = αN ,β , applying the Leckband inequality to f (t ) = ψ ′ (t )
t β NN−1 , 1−β
ϕ(t ) =
t β 1−NN 1−β
where ψ(t ) ∈ C 1 ([0, ∞)) is given by (4). Then ∞
|f (t )| dt = N
0
t β(N −1)
∞
|ψ ′ (t )|N 0
(1 − β)N −1
dt = 1
by (5). Furthermore, with this choice of ϕ one has G(x) =
x
ϕ
N N −1
NN−1
(y)dy
=
x
0
0
1−β
N −1 N
tβ
= x(1−β)
N −1 N
and F (x) =
x
f (y)ϕ(y)dy = 0
x
ψ ′ (y)dy = ψ(x). 0
Finally, set N
M (s) = s (N −1)(1−β) ,
Φ (s) = e−s
Then we get M (G(t )) = t ,
N
M (F (t )) = ψ (N −1)(1−β) (t ) = ψ γ (t ).
and
Hence, with these particular choices, the Leckband inequality becomes +∞
γ eψ (t )−t dt ≤ C
0
+∞
e−s ds = C ,
for all ψ with
0
+∞
|ψ ′ (t )|N 0
t β(N −1)
(1 − β)N −1
dt ≤ 1.
This is precisely inequality (6) for α¯ = 1, which we wanted to prove. Proof of Theorem 4. For the proof of the double exponential case (Theorem 4), we apply Leckband’s inequality to the functions f (t ) = ψ ′ (t )(1 + t )
N −1 N
,
ϕ(t ) = (1 + t )
Then ∞
|f (t )|dt = 0
∞
|ψ ′ |N (1 + t )N −1 dt = 1 0
1−N N
.
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by (8), and by the choice of ϕ one has G(x) =
x
NN−1 N ϕ N −1 (y)dy =
0
x
1 y+1
0
NN−1
dy
N −1 = log(1 + x) N
and F (x) =
x
f (y)ϕ(y)dy =
x
ψ ′ (y)dy = ψ(x). 0
0
Finally, setting M (t ) = et
N N −1
− 1,
Φ (s) = e−Ns
the Leckband inequality becomes in this particular case +∞
eNe
′ ψ N (t ) −Nt
+∞
e−Ns ds = CN ,
dt ≤ C
0
for all ψ with
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1 0
0
and this is inequality (8).
+∞
1
−1 Remark 11. We point out that if b < ωNN− 1 then the nonlinearity
ea e
′ b |u|N 1
−1 is subcritical for any a > 0, and if b > ωNN− 1 then it is supercritical, for any a > 0.
5. A remark on the supercritical case 1,N
In this section we do not restrict to the radial case, and consider all functions in W0 (B, w). We prove that in this case we cannot go beyond the classical TM inequality. Indeed, in the supercritical case with respect to the Moser inequality, 1 ,N i.e. γ = NN−1 , α > N ωN −1 , the supremum in the whole space W0 (B, w) turns out to be infinite. So the presence of the weight has no effect with regard to criticality. We will prove this only in the case w(x) = log |xe| being similar.
β(N −1)
, the other cases
1
Proposition 12. Let α > N wnN−−11 . Then
eα|u|
sup
N N −1
dx = +∞.
∥u∥w =1 B(0)
Proof. It is sufficient to evaluate the functional 1
e
N
N −1 N ωN −1 (1+δ)β |u| N −1
dx,
δ>0
B
on a suitable translation of the following Moser sequence
wk (x) =
1
ω
1/N N −1
N −1 (log k) N
|x| ≤
− log |x| (log k)1/N
| x| ≥
1 k 1 k
.
Indeed, consider
wk,a (x) =
1
N −1 (log k) N a
|x − xa | <
a k
log |x−x | a
a
0
|x − xa | ≥ a
1/N ωN −1 1/N (log k)
k
≤ |x − xa | < a
where xa = (1 − a, 0), 0 < a < 1/2, k > 2. We observe that ∥∇wk,a ∥LN = 1. Indeed
|∇wk,a (x)|N dx = B
1 log k
a
1
a/k
r
dr = 1.
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It is now sufficient to choose a <
uk,a (x) = log
e
1 − 2a
1 2
(1 −
1 eδ
)
–
9
) and test the functional on the following sequence
−β NN−1 wk,a (x).
This sequence satisfies
log e β(N −1) | x| |∇wk,a (x)|N dx log e B 1−2a ≤ |∇wk,a (x)|N dx ≤ 1
∥uk,a ∥Nw =
B
since |x| ≥ 1 − 2a, recalling that the support of this sequence is contained in a ball of radius a centered in (1 − a, 0). Moreover
B
since
1
−1 exp N ωNN− 1
1 + δ β N e wkN,a−1 dx → +∞, log
1+δ > 1 by the choice of a.
log 1−e2a
k → +∞
1−2a
References [1] D.R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (2) (1988) 385–398. [2] Adimurthi, O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality, Comm. Partial Differential Equations 29 (1) (2004) 295–322. [3] Adimurthi, K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl. 13 (2007) 585–603. [4] M. Calanchi, B. Ruf, On a Trudinger–Moser type inequality with logarithmic weights, J. Differential Equations 258 (2015) 1967–1989. [5] M. Calanchi, E. Terraneo, Non-radial maximizers for functionals with exponential nonlinearity in R2 , Adv. Nonlinear Stud. 5 (2005) 337–350. [6] L. Carleson, S.A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (2) (1986) 113–127. [7] R. Černý, S. Maškowá, A sharp form of an embedding into multiple exponential spaces, Czechoslovak Math. J. 60 (135) (2010) 751–782. [8] A. Cianchi, A sharp embedding theorem for Orlicz–Sobolev spaces, Indiana Univ. Math. J. 45 (1996) 39–65. [9] A. Cianchi, Moser–Trudinger inequalities without boundary conditions and isoperimetric problems, Indiana Univ. Math. J. 54 (3) (2005) 669–705. [10] J.F. De Oliveira, J.M. do Ó, Trudinger–Moser type inequalities for weighted spaces involving fractional dimentions, Proc. Amer. Math. Soc. 142 (8) (2014) 2813–2828. [11] D.E. Edmunds, H. Hudzik, M. Krbec, On weighted critical imbeddings of Sobolev spaces, Math. Z. 268 (1–2) (2011) 585–592. [12] M. Flucher, Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (3) (1992) 471–497. [13] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (3) (1993) 415–454. [14] S. Hencl, A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal. 204 (2003) 196–227. [15] M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in RN , Math. Ann. 351 (4) (2011) 781–804. [16] M. Krbec, T. Schott, Embeddings of weighted Sobolev spaces in the borderline case, Real Anal. Exchange 23 (2) (1997/98) 395–420. [17] A. Kufner, L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003. [18] M.A. Leckband, An integral inequality with applications, Trans. Amer. Math. Soc. 283 (1) (1984) 157–168. [19] Y. Li, B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in Rn , Indiana Univ. Math. J. 57 (1) (2008) 451–480. [20] Kai-Ching Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (7) (1996) 2663–2671. [21] D.E. Marshall, A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1) (1989) 131–137. [22] N. Masmoudi, F. Sani, Adams’ inequality with the exact growth condition in R4 , Comm. Pure Appl. Math. 67 (8) (2014) 1307–1335. [23] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71) 1077–1092. [24] C.J. Neugebauer, Some inequalities related to Hölder’s inequality, Proc. Amer. Math. Soc. 82 (4) (1981) 560–564. [25] J. Peetre, Espaces d’ interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966) 279–317. [26] S.I. Pohozaev, The Sobolev embedding in the case pl = n, in: Proc. of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Section, Energet. Inst., Moscow, Moskov, 1965, pp. 158–170. [27] B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in R2 , J. Funct. Anal. 219 (2) (2005) 340–367. [28] B. Ruf, F. Sani, Sharp Adams-type inequalities in Rn , Trans. Amer. Math. Soc. 365 (2) (2013) 645–670. [29] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483. [30] V.I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961) 805–808; Translated in Soviet Math. Dokl. 2 (1961) 746–749.
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Trudinger–Moser type inequalities with logarithmic weights in dimension N Marta Calanchi, Bernhard Ruf ∗ Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy
article
abstract
info
We consider borderline embeddings of Trudinger–Moser type for weighted Sobolev spaces in bounded domains in RN . The embeddings go into Orlicz spaces with exponential growth functions. It turns out that the most interesting weights are powers of the logarithm, for which an explicit dependence of the maximal growth functions can be established. Corresponding Moser type results are also proved, with explicit sharp exponents. In the particular case of a logarithmic weight with the limiting exponent N − 1, a maximal growth of double exponential type is obtained, while for any larger exponent the embedding goes into L∞ . © 2015 Elsevier Ltd. All rights reserved.
Communicated by S. Carl Dedicated to Enzo Mitidieri on the occasion of his 60th birthday Keywords: Trudinger–Moser inequality Limiting Sobolev embedding Weighted Sobolev space Orlicz space
1. Introduction 1,N
(Ω ) the standard first order Sobolev space given by N1 |∇ u|N dx .
Let Ω ⊂ RN denote a bounded domain in RN , and denote with W0
1 ,N W0 (Ω ) = cl u ∈ C0∞ |∇ u|N dx < ∞ ,
Ω
∥u∥ :=
Ω
1 ,N
This space is a limiting case for the Sobolev embedding theorem, which yields W0 easy examples show that
1,N W0
⊂ Lp (Ω ), for all 1 ≤ p < ∞, but
̸⊂ L (Ω ). One knows from the works by Yudovich [30], Peetre [25], Pohozaev [26] and ∞
′
1 ,N W0
N Trudinger [29] that (Ω ) embeds into the Orlicz space Lϕ (Ω ), with N-function ϕ(t ) = e|t | − 1, where N ′ = embedding was made more precise by J. Moser [23] by the following sharp inequality
sup ∥∇ u∥N ≤1 Ω
N′
N . N −1
This
1
−1 eα|u| dx < ∞ ⇐⇒ α ≤ N ωNN− 1,
(1)
where ωN −1 denotes the area of the unit sphere in RN . This result has lead to a very rich literature on related results; we just mention a few: – – – – –
Extension of Moser’s inequality to unbounded domains, see Ruf [27], Li–Ruf [19], and Ishiwata [15] Attainability of the supremum in (1): positive answers have been given by Carleson–Chang [6], Flucher [12], and Lin [20] Trudinger–Moser inequalities on manifolds, see L. Fontana [13] Improvements and sharpening of Moser’s inequality: Adimurthi–Druet [2], and Masmoudi–Sani [22] Moser inequalities with other boundary conditions, see Cianchi [9]
∗
Corresponding author. E-mail addresses: [email protected] (M. Calanchi), [email protected] (B. Ruf).
http://dx.doi.org/10.1016/j.na.2015.02.001 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
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M. Calanchi, B. Ruf / Nonlinear Analysis (
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– Extension of the Trudinger–Moser inequality to higher order Sobolev spaces: Adams [1], and Ruf–Sani [28] – TM-inequalities with weights, see Calanchi–Terraneo [5], and Adimurthi–Sandeep [3], see also [10,11,16] In the papers by Calanchi–Terraneo [5] and Adimurthi–Sandeep [3] the influence of weights in the integral term on the maximal growth was studied. In the present paper, we take a different interest, and study the influence of weights in the Sobolev norm; that is, we change the function space in which we work by adding some weights. More precisely, let w ∈ C (Ω ) be a non-negative function, and consider the weighted Sobolev space 1 ,N
W0
|∇ u|N w(x)dx < ∞ . (Ω , w) = cl u ∈ C0∞ (Ω ); Ω
For a general embedding theory for weighted Sobolev spaces, see Kufner–Persson [17]. For Trudinger–Moser type embeddings of weighted Sobolev spaces, the following observations are in order: (a) it turns out that the most interesting weights are of logarithmic type; (b) an effect on the maximal growth is only present if we restrict attention to radially symmetric functions. We are therefore lead to consider problems of the following type: let B ⊂ RN denote the unit ball in RN , and consider the weighted Sobolev space of radial functions
1,N
N W0,rad (B, w) = cl u ∈ C0∞ ,rad (B); ∥u∥w :=
|∇ u|N w(x)dx < ∞
B
where
1 β(N −1) e β(N −1) w(x) = log or w(x) = log , |x| | x|
β ≥ 0.
(2)
We will prove the following. Theorem 1. Let β ∈ [0, 1) and let w(x) be given by (2). Then
(a)
γ
B(0)
e|u| dx < +∞,
1,N
for all u ∈ W0,rad (B, w),
if and only if
γ ≤ γN ,β :=
N
(N − 1)(1 − β)
=
N′ 1−β
,
and
(b)
sup
∥u∥w ≤1,rad B(0)
γN ,β
eα|u|
dx < +∞
if and only if 1 1−β 1/N −1 α ≤ αN ,β = N ωN −1 (1 − β)
(critical growth),
(3)
where ωN −1 is the area of the unit sphere SN −1 in RN . Remark 2. (a) gives the generalization of Trudinger’s theorem to the weighted case (for radial functions), while (b) extends the Trudinger–Moser inequality; indeed, for β = 0 we recover the classical Trudinger–Moser inequality.
β(N −1) 1,N Next, we consider the case β > 1: then, considering the weight w1 (x) = log |xe| , the functions in W0,rad (B, w1 ) are bounded. Theorem 3. Let β > 1. Then we have the following embedding 1 ,N W0,rad (B, w1 ) ↩→ L∞ (B).
For the borderline case β = 1, one has a critical growth of double exponential type, as described in the following. Theorem 4. Let w1 (x) = | log |xe| |N −1 (i.e. β = 1). Then, setting N ′ =
(a)
B(0)
|u|N
ee
′
dx < +∞,
N ∀ u ∈ W01,,rad (B, w1 ),
N N −1
M. Calanchi, B. Ruf / Nonlinear Analysis (
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while 1
(b)
sup
∥u∥w1 ≤1,rad B(0)
e
ω N −1 |u|N a e N −1
′
dx < +∞ ⇐⇒ a ≤ N .
We point out that the special case N = 2, i.e. the corresponding inequality for W 1,2 (B, w), B ⊂ R2 was studied in [4]. The proof is based on ideas by Leckband [18]. Indeed, in [18] a very general optimal integral inequality is proved. The proof in [4] specializes (and simplifies) the proof of Leckband to the particular situation of W 1,2 (B, w). In this note we show that after some suitable manipulations, the results in Theorems 1 and 4 can be directly deduced from Leckband’s inequality. A simplified direct proof in the spirit of [4] seems also possible. 2. Trudinger–Moser embeddings with weights In order to prove Theorem 1 we start with a Radial Lemma which is of crucial importance in the sequel. Lemma 5. Let u be a radially symmetric C01 function on the ball B = B(0) ⊂ RN . Then one has
1−β log |x| N ′
∥u∥w , ∀ β < 1, (ωN −1 )1/N (1 − β)1/N ′ β(N −1) for w = w0 (x) = log |x| ; while for w = w1 (x) = | log( |xe| )|β(N −1) e 1−β 1/N ′ log − 1 | x| (ii) |u(x)| ≤ ∥u∥w , β ̸= 1 (ωN −1 )1/N |1 − β|1/N ′ 1 ′ (iii) |u(x)| ≤ 1/N log1/N log(e/|x|) ∥u∥w , β = 1. ωN −1 (i) |u(x)| ≤
Proof. (i) Let v(|x|) = u(x). Then, by Hölder’s inequality
|u(x)| = v(|x|) − v(1) = 1
|v ′ (t )| t
=
N −1 N
| x| 1
v ′ (t )dt
| log t |β/N t − ′
N −1 N
′
| log t |−β/N dt
| x| 1
|v (t )| t ′
≤
N
N −1
β(N −1)
| log t |
dt
1/N
|x|
|x|
1
1
1
β(N −1)
1
dt β
NN−1
t | log t |
1/N
| log t |1−β |x| 1 1−β
ωN −1 |v (t )| t | log t | dt 1/N ωN −1 | x| 1−β log |x| N ′ ∥ u∥ w . ≤ (ωN −1 )1/N (1 − β)1/N ′ By a similar procedure, substituting | log t | with | log(e/t )|, one obtains (ii). =
′
N
N −1
For completeness we prove (iii): as above we get
|x| |u(x)| = v ′ (t )dt 1 1 1/N ′ − N −1 −1/N ′ N −1 ≤ |v ′ (t )| t N log(e/t ) t N log(e/t ) dt | x|
1
|v (t )| t ′
≤
N
N −1
(N −1) log(e/t ) dt
1/N
|x|
=
ω N −1
1
(N −1) |v ′ (t )|N t N −1 log(e/t ) dt
| x|
log1/N log(e/|x|) ′
≤
1/N
ω N −1
∥ u∥ w .
NN−1 dt |x| t log(e/t ) 1/N 1/N ′ log log(e/|x|) 1
1
1/N
ωN −1
NN−1
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M. Calanchi, B. Ruf / Nonlinear Analysis (
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β(N −1) Proof of Theorem 1. For the sufficiency part it is enough to consider only the case w(x) = w0 (x) = log |x| . Indeed,
β(N −1) observe that for β ∈ [0, 1) one has, with w1 (x) = log |xe|
H01 (B, w1 ) ↩→ H01 (B, w0 ). We prove the theorem in the critical case, i.e. for γ = γN ,β , the subcritical case is then a direct consequence. Since C0∞ (Ω ) is dense in H01 (B, w0 ), we may assume that u ∈ C0∞ (Ω ) and u ≥ 0 (otherwise replace u by |u|). Since the problem is radially symmetric we introduce the variable t by
|x| = e−t /N and set 1/N
ψ(t ) = ωN −1 N
N −1 (1−β) N
(1 − β)
N −1 N
u(x).
(4)
Then the weighted Sobolev norm becomes
β(N −1) |∇ u| log |x| dx = N
B(0)
+∞
0
|ψ ′ |N t β(N −1) dt (1 − β)N −1
(5)
and the exponential integral
N
ω N −1
α|u|γ
e
+∞
γ −t ¯ eα|ψ| dt
dx =
B
(6)
0
where
γ = γN ,β =
N
(N − 1)(1 − β)
,
and
α , αN ,β
α¯ =
where αN ,β is the critical growth given in (3). We first prove (a), that is we show that the integral in (6), namely
+∞ 0
|ψ ′ |N t β(N −1) (1−β)N −1
+∞ 0
γ −t ¯ eα|ψ| dt, is finite for any ψ satisfying
dt < +∞.
Arguing as in the paper of Moser [23], for all ε > 0 there exists T = T (ε) such that +∞
T
|ψ ′ |N t β(N −1) dt < ε N . (1 − β)N −1
Hence, by Hölder’s inequality
ψ(t ) = ψ(T ) +
t
ψ ′ (s) ds = ψ(T ) +
T
≤ ψ(T ) +
t
|ψ ′ (s)| sβ/N s−β/N ds ′
′
T t
1/N
t
|ψ ′ (s)|N sβ(N −1) ds
T
NN−1
s−β ds T
1/N NN−1 |ψ ′ (s)|N sβ(N −1) 1−β 1−β = ψ(T ) + ds t − T (1 − β)N −1 T NN−1 ≤ ψ(T ) + ε t 1−β − T 1−β for all t ≥ T . t
This implies that there exists T¯ such that N
α¯ ψ (1−β)(N −1) (t ) ≤
1 2
t,
for all t ≥ T¯ .
This is sufficient to guarantee the existence of the integral in (6). In order to prove the uniform estimate Theorem 1(b), remark that by the radial lemma +∞
if 0
|ψ ′ |N t β(N −1) dt ≤ 1, (1 − β)N −1
then ψ(t ) ≤ t
Hence, if α¯ < 1, it follows easily by (7) that +∞
0
γ −t ¯ eα|ψ| dt ≤
+∞
0
eα¯ t −t dt < +∞.
N −1 (1−β) N
, ∀ t > 0.
(7)
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Sharpness: Now we prove that the theorem is sharp in the sense that if α > αN ,β Theorem 1(b) is infinite.
5
⇐⇒ α¯ > 1, then the supremum in
β(N −1) Again we consider first the case w = w0 (x) = log |x| . It is sufficient to test
+∞
γ −t ¯ eα|ψ| dt
0
on the family of functions of Moser type
ηk (t ) =
1−β t
t ≤ k, 1−β k(1N−β)(N −1) N
t ≥ k.
k
One easily sees that Then, with γ = +∞
|ηk′ |N t β(N −1)
+∞
0 (1−β)N −1 N (N −1)(1−β)
¯ k | γ −t eα|η dt ≥
0
+∞
dt = 1.
eα¯ k−t dt → +∞,
if α¯ > 1.
k
β(N −1) For the case w(x) = log |xe| the family of functions becomes (with the change of variables t = N − N log |x|)
t 1−β − N 1−β 1/N , ξk (t ) = (k + N )1−β − N 1−β N −1 (k + N )1−β − N 1−β N ,
N ≤t ≤k+N t ≥ k + N.
Then ξk has norm 1, and +∞
¯ k | γ −t eα|ξ dt ≥
N
+∞
1 1−β −N 1−β ] 1−β
¯ k+N ) eα[(
−t
dt
k+N 1 1−β −N 1−β ] 1−β
¯ k+N ) = eα[(
−(k+N )
→ +∞,
as k → ∞, if α¯ > 1.
The critical case α¯ = 1. This case is more delicate. We will complete the proof in Section 4, applying a result by Leckband [18], which we will also use for the double exponential case.
β(N −1) Proof of Theorem 3. In the next result we show that in the case of w1 (x) = log |xe| and β > 1 the functions in 1 ,N W0,rad (B) are bounded. This follows easily by the radial lemma. In fact by (ii) in Lemma 5 we have |u(x)| ≤
1/N ′ e 1−β log − 1 |x|
∥ u∥ w ≤
(ωN −1 )1/N |1 − β|1/N ′
2∥u∥w . (ωN −1 )1/N |1 − β|1/N ′
3. A limiting case with double exponential growth In this section we consider the limiting case β = 1, i.e. weights of the form (log |xe| )N −1 . As stated in Theorem 4 we obtain a critical growth of double exponential type. We remark that embeddings into Orlicz spaces with double exponential growth have recently been obtained by Cianchi [8] and belonging to Sobolev–Zygmund spaces WLn Log α L(Ω ), Ω ⊂ RN , i.e. for functions α Hencl [14] for functions u such that Ω |∇ u|N log |∇ u| dx is bounded. See also the extensions by Černý–Maškowá [7] to the case of multiple exponential growths. These results are of different type than the ones in Theorem 4: they involve a logarithmic Sobolev norm, while our results are based on standard Sobolev norms with logarithmic weights. Proof of Theorem 4 (Subcritical and Supercritical Cases). (a) can be proved with the same argument used in the first part of Theorem 1. 1/N In order to prove (b), set ψ(t ) = ωN −1 u(x) with |x| = e−t . Then
e N −1 |∇ u(x)|N log dx = |x| B
+∞
|ψ ′ (t )|N (1 + t )N −1 dt 0
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and 1
e
ω N −1 |u|N ae N −1
′
dx =
B
ωN −1
+∞
N
eae
′ |ψ|N (t ) −Nt
dt .
0
Thus, we need to prove that
ω N −1
N
+∞
′ |ψ|N (t ) −Nt
eae
for all ψ with
dt ≤ C
+∞
0
|ψ ′ (t )|N (1 + t )N −1 dt = 1.
(8)
0
By (iii) of the Radial Lemma one has +∞
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1 ⇒ |ψ(t )| ≤ log
N −1 N
(1 + t ),
∀ t > 0.
0
Therefore +∞
ae|ψ|
e
N ′ (t )
−Nt
e
+∞
dt ≤
0
ea(1+t )−Nt dt < +∞,
if a < N .
0
Sharpness. The necessary condition If a > N it is sufficient to test the functional on the following sequence
log(1 + t ) 0≤t ≤k ψk (t ) = log1/N (1 + k) N −1 log N (1 + k) t ≥ k. We have +∞
|ψk′ (t )|N (1 + t )N −1 dt = 1 0
and +∞
eae
′ ψkN (t )
e−Nt dt ≥
+∞
eae
′ ψkN (t )
e−Nt dt
k
0
+∞
ea+ak−Nt dt → +∞,
=
as k → +∞.
k
Critical exponent: If a = N the estimate is more delicate. We have to prove that there exists C > 0 such that +∞
eNe
′ |ψ|N (t ) −Nt
dt < C ,
∀ ψ with
0
+∞
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1. 0
In order to prove this, we make use of a result due to Leckband, which we explain in the following section. 4. The Leckband–Neugebauer result and the proofs of the critical cases We begin with the definition of a special class of functions, the so-called C ∗ -functions, see [24,18]. Definition 6. A continuous function ρ : [0, +∞) → [0, +∞) is called a C ∗ -function if there is a constant Cρ > 0 such that for every 0 < d < +∞, there exists a constant D > 0 with
ρ((l + d)s) ≤ Cρ ρ (l − 1)s ,
for all l ≥ D and 0 < s < +∞.
One easily notes that Example 7. Any concave function is a C ∗ -function. The product of C ∗ -functions is a C ∗ -function. Next, we introduce the notion of a C ∗ -convex function by the following implicit definition. Definition 8. A C 1 -function M : [0, +∞) → [0, +∞) is called a C ∗ -convex function if M is convex, and the function ρ defined by the differential equation ρ(M (t )) = M ′ (t ) is a C ∗ -function.
M. Calanchi, B. Ruf / Nonlinear Analysis (
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Some typical examples of C ∗ -convex functions: p
Example 9. M1 (t ) = t p and M2 (t ) = et − 1 with p > 1 are C ∗ -convex functions. Indeed their respective C ∗ -functions are
ρ1 (s) = ps
p−1 p
and ρ2 (s) = p(s + 1) log
p−1 p
(s + 1).
We now state a general functional inequality by Leckband [18]. This quite unimpressive looking inequality is actually quite magic, and contains many of the known borderline inequalities. It is a generalization of a previous result by Neugebauer [24], which in turn is a generalization of Moser’s integral inequality [23], see also [21]. Theorem 10 (Leckband’s Inequality [18]). Let f ∈ LN ([0, +∞)) such that ∥f ∥N = 1, ϕ : R+ → R+ with ϕ ≥ 0 is locally integrable, and set G(x) =
x
ϕ
N N −1
NN−1
(y)dy
F (x) =
and
x
f (y)ϕ(y)dy. 0
0
Let Φ ≥ 0 be a nonincreasing function on [0, +∞), and M (t ) a C ∗ -convex function. Then there exists a constant C > 0 +∞
Φ M G(t ) − M F (t ) d M G(t ) ≤ C ∥Φ ∥1 .
(9)
0
Proof of Theorem 1 (Critical Case). We now complete the proof of Theorem 1 in the critical case α¯ = 1, i.e. α = αN ,β , applying the Leckband inequality to f (t ) = ψ ′ (t )
t β NN−1 , 1−β
ϕ(t ) =
t β 1−NN 1−β
where ψ(t ) ∈ C 1 ([0, ∞)) is given by (4). Then ∞
|f (t )| dt = N
0
t β(N −1)
∞
|ψ ′ (t )|N 0
(1 − β)N −1
dt = 1
by (5). Furthermore, with this choice of ϕ one has G(x) =
x
ϕ
N N −1
NN−1
(y)dy
=
x
0
0
1−β
N −1 N
tβ
= x(1−β)
N −1 N
and F (x) =
x
f (y)ϕ(y)dy = 0
x
ψ ′ (y)dy = ψ(x). 0
Finally, set N
M (s) = s (N −1)(1−β) ,
Φ (s) = e−s
Then we get M (G(t )) = t ,
N
M (F (t )) = ψ (N −1)(1−β) (t ) = ψ γ (t ).
and
Hence, with these particular choices, the Leckband inequality becomes +∞
γ eψ (t )−t dt ≤ C
0
+∞
e−s ds = C ,
for all ψ with
0
+∞
|ψ ′ (t )|N 0
t β(N −1)
(1 − β)N −1
dt ≤ 1.
This is precisely inequality (6) for α¯ = 1, which we wanted to prove. Proof of Theorem 4. For the proof of the double exponential case (Theorem 4), we apply Leckband’s inequality to the functions f (t ) = ψ ′ (t )(1 + t )
N −1 N
,
ϕ(t ) = (1 + t )
Then ∞
|f (t )|dt = 0
∞
|ψ ′ |N (1 + t )N −1 dt = 1 0
1−N N
.
8
M. Calanchi, B. Ruf / Nonlinear Analysis (
)
–
by (8), and by the choice of ϕ one has G(x) =
x
NN−1 N ϕ N −1 (y)dy =
0
x
1 y+1
0
NN−1
dy
N −1 = log(1 + x) N
and F (x) =
x
f (y)ϕ(y)dy =
x
ψ ′ (y)dy = ψ(x). 0
0
Finally, setting M (t ) = et
N N −1
− 1,
Φ (s) = e−Ns
the Leckband inequality becomes in this particular case +∞
eNe
′ ψ N (t ) −Nt
+∞
e−Ns ds = CN ,
dt ≤ C
0
for all ψ with
|ψ ′ (t )|N (1 + t )N −1 dt ≤ 1 0
0
and this is inequality (8).
+∞
1
−1 Remark 11. We point out that if b < ωNN− 1 then the nonlinearity
ea e
′ b |u|N 1
−1 is subcritical for any a > 0, and if b > ωNN− 1 then it is supercritical, for any a > 0.
5. A remark on the supercritical case 1,N
In this section we do not restrict to the radial case, and consider all functions in W0 (B, w). We prove that in this case we cannot go beyond the classical TM inequality. Indeed, in the supercritical case with respect to the Moser inequality, 1 ,N i.e. γ = NN−1 , α > N ωN −1 , the supremum in the whole space W0 (B, w) turns out to be infinite. So the presence of the weight has no effect with regard to criticality. We will prove this only in the case w(x) = log |xe| being similar.
β(N −1)
, the other cases
1
Proposition 12. Let α > N wnN−−11 . Then
eα|u|
sup
N N −1
dx = +∞.
∥u∥w =1 B(0)
Proof. It is sufficient to evaluate the functional 1
e
N
N −1 N ωN −1 (1+δ)β |u| N −1
dx,
δ>0
B
on a suitable translation of the following Moser sequence
wk (x) =
1
ω
1/N N −1
N −1 (log k) N
|x| ≤
− log |x| (log k)1/N
| x| ≥
1 k 1 k
.
Indeed, consider
wk,a (x) =
1
N −1 (log k) N a
|x − xa | <
a k
log |x−x | a
a
0
|x − xa | ≥ a
1/N ωN −1 1/N (log k)
k
≤ |x − xa | < a
where xa = (1 − a, 0), 0 < a < 1/2, k > 2. We observe that ∥∇wk,a ∥LN = 1. Indeed
|∇wk,a (x)|N dx = B
1 log k
a
1
a/k
r
dr = 1.
(10)
M. Calanchi, B. Ruf / Nonlinear Analysis (
It is now sufficient to choose a <
uk,a (x) = log
e
1 − 2a
1 2
(1 −
1 eδ
)
–
9
) and test the functional on the following sequence
−β NN−1 wk,a (x).
This sequence satisfies
log e β(N −1) | x| |∇wk,a (x)|N dx log e B 1−2a ≤ |∇wk,a (x)|N dx ≤ 1
∥uk,a ∥Nw =
B
since |x| ≥ 1 − 2a, recalling that the support of this sequence is contained in a ball of radius a centered in (1 − a, 0). Moreover
B
since
1
−1 exp N ωNN− 1
1 + δ β N e wkN,a−1 dx → +∞, log
1+δ > 1 by the choice of a.
log 1−e2a
k → +∞
1−2a
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