Moser–Trudinger inequality for functions with mean value zero

Nonlinear Analysis 66 (2007) 2742–2755 www.elsevier.com/locate/na

Moser–Trudinger inequality for functions with mean value zero Yunyan Yang Department of Mathematics, Information School, Renmin University of China, Beijing 100872, PR China Received 16 January 2006; accepted 5 April 2006

Abstract Let Ω be a bounded smooth domain in Rn (n ≥ 2). This paper deals with the Moser–Trudinger inequality for functions with mean value zero. Using blowing up analysis, the author proves that     n n−1 sup eα|u| dx : u ∈ H 1,n (Ω ), |∇u|n dx = 1, udx = 0 Ω

Ω

Ω

is attained for any α ≤ αn , and the supremum is infinity for any α > αn , where αn = n(ωn−1 /2)1/(n−1) , and ωn−1 is the area of the unit sphere in Rn . c 2006 Elsevier Ltd. All rights reserved.  MSC: 46E35 Keywords: Moser–Trudinger inequality; Blowing up analysis; Extremal function

1. Introduction In 1988, Chang and Yang [3] proved the following result: Theorem A (Chang–Yang). Suppose that D is a piecewise C 2 , bounded, finitely connected domain in the plane with a finite number of vertices. Let θ D be the minimum interior angle at at the vertices of D. There exists a constant c D such that, for all u ∈ C 1 (D) with  the plane  2 D |∇u| dx ≤ 1, and D udx = 0, we have  2 e2θ D u dx ≤ c D . D

E-mail address: yunyan [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2006.04.004

Y. Yang / Nonlinear Analysis 66 (2007) 2742–2755

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If we replace 2θ D with any positive β, the integral is still finite, but if β > 2θ D , it can be made arbitrarily large by an appropriate choice of u. Later, Fontana [6] conjectured that there should be some extension of Theorem A in high dimension: Conjecture (Fontana). Let D be a domain in Rn with compact closure. Suppose that the limit |∂ Br (P) ∩ D| (1.1) r n−1 exists for any P ∈ ∂ D, where the numerator denotes the n − 1 surface measure of ∂ Br (P) ∩ D.  Then there must be a constant C such that, for all f ∈ C 1 (D) satisfying D |∇ f |n dx ≤ 1 and  1/(n−1) : D f dx = 0, the following holds for α = nθ D  exp[α| f |n/(n−1) ]dx ≤ C. θ (P) = lim

r→0

D 1/(n−1)

For any α > nθ D

, no constant C can bound the integral uniformly with respect to f .

Let Ω be a smooth bounded domain in Rn , and H 1,n (Ω ) the completion of C ∞ (Ω ) in the norm 1/n   n  |u| + |∇u|n dx .

u H 1,n (Ω ) = Ω

Define a function space     H = u ∈ H 1,n (Ω ) : |∇u|n dx = 1, udx = 0 . Ω

Ω

Then we can state our main result in this paper as follows: Theorem 1.1. Let Ω be a smooth bounded domain in Rn , and H be as above. Then we have  n/n−1 sup eα|u| dx < +∞ (1.2) u∈H Ω

for any α ≤ αn = n(ωn−1 /2)1/(n−1), and the supremum is infinity for any α > αn , where ωn−1  n/(n−1) is the area of the unit sphere in Rn . Furthermore, supu∈H Ω eαn |u| dx is attained. Remark 1.2. By the definition of θ (see 1.1), we have, for any p ∈ ∂Ω , θ ( p) = ωn−1 /2. Hence θΩ = ωn−1 /2, whence Theorem 1.1 gives more information than Fontana’s conjecture in the case that ∂Ω is smooth. Remark 1.3. When Ω = B1 = {x ∈ Rn : |x| ≤ 1}, Leckband [8] obtained the inequality (1.2) via the method of Carleson–Chang [2]. Remark 1.4. When n = 2, we have proved Theorem 1.1 in [19].  n/(n−1) We write Jα (u) = Ω eα|u| dx for simplicity. To prove Theorem 1.1, we first find a maximizer u  ∈ H of the subcritical functional Jαn − for any  > 0, and give the corresponding Euler–Lagrange equation, which is a nonlinear elliptic equation. Second, we use blowing up analysis to understand the asymptotic behavior of u  . Third, using the capacity technique, we can

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derive an upper bound of Jαn under the assumption that blowing up occur. Finally we construct a function sequence to show that the upper bound of Jαn is strictly great than the one derived in the third step. This contradiction indicates that no blowing up occurs, whence Theorem 1.1 holds by elliptic estimates. Note that the main idea of the proof proceeds similarly as Li did in [11]; some modifications are needed, and some technical difficulty will be smoothed in our situation. For extremal functions for the Moser–Trudinger inequality, we would like to mention Carleson–Chang [2], Flucher [5], Lin [13], Li [9,10], Li–Liu [12], Yang [20,21]. We organize this paper as follows. In Section 2, we give the best constant for inequality (1.2). Section 3 contributes to the asymptotic behavior of maximizers of subcritical functionals Jαn − . An upper bound of Jαn is derived under the assumption that blowing up occurs in Section 4. In the last section, we construct a function sequence to show that the upper bound of Jαn is in fact greater than the one that we derived in Section 5, and this contradiction completes the proof of Theorem 1.1. 2. The best constant In this section, we will show that αn = n(ωn−1 /2)1/(n−1) is the best constant for inequality (1.2). Here, the best means for any α < αn is supu∈H Jα (u) < +∞, while for any α > αn , supu∈H Jα (u) = +∞. We first state a well-known lemma: Lemma 2.1. There exists an α0 > 0 such that supu∈H Jα0 (u) < +∞. Denote α˜ = sup{α : supu∈H Jα (u) < +∞}. Then we have the following: Lemma 2.2. α˜ = n(ωn−1 /2)1/(n−1). Proof. In [6], Fontana constructed a blowing up sequence to show that α˜ cannot exceed n(ωn−1 /2)1/(n−1). We only need to show that α˜ ≥ n(ωn−1 /2)1/(n−1). Supposing not, then there exists a sequence u  ∈ H such that α˜ +  < n(ωn−1 /2)1/(n−1) and Jα+ ˜ (u  ) → +∞.

(2.1)

Then, passing to a subsequence, we can assume that u  u0 u → u0

weakly in H 1,n (Ω ) strongly in L 2 (Ω ).

We first claim that u 0 = 0. If u 0 = 0, then, by the Brezis–Lieb Lemma [14], we have   n |∇(u  − u 0 )| dx → 1 − |u 0 |n dx < 1. Ω

Ω

Hence there exists α1 > α˜ such that Jα1 (u  − u 0 ) ≤ C for sufficiently small . It is not difficult to derive that  1/ p  1/ p  n n n eα2 |u  | dx ≤ eα1 |u  −u 0 | dx ec|u 0 | dx Ω

Ω

Ω

for some constants α˜ < α2 < α1 , 1/ p + 1/ p = 1 and c depending only on α1 , α2 . Therefore Jα2 (u  ) is bounded, which contradicts (2.1), whence u 0 = 0. Once u 0 = 0, it is routine that |∇u  |n dx  δ p for some p ∈ Ω , whence  n ˜  | n−1 e(α+)|u dx → +∞ (2.2) Br ( p)∩Ω

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n

for any r > 0. If p ∈ Ω , the usual truncation gives that e|u  | n−1 is bounded in L α (Br ( p)) for some α > n(ωn−1 /2)1/(n−1) and r > 0. This contradicts (2.2), whence p ∈ ∂Ω . Take a normal coordinate system (V, φ) near p such that φ( p) = 0, φ : Ω ∩ V → Rn + = {x ∈ Rn : x 1 > 0}, φ : ∂Ω ∩ V → ∂Rn + = {x ∈ Rn : x 1 = 0}. Assume that φ −1 (Bδ (0)) ⊂ V . Choose a cutoff function η ∈ C0∞ (Bδ (0)), η ≡ 1 on Bδ/2 (0). Set  η(u  ◦ φ −1 )(x 1 , x ), x 1 ≥ 0, u˜  = η(u  ◦ φ −1 )(−x 1 , x ), x 1 < 0. Then u˜  ∈ H01(Bδ (0)). One can easily see that  lim |∇ u˜  |n dx = 2 + oδ (1), →0

Bδ (0)

where oδ (1) → 0 as δ → 0. By the classical Moser’s result [15], we have  n n−1 eα|u˜  | dx < +∞ lim sup →0

Bδ (0)

for some α : α˜ < α < n(ωn−1 /2)1/(n−1), provided that δ is sufficiently small. Hence  n n−1 eα|u  | dx < +∞, lim sup →0

φ −1 (Bδ/2 (0))∩Ω

which contradicts (2.2). Therefore we have α˜ ≥ n(ωn−1 /2)1/(n−1).



3. Blowing up analysis In this section, we use the method of blowing up analysis to understand the asymptotic behavior of the maximizers of the subcritical functional Jαn − . Lemma 3.1. For any  > 0, Jαn − defined on H admits a maximizer u  ∈ H ∩ C 1 (Ω). Proof. It is well known that there exists u  ∈ H such that Jαn − (u  ) = sup Jαn − (u). u∈H

The Euler–Lagrange equation is n ⎧ 2−n 1 μ n−1 n−1 e(αn −)|u  | ⎪ −Δ u = u |u | − n    ⎪ ⎪ λ λ ⎪  ⎪ ⎪ ⎪ ∂u  ⎪ = 0 on ∂Ω ⎪ ⎪ ⎪ ∂n ⎪   ⎪ ⎨ |∇u  |n dx = 1, u  dx = 0 ⎪ Ω  Ω ⎪ n ⎪ n ⎪ n−1 ⎪ n−1 e(αn −)|u  | = |u | dx λ ⎪   ⎪ ⎪ ⎪ Ω ⎪  ⎪ n ⎪ 2−n ⎪ n−1 ⎩μ = 1 u  |u  | n−1 e(αn −)|u  | dx,  |Ω | Ω

in Ω

where Δn u  = div(|∇u  |n−2 ∇u  ). By elliptic estimates, we have u  ∈ C 1 (Ω).

(3.1)



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Lemma 3.2. lim→0 Jαn − (u  ) = supu∈H Jαn (u). Proof. Obviously, lim sup→0 Jαn − (u  ) ≤ supu∈H Jαn (u). By the definition of u  , we have, for any fixed u ∈ H,   n n n−1 αn |u| n−1 e dx ≤ lim inf e(αn −)|u| dx →0 Ω Ω n n−1 ≤ lim inf e(αn −)|u  | dx, →0

whence



sup

u∈H Ω

Ω



n

e

αn |u| n−1

dx ≤ lim inf →0

Ω

e(αn −)|u  |

Hence lim→0 Jαn − (u  ) = supu∈H Jαn (u).

n n−1

dx.



Lemma 3.3. lim inf→0 λ > 0. Proof. Using the inequality et ≤ 1 + tet for t ≥ 0, one has  n n−1 e(αn −)|u  | dx ≤ |Ω | + (αn − )λ .

(3.2)

By Lemma 3.2,   n n n−1 n−1 lim e(αn −)|u  | dx = sup eαn |u| dx > |Ω |.

(3.3)

Ω

→0 Ω

u∈H Ω

Combining (3.2) and (3.3), one gets the result.



Lemma 3.4. μ /λ is bounded. Proof. The following estimates are straightforward:  n 1 1 1 n−1 |μ |/λ ≤ |u  | n−1 e(αn −)|u  | dx |Ω | Ω λ n  n 1 αn |u  | n−1 (αn −)|u  | n−1 1 ≤ e + e dx λ |Ω | |u  |≥1 λ 1 1 αn , e + ≤ λ |Ω | which, together with Lemma 3.3, gives the result.  Denote c = |u  |(x  ) = maxx∈Ω |u  (x)|. If c is bounded, Theorem 1.1 holds by applying elliptic estimates to Eq. (3.1). In the following, without loss of generality, we assume that u  (x  ) → +∞; otherwise, we consider −u  instead of u  . Passing to a subsequence, we may assume that x  → p ∈ Ω . Using the same argument in the proof of Lemma 2.2, we have u  → 0 strongly in L n (Ω ), |∇u  |n dx  δ p weakly in the sense of measure, and p ∈ ∂Ω . Let (V, φ) be a normal coordinate system around p such that φ( p) = 0, φ(∂Ω ∩ V ) = {y ∈ Rn : y1 = 0} ∩ B1 (0) and φ(V ∩ Ω ) = {y ∈ Rn : y1 > 0} ∩ B1 (0). Write k k n  ∂φ −1 ∂φ −1 gi j = gi j (y) = ∂yi ∂y j k=1

g = det(gi j );

(g i j )n×n = (gi j )−1 n×n ,

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where (gi j )−1 n×n denotes the inverse of the matrix (gi j )n×n . In this coordinate system, ∂gi j (0) = 0, ∂y l

gi j (0) = δi j ,

for i, j, l = 1, 2, . . . , n, where δi j = 0 for i = j , and δi j = 1 for i = j . Set  u ◦ φ −1 (y1 , y ) for y1 ≥ 0 u˜  (y) =  for y1 < 0, u  ◦ φ −1 (−y1 , y ) which is defined on B1 (0). We still denote φ(x  ) by x  for simplicity. Let ⎧ 1 − 1 n/(n−1) ⎪ ⎪ r = λn c n−1 e−(αn −)/nc ⎪ ⎨ 1 ψ (y) = u˜  (x  + r y) ⎪ c  ⎪ ⎪ 1 ⎩ ϕ (y) = cn−1 (u˜  (x  + r y) − c ),

(3.4)

where ψ and ϕ are defined on Ω = {x ∈ Rn : x  + r y ∈ B1 (0)}. Then we have ⎛ ⎞  n−2 1 ∂ ∂ u ˜ ∂ u ˜ ∂ u ˜ √    ⎝g i j g g kl ⎠. Δn u  (x) = Δg,n u˜  (y) = √ g ∂yi ∂y k ∂y l ∂y j Hence we have, by (3.1), −Δg,n u˜  (y) =

n 2−n 1 μ n−1 u  |u  | n−1 e(αn −)|u  | − . λ λ n/(n−1)

Lemma 3.5. For any 0 < α < αn , we have rn eαc

→ 0 as  → 0.

Proof. By the definition of r , we have, for any 0 < α < αn , n/(n−1)

rn eαc

n/(n−1)

= λ c−n/(n−1) e−(αn −)c

n/(n−1)

n/(n−1)

eαc 

n

= c−n/(n−1) e−(αn −−α)c |u  | n−1 e(αn −)|u  | Ω  n n n−1 ≤ c−n/(n−1) |u  | n−1 eα|u  | dx

n n−1

dx

Ω

n

for sufficiently small . Clearly, |u  | n−1 eα|u  | we get the result. 

n n−1

is bounded in L β (Ω ) for some β > 1. Hence

By Eq. (3.1), we have 2−n

−Δg,n ψ (y) = c1−n ψ |ψ | n−1 e(αn −)(|u˜  |

n n n−1 −cn−1 )

− c−n rn

μ , λ

(3.5)

and 2−n

−Δg,n ϕ (y) = ψ |ψ | n−1 e(αn −)(|u˜  |

n n n−1 −cn−1 )

− c rn

μ . λ

(3.6)

By the standard elliptic estimates and the Liouville Theorem for n-harmonic functions, ψ → 1

1 in Cloc (Rn ).

(3.7)

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On the other hand, for any fixed R > 0, we have, by (3.7), n

n

n

n

|u˜  | n−1 − cn−1 = cn−1 (|ψ | n−1 − 1)  n n (|ψ | − 1) + O((|ψ | − 1)2 ) = cn−1 n−1 n = ϕ (1 + O((ψ − 1)2 )) n−1 in B R as  → 0. Applying the Harnack inequality for an n-Laplace equation [16] and Lemmas 3.4 and 3.5 to Eq. (3.6), one can see that ϕ is bounded in L ∞ (B R ). Then elliptic estimates [18] imply that ϕ is bounded in C 1,γ (B R/2) for some 0 < γ < 1, whence ϕ → ϕ in C 1 (B R/4 ). Clearly, we have   n n n n−1 n−1 αn ϕ(y) n−1 e dy ≤ lim c−1/(n−1)u˜ 1/(n−1) e(αn −)(|u˜  | −c ) dy →0

B R/4

B R/4



≤ 2(1 + O(δ)) lim

→0

≤ 2(1 + O(δ)).

n n 1 n−1 |u  | n−1 e(αn −)|u  | dx Bδ ( p)∩Ω λ

Letting R → +∞ first, and then δ → 0, we have ⎧ n αn ϕ ⎪ in Rn ⎪−Δn ϕ = e n−1 ⎪ ⎨ϕ(0) = sup ϕ = 0 Rn ⎪ n ⎪ ⎪ ⎩ e n−1 αn ϕ dx ≤ 2.

(3.8)

Rn

Let Vt = {x ∈ Rn : ϕ(x) > t}. Using the same argument as the proof of Ding’s Lemma [4],   n n one can show that Rn e n−1 αn ϕ dx ≥ 2. Hence we have, by (3.8), Rn e n−1 αn ϕ dx = 2, and ∂Vt are all spheres, which implies that ϕ is radially symmetric and   1 ω n n−1 n−1 n−1 n−1 ϕ(x) = − . (3.9) log 1 + |x| αn 2n In fact, we have proved the following: 1 (Rn ), where ψ , ϕ and ϕ are as above. Lemma 3.6. ψ → 1, and ϕ → ϕ in Cloc  

Defining u ,β = min{u  , βc }, we have the following:  Lemma 3.7. lim→0 Ω |∇u ,β |n dx = β, ∀0 < β < 1. 

Proof. The proof is completely analogous to [11], so we omit it. n/(n−1)

Lemma 3.8. lim→0 Jαn − (u  ) ≤ |Ω | + lim sup→0 λ /c

.

Proof. We have, for any 0 < β < 1,   n n n−1 n−1 Jαn − (u  ) = e(αn −)|u  | dx + e(αn −)|u  | dx u <βc u  ≥βc    n n−1 ≤ e(αn −)|u ,β | dx + β −n/(n−1) λ /cn/(n−1) . Ω

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By Lemma 3.7, one has  n n−1 e(αn −)|u ,β | dx → |Ω |. Ω

Hence Jαn − (u  ) ≤ |Ω | + β −n/(n−1) λ /cn/(n−1) + o (1). Letting  → 0 first, then β → 1, we get the result.



An immediate result of Lemma 3.8 is the following: Corollary 3.9. lim→0 c /λ = 0. Lemma 3.10. ∀φ ∈ C ∞ (Ω ), we have  n 2−n 1 n−1 lim φ c u  |u  | n−1 e(αn −)|u  | dx = φ( p). →0 Ω λ Proof. We divide Ω into three parts: Ω = ({u  > βc } \ B Rr (x  )) ∪ {u  ≤ βc } ∪ (Ω ∩ B Rr (x ) ) for some 0 < β < 1. Denote the integrals on the above three domains by I1 , I2 and I3 , respectively:  n n 1 n−1 c u n−1 e(αn −)|u  | dx |I1 | ≤ sup |φ| {u  >βc }\B Rr (x  ) λ Ω    n n 1 n−1 (αn −)|u  | n−1 ≤ sup |φ| 1 − c u  e dx Ω ∩B Rr (x ) λ Ω    n α ϕ = sup |φ| 1 − e n−1 n dx + o (R) . Ω

B+ R

Letting  → 0 first, then R → +∞, we have I1 → 0:  n n c n−1 |u  | n−1 e(αn −)|u ,β | dx, |I2 | ≤ sup |φ| λ Ω Ω which, together with Lemma 3.7 and Corollary 3.9, gives I2 → 0. It is easy to see that   n e n−1 αn ϕ dx + o (R) I3 = φ(x  + r ξ ) B+ R

for some ξ ∈ B R+ (0). Letting  → 0 first, then R → +∞, we have I3 → φ( p). Combining all the above estimates, we finish the proof of the lemma.  The following phenomenon was first discovered by Brezis–Merle [1], developed by Struwe [17], and generalized by Li [11] on Riemannian manifolds:

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Lemma 3.11. Let u ∈ C 1 (Ω ) be a weak solution of ⎧ −Δn u = f in Ω ⎪ ⎪ ⎨ ∂u   =0 ⎪ ∂n ∂ Ω ⎪ ⎩

u L n (Ω ) ≤ c0 .

(3.10)

1  If f ∈ L 1 (Ω ), then we have, for any q: 1 < q < n, ( Ω |∇u|q dx)1/q ≤ C f Ln−1 1 (Ω ) for some constant C depending only on q and c0 .

Using the same method of proving Theorem 4.7 in [11], we immediately have: 1

Lemma 3.12. cn−1 u   G weakly in H 1,q (Ω ) for any 1 < q < n, where G is a Green function satisfying ⎧ −Δn G = δ p − 1/|Ω | in Ω ⎪ ⎪ ⎪ ⎪ ⎨∂G = 0 on ∂Ω \ { p} (3.11) ∂n  ⎪ ⎪ ⎪ ⎪ ⎩ Gdx = 0. Ω

1

Furthermore, cn−1 u  → G in C 1 (Ω ) for any domain Ω ⊂⊂ Ω \ { p}. It is well known that G takes the following form: G(x) = −(2/ωn−1 )1/(n−1) log |x − p| + A p + v(x),

(3.12)

where A p is a constant, and v(x) ∈ C 0 (Ω) ∩ C 1 (Ω \ { p}). 4. The upper bound of Jαn In this section, we use the capacity technique to derive the upper bound of Jαn . As in Section 3, choose a normal coordinate system (V, φ) around p such that φ( p) = 0 and φ : V ∩ Ω → {x = (x 1 , x ) ∈ Rn : x 1 > 0}. Denote φ(x  ) by (x 1 , x  ), and φ −1 (0, x  ) by x  . Then x  ∈ ∂Ω , and we have x 1 /r → 0 by Lemma 3.6. Let G  (x) be a distributional solution of ⎧ −Δn G  (x) = δx  ⎪ ⎪ n ⎪ ⎨ G | log δ ∂ Bδ (x  )∩Ω = − α (4.1) n  ⎪  ∂ G ⎪  ⎪  = 0. ⎩ ∂n ∂ Ω ∩Bδ (x  ) According to Kichenassamy–Veron [7] and Li [11], using a reflection argument, one can obtain the existence of G  , which can be represented by n G  (x) = − log |x − x  | + v (x), αn where v = O(δ) uniformly with respect to .

Y. Yang / Nonlinear Analysis 66 (2007) 2742–2755

Defining a space of functions  Λ = u ∈ H 1,n ({x ∈ Ω : c1 ≤ G  (x) ≤ c2 }) :   ∂u  u|G  =c1 = a, u|G  =c1 = b, = 0 . ∂n ∂ Ω  It can be seen that infu∈Λ c1 ≤G  ≤c2 |∇u|n dx is attained by a function Ψ satisfying ⎧ ⎪ ⎨ Δn Ψ = 0 Ψ |G  =c1 = a, Ψ |G  =c2 = b ⎪ ⎩∂Ψ /∂ n| = 0.

2751

(4.2)

∂ Ω ∩Bδ (x  )

Eq. (4.2) has a unique solution Ψ=

b(G  − c1 ) − a(G  − c2 ) , c2 − c1

(4.3)

which, together with (4.1), gives  (b − a)n |∇Ψ |n dx = . (c2 − c1 )n−1 c1 ≤G  ≤c2

(4.4)

Choose y ∈ Ω ∩ Bδ (x  ) such that |y − x  | = Rr , where r has been defined in (3.4). Set S = {x ∈ Ω ∩ Bδ (x  ) : G  (x) = G  (y )}. One can see that S ⊂ Ω ∩ (Becδ Rr (x  ) \ Be−cδ Rr (x  )) for some universal constant c and sufficiently small . By Lemma 3.6, one has u  | S  ≥ b  = c +

ϕ(ecδ R) + o (1) 1

,

(4.5)

cn−1

where o (1) → 0 as  → 0. By Lemma 3.12, one gets sup

u  |Ω ∩∂ Bδ (x  ) ≤ a =

Ω ∩∂ Bδ (x  )

G + o (1) 1

,

(4.6)

cn−1

where G is defined in (3.12). Denote G = {x ∈ Ω ∩ Bδ (x  ) : G  (x) < G  (y )}. Set u  = min{max{u  , a }, b }, then one has u  ∈ Λ and    |∇u  |n dx ≤ 1 − |∇u  |n dx − |∇u  |n dx. (4.7) G

By (4.4), we have  |∇u  |n dx ≥ G

Ω \Bδ (x  )

(b − a )n . (G  (y ) + αnn log δ)n−1

Ω ∩Bδ (x  )\G

(4.8)

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Applying the divergence theorem to Eq. (3.11), we have  n |∇G|n dx = − log δ + A p + O(δ) + o (δ), α n Ω \Bδ (x  ) where o (δ) → 0 for any fixed δ > 0 as  → 0. Hence, by Lemma 3.12,   1 n |∇u  |n dx = n − log δ + A p + O(δ) + o (δ) . αn Ω \Bδ (x  ) cn−1

(4.9)

(4.10)

A direct calculation shows that  1 1    n−1 n − 1 n−1 1 2 1 2 n |∇ϕ| dx = − 1+ + ···+ + ωn−1 n 2 n−1 ωn−1 B+ R (0)  1   n n ωn−1 n−1 n−1 log 1 + × R n−1 + O(R − n−1 ). (4.11) n 2n 1

1 (Rn + ), we have Hence, by the fact that cn−1 (u  (x  + r x) − c ) → ϕ in Cloc

 1 1 n−1 1 + + ···+ ωn−1 n 2 n−1  ω  1 n n−1 n−1 −cδ − log 1 + (e R) n−1 2n  + o R (δ) + o (1) + o (R) ,



n − n−1

Ω ∩Bδ (x  )\G



|∇u  |n dx ≥ −c

2



1 n−1

(4.12)

where o R (δ) → 0 for any fixed δ as R → +∞, and o (R) → 0 for any fixed R as  → 0. On the other hand,    n − n−1

(b − a )n ≥ cn 1 − nc

sup

Ω ∩∂ Bδ (x )

G − ϕ(ecδ R) + o (1)

for sufficiently small , and   n−1   n n  n−1 n δ n − n−1 log δ ≤ 1− c 1 + c log G  (y ) + αn αn αn R  1 λ − log n/(n−1) + O(δ) + o (1) . αn c

(4.13)

(4.14)

Adding (4.10) and (4.12)–(4.14) to (4.7) and (4.8), letting  → 0 first, then R → +∞ and δ → 0, we have lim sup →0

λ n/(n−1) c

≤

ωn−1 αn A p +1+ 1 +···+ 1 2 n−1 . e 2n

(4.15)

By Lemma 3.8, lim sup Jαn − (u  ) ≤ |Ω | + →0

ωn−1 αn A p +1+ 1 +···+ 1 2 n−1 . e 2n

(4.16)

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By Lemma 3.2, we have proved the following: Proposition 4.1. Under the assumption that c → +∞, there holds that sup Jαn (u) ≤ |Ω | +

u∈H

ωn−1 αn A p +1+ 1 +···+ 1 2 n−1 . e 2n

(4.17)

5. The existence result In this section, we will finish the proof of Theorem 1.1. Let G be as in (3.12), and Gt = {x ∈ Ω : G > t}, then we have the following: n

2

Lemma 5.1. (i) |∂Gt | n−1 ≥ αn (1 + O(e− n αn t ))|Gt |;  n 2 1 (ii) ∂ Gt |∇G| dσt ≥ (ωn−1 /2) n−1 (1 + O(e− n αn t ))e−αn t +αn A p . Proof. To prove (i), we choose a normal coordinate system (U, φ) near p ∈ ∂Ω such that n  φ( p) = 0, ds 2 = (δi j + O(|y|2 ))dyi dy j , φ(∂Ω ∩ U ) ⊂ {y ∈ Rn : y1 = 0}, and i, j =1

ψ(U ) ⊂ {y ∈ Rn : y1 > 0}. Using the standard isoperimetric inequality, we have n

|φ(∂Gt )| n−1 ≥ αn |φ(Gt )|. On the other hand, note that, for x ∈ ∂Gt , |x − p| = O(e− |∂Gt | = (1 + O(e−2 |Gt | = (1 + O(e

αn n

−2 αnn

t

t

αn n

t

), we have

))|φ(∂Gt )|,

))|φ(Gt )|.

Hence we have n

|∂Gt | n−1 ≥ αn (1 + O(e−2

αn n

t

))|Gt |.

A slight modification of the proof of Lemma 3.7 in [11] gives (ii). Similarly to [11], we take  ⎧   αn 1 n n−1 ⎨C − C − n−1 log 1 + cn  − n−1 e− n−1 t + B f  (t) = αn n ⎩ − n−1 t C for t < t ,



for t ≥ t

where cn = (ωn−1 /(2n))1/(n−1), t = − αnn log(R), R depends only on  such that R → 0 and R → +∞ as  → 0, and both B and C are constants depending only on . Set φ (x) = f  (G(x)),

∀x ∈ Ω ,

where G is defined by (3.11). To ensure φ ∈ H 1,n (Ω ), we assume  1 n 1 n−1 n − n−1 n−1 C −C log(1 + cn R ) + B = − C − n−1 log(R). αn αn By a straightforward calculation, we have

(5.1)

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Y. Yang / Nonlinear Analysis 66 (2007) 2742–2755



 1 1 n−1 − n n−1 1 + + ···+ |∇φ | dx = − C αn 2 n−1 Ω n n n−1 − n n 1 + C n−1 log(1 + cn R n−1 ) + C − n−1 log αn αn R n

+ C − n−1 (O((R)n log(R)) + O((R)n log R) + O(R − n−1 )). n

n

 Setting Ω |∇φ |n dx = 1, we have  n n 1 n−1 1 n−1 n 1 n−1 + C 1 + + ··· + =− log(1 + cn R n−1 ) + log αn 2 n−1 αn αn R n

+ O((R)n log(R)) + O((R)n log R) + O(R − n−1 ),

(5.2)

which, together with (5.1), gives  1 n−1 1 B =− 1 + + ··· + + O((R)n log(R)) αn 2 n−1 n

+ O((R)n log R) + O(R − n−1 ). It is easy to see that  1 1 φ = φ dx = C − n−1 (O((R)n log(R)) + O((R)n log R)). |Ω | Ω

(5.3)

(5.4)

Applying Lemma 5.1, by a delicate but straightforward calculation, we obtain ωn−1 αn A p +1+ 1 +···+ 1 2 n−1 e Jαn (φ − φ  ) > |Ω | + 2n for sufficiently small  > 0, which gives that ωn−1 αn A p +1+ 1 +···+ 1 2 n−1 . e sup Jαn (u) > |Ω | + 2n u∈H

(5.5)

The contradiction between (5.5) and (4.17) implies that c = |u  (x  )| must be bounded and the proof of Theorem 1.1 is completely finished.  References [1] H. Brezis, F. Merle, Uniform estimates and blow up behavior for solutions of −u = V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223–1253. [2] L. Carleson, A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. 110 (1986) 113–127. [3] A. Chang, P. Yang, Conformal deformation of metrics on S 2 , J. Differential Geom. 27 (1988) 259–296. [4] W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615–622. [5] M. Flucher, Extremal functions for Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992) 471–497. [6] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (1993) 415–454. [7] S. Kichenassamy, L. Veron, Singular solution of the p-Laplace equation, Math. Ann. 275 (1986) 599–615. [8] M. Leckband, Moser’s inequality on the ball B n for functions with mean value zero, Comm. Pure Appl. Math. LVIII (2005) 789–798. [9] Y. Li, Moser–Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001) 163–192.

Y. Yang / Nonlinear Analysis 66 (2007) 2742–2755

2755

[10] Y. Li, Remarks on the extremal functions for the Moser–Trudinger inequality, Acta Math. Sin. Engl. Ser. 22 (2006) 545–550. [11] Y. Li, The existence of the extremal function of Moser–Trudinger inequality on compact Riemannian manifolds, Sci. China Ser. A 48 (2005) 618–648. [12] Y. Li, P. Liu, Moser–Trudinger inequality on the boundary of compact Riemannian surface, Math. Z. 250 (2005) 363–386. [13] K. Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (1996) 2663–2671. [14] P.L. Lions, The concentration-compactness principle in the calculus of variation, the limit case, part I, Rev. Mat. Iberoamericana 1 (1985) 145–201. [15] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971) 1077–1091. [16] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964) 248–302. [17] M. Struwe, Positive solution of critical semilinear elliptic equations on non-contractible planar domain, J. Eur. Math. Soc. 2 (2000) 329–388. [18] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984) 126–150. [19] Y. Yang, Extremal functions for Moser–Trudinger inequalities on 2-dimensional compact Riemannian manifolds with boundary, Internat. J. Math. 17 (2006) 313–330. [20] Y. Yang, Extremal functions for a sharp Moser–Trudinger inequality, Internat. J. Math. 17 (2006) 331–338. [21] Y. Yang, A sharp form of Moser–Trudinger inequality on compact Riemannian surface, Trans. Amer. Math. Soc. (in press).