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A critical problem on the Hardy–Sobolev inequality in boundary singularity case
Nonlinear Analysis 160 (2017) 70–78
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Nonlinear Analysis www.elsevier.com/locate/na
A critical problem on the Hardy–Sobolev inequality in boundary singularity case Masato Hashizume Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585, Japan
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Article history: Received 26 January 2017 Accepted 8 May 2017 Communicated by Enzo Mitidieri Keywords: boundary singularity Hardy–Sobolev inequality minimization problem Neumann problem
abstract We study a Neumann problem with the Hardy–Sobolev nonlinearity. In boundary singularity case, the impact of the mean curvature at singularity on existence of least-energy solution is well known. Existence and nonexistence of least-energy solution is studied by Hashizume (2017) except for lower dimension case. In this paper, we improve this previous work. More precisely, we study four dimensional case and show existence of minimizer in critical case in some sense. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Let N ≥ 3, Ω be smooth bounded domain in RN , 0 < s < 2, 2∗ (s) = 2(N − s)/(N − 2) and λ > 0. The ∗ −s bounded embedding from H 1 (Ω ) to L2 (s) (Ω , |x| dx) leads to the Hardy–Sobolev inequality (∫ )2/2∗ (s) ∫ 2∗ (s) |u| 2 N µs,λ (Ω ) ≤ (|∇u| + λu2 )dx, s dx |x| Ω Ω where the constant µN s,λ (Ω ) is the largest possible constant defined by ∫ 2 (|∇u| + λu2 )dx N Ω µs,λ (Ω ) = inf ( )2/2∗ (s) . u∈H 1 (Ω)\{0} ∫ |u|2∗ (s) dx |x|s Ω The Dirichlet case, that is, the minimization problem of ∫ µD s (Ω )
2
|∇u| dx = inf (∫ )2/2∗ (s) ∗ (s) 1 2 u∈H0 (Ω)\{0} |u| dx s |x| Ω
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2017.05.003 0362-546X/© 2017 Elsevier Ltd. All rights reserved.
Ω
M. Hashizume / Nonlinear Analysis 160 (2017) 70–78
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is studied by many researchers in both interior singularity case and boundary singularity case. In interior singularity case, properties of µD s (Ω ) are similar to the best constant of the Sobolev inequality. More precisely, µD (Ω ) is independent of Ω and never achieved. In boundary singularity case, [6–8] showed that minimizer s exists when the mean curvature of ∂Ω at 0 is negative. In addition, [6] prove the nonexistence result under N the assumption T (Ω ) ⊂ RN + for some rotation T , where R+ is a half-space. After these works [11] investigated D a generalized minimization problem concerning µs (Ω ). Existence and nonexistence of minimizer of µN s,λ (Ω ) have been studied by [6,9]. In [6], they showed the existence of minimizer under the positivity of the mean curvature at 0. However the nonpositive mean curvature case was not dealt with in [6]. Recently, some part of this problem have been clarified by [9]. The result of nonpositive mean curvature case is completely different from that of positive mean curvature case. Concerning existence of minimizer of µN s,λ (Ω ) in nonpositive mean curvature case we obtained the following result; Theorem 1.1 ([9]). Assume that N ≥ 5 and the mean curvature of ∂Ω at 0 is nonpositive. Then there exists a positive constant λ∗ = λ∗ (Ω ) such that (i) If 0 < λ < λ∗ then µN s,λ (Ω ) is attained. (ii) If λ > λ∗ then µN (Ω ) is not attained. s,λ This situation is closed to the three dimensional case of the minimization problem introduced by Brezis and Nirenberg [3]: ∫ 2 (|∇u| + λu2 )dx Ω Sλ (Ω ) := inf )N/N −2 , λ ∈ R. ( 2N u∈H01 (Ω)\{0} ∫ N −2 dx |u| Ω ˜ ∗ (Ω ) < 0 such that Sλ (Ω ) is attained when λ < λ ˜ ∗ (Ω ) and Sλ (Ω ) is In [3] they proved that there exists λ ˜ not attained when λ > λ∗ (Ω ). In addition, it is also proved that if Ω is a ball, existence of a minimizer is equivalent to λ < λ∗ (Ω ). After that, by [5] this result was extended to the general bounded domain case. Our main purpose of this paper is to improve Theorem 1.1. More precisely, we investigate the case when N = 4 and the case when λ = λ∗ . What is related to the minimization problem µN s,λ (Ω ) is the following elliptic equation: ⎧ 2∗ (s)−1 ⎪ ⎨−∆u + λu = u , u > 0 in Ω , s |x| (1) ⎪ ⎩ ∂u = 0 on ∂Ω . ∂ν Least-energy solution of (1) is defined by solution of (1) attaining µN s,λ (Ω ) and thus existence of least-energy solution of (1) is equivalent to existence of minimizer of µN (Ω ). s,λ Asymptotic behavior of least-energy solution of (1) and µN s,λ (Ω ) as λ → ∞ have been studied in [9]. These studies are natural because a least-energy solution exists for any λ when the mean curvature at 0 is positive as we know. However, not only that, these studies play a crucial role in studying the minimization problem µN s,λ (Ω ). Theorem 1.1 asserts that least-energy solution of (1) does not exist for sufficiently large λ when the mean curvature at 0 is nonpositive. In order to prove this fact, we need to investigate the asymptotic behavior of least-energy solution and µN s,λ (Ω ) as λ → ∞ under the assumption of existence of least-energy solution for any λ. This technique of the asymptotic analysis in [9] is used everywhere in this paper. Our main results is as follows: Theorem 1.2. Assume N = 4 and the mean curvature of ∂Ω at 0 is nonpositive. Then there exists a positive constant λ∗ = λ∗ (Ω ) such that
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(i) If 0 < λ < λ∗ then µN s,λ (Ω ) is attained. N (ii) If λ > λ∗ then µs,λ (Ω ) is not attained. Theorem 1.3. Assume N ≥ 4, the mean curvature at 0 is negative, and λ∗ is a constant obtained by Theorem 1.1 or Theorem 1.2. Then µN s,λ∗ (Ω ) is attained. The approaches to prove these theorems are based on [1,2,9,15–17]. The blow-up analysis for semilinear Neumann problem involving the Sobolev critical exponent was started by [1,2]. After these [15–17] studied the best constant of the Sobolev inequality, that is, they studied µN s,λ (Ω ) in the case when s = 0, and [9] N studied µs,λ (Ω ) in the case when s ∈ (0, 2). In [15] the author studied the asymptotic analysis for the Neumann problem involving the mean curvature on the boundary condition. [16] studied minimization problem on exterior domain by using the techniques in [15]. In [17] they expanded some parts of the results of [15] into the four dimensional case. Finally [9] investigated the asymptotic analysis related to µN s,λ (Ω ) as we mentioned before. This paper is organized as follows. In Section 2 we introduce some useful facts to prove Theorems 1.2 and 1.3. In Section 3 we prove Theorem 1.2. In Section 4 we prove Theorem 1.3. 2. Preliminaries In this section we prepare some facts in advance to prove Theorems 1.2 and 1.3. µs denotes the best constant of the Hardy–Sobolev inequality on RN , that is, µs is defined by } {∫ ∫ 2∗ (s) ⏐ |u| ⏐ 2 1,2 N µs := inf |∇u| dx⏐u ∈ D (R ), s dx = 1 . |x| RN RN
(2)
We set ( U (x) =
2−s
|x| 1+ (N − s)(N − 2)
−2 )− N2−s
,
which is a minimizer for µs . In addition, for ε > 0 we define a function by (x) N −2 . Uε (x) = ε− 2 U ε The following lemmas are introduced in [9]. Lemma 2.1. (i) µN s,λ (Ω ) is continuous and non-decreasing with respect to λ. (2−s)/(N −s) (ii) For any λ > 0, µN . s,λ (Ω ) ≤ µs /2 N (iii) limλ→0 µs,λ (Ω ) = 0. Lemma 2.2. We have either ˜ such that for λ ≥ λ ˜ (i) there exists λ µN s,λ (Ω ) = or
( ) 2−s 1 N −s µs , 2
(3)
(4)
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(ii) for all λ the equality of (i) does not hold and lim µN s,λ (Ω ) =
λ→∞
( ) 2−s 1 N −s µs . 2
Lemma 2.3. (2−s)/(N −s) (i) If µN then µN s,λ (Ω ) < µs /2 s,λ (Ω ) is attained. ˜ such that µN (Ω ) = µs /2(2−s)/(N −s) then µN (Ω ) is not attained (ii) If there exists a positive constant λ ˜ s,λ s,λ ˜ for all λ > λ.
We recall some facts about a diffeomorphism straightening a boundary portion around a point P ∈ ∂Ω , which was introduced in [10,12–14]. Through translation and rotation of the coordinate system we may assume that P is the origin and inner normal to ∂Ω at P is pointing in the direction of the positive xN -axis. In a neighborhood N around P , there exists a smooth function ψ(x′ ), x′ = (x1 , . . . , xN −1 ) such that ∂Ω ∩ N can be represented by xN = ψ(x′ ) =
N −1 1 ∑ 2 αi x2i + o(|x′ | ) 2 i=1
where α1 , . . . , αN −1 are the principal curvatures of ∂Ω at P . For y ∈ RN with |y| sufficiently small, we define a mapping x = Φ(y) = (Φ1 (y), . . . , ΦN (y)) by ⎧ ⎪ ⎨yj − yN ∂ψ (y ′ ) j = 1, . . . , N − 1 ∂xj Φj (y) = ⎪ ⎩y + ψ(y ′ ) j = N. N
The differential map DΦ is ⎛ ∂2ψ ′ ⎜δij − ∂x ∂x (y )yN i j ⎜ DΦ(y) = ⎜ ⎝ ∂ψ ′ (y ) ∂xj
−
⎞ ∂ψ ′ (y )⎟ ∂xi ⎟ ⎟ ⎠ 1 1≤i,j≤N −1
and near y = 0 2
|JΦ(y)| = |det DΦ(y)| = 1 − (N − 1)H(P )yN + O(|y| ). We write as Ψ (x) = (Ψ1 (x), . . . , ΨN (x)) instead of the inverse map Φ −1 (x). Br (a) denotes an open ball with center a and radius r. In addition, suppose Br = Br (0) and Br+ = {y ∈ Br |yN > 0}. Define N0 as a neighborhood around 0 such that Φ(Bδ ) = N0 . 3. Proof of Theorem 1.2 In this section we prove Theorem 1.2. In order to prove Theorem 1.2 we need the following proposition: Proposition 3.1. Suppose that there exists a least-energy solution of (1) for all λ. Then there exist positive constants C1 and C2 such that as λ → ∞ ( ) 2−s ( ) 1 4−s 1 1 N 2 2 2 µs,λ (Ω ) = µs − C1 H(0)ε + C2 λε log 1/2 + O λε + ε log 1/2 , 2 λ ε λ ε where H(0) is the mean curvature at 0 and 0 < ε = o(1/λ1/2 ).
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Assuming that Proposition 3.1 holds, we obtain Theorem 1.2 immediately. Indeed, by Proposition 3.1 N (2−s)/(N −s) minimizer of µN s,λ (Ω ) does not exist for λ sufficiently large (if the minimizer exists, µs,λ (Ω ) > µs /2 and this contradicts (ii) of Lemma 2.1). This implies the existence of λ∗ = λ∗ (Ω ) such that part (i) of ˜ = λ∗ . As a consequence (i) and (ii) follow from Lemma 2.3. Lemma 2.2 holds as λ Proof of Proposition 3.1. We note that 2∗ (s) = 4 − s. Suppose that vλ is a least-energy solution of (1), that is, vλ satisfies ⎧ v 3−s ⎪ ⎨−∆vλ + λvλ = λ s , vλ > 0 in Ω , |x| (5) ⎪ ⎩ ∂vλ = 0 on ∂Ω . ∂ν We define αλ and βλ as αλ = ∥vλ ∥L∞ (Ω) ,
βλ =
From Theorem 3.1 in [9] we have ∫ 1 4−s 2 |∇vλ | dx = µs2−s , lim λ→∞ Ω 2
1 . αλ
∫ lim λ
λ→∞
vλ2 dx = 0.
(6)
Ω
We set a cut-off function ηλ (y) = ηλ (|y|) such that support of ηλ is in Bδ/λ1/2 and ηλ = 1 in Bδ/2λ1/2 . We may assume that |∇ηλ | ≤ C/λ1/2 , |D2 ηλ | ≤ C/λ. For simplicity we write η. For η, a positive constant ε, and Uε which is defined in (4) we set { η(Ψ (x))Uε (Ψ (x)) (x ∈ N0 ), Vε (x) = 0 (x ∈ RN \ N0 ). Set ∫ ⟨ϕ, ψ⟩λ =
∫ ∇ϕ∇ψdx + λ
ϕψdx,
Ω
Ω
2
∥ϕ∥λ = ⟨ϕ, ϕ⟩λ .
In addition to (6), we have lim ∥vλ − Vβλ ∥λ = 0.
λ→∞
Let { ⏐ } ⏐ M = cVε ⏐c ∈ R+ , 0 < ε ≤ 1 ,
dist(u, M ) = inf ∥u − ϕ∥λ , ϕ∈M
and { E(ε, λ) =
⟩ } ⟨ ⏐ ∂ ⏐ =0 . ϕ ∈ H 1 (Ω )⏐⟨ϕ, Vε ⟩λ = ϕ, Vε ∂ε λ
We have the following lemma corresponding to Lemma 4.2 in [9]. Lemma 3.2. If λ is sufficiently large dist(vλ , M ) is attained by cVε , where c = c(λ), ε = ε(λ). Moreover, ε →1 βλ as λ → ∞.
and
c→1
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From this lemma there exists ωλ ∈ Eε,λ such that 2
2
2
∥vλ ∥λ = ∥cVε ∥λ + ∥ωλ ∥λ ,
vλ = cVε + ωλ ,
∥ωλ ∥λ = o(1).
(7)
By applying the proof of Lemma 4.3 in [9] we have the following lemma: Lemma 3.3. We assume ε = ε(λ) is given in Lemma 3.3. Then there exists σ > 0 and λ0 such that for all ω ∈ E(ε, λ) and λ > λ0 we have ∫ Vε2−s ω 2 2 dx ≤ ∥ω∥λ . (3 − s + σ) s |x| Ω
Multiplying (5) by ωλ and integrating on Ω by parts, we have ∫ (cVε + ωλ )3−s wλ 2 ∥ωλ ∥λ = dx. s |x| Ω For the right hand side we have ∫ ∫ ∫ Vε3−s ωλ Vε2−s ωλ2 (cVε + ωλ )3−s wλ γ 3−s 2−s dx = c dx + (3 − s)c dy + O(∥ωλ ∥λ ), s s s |x| |x| |x| Ω Ω Ω where γ = min {3, 4 − s}. Thus we have ∫ ∫ Vε3−s ωλ Vε2−s ωλ2 γ 2 3−s dx = c dx + O(∥ωλ ∥λ ). ∥ωλ ∥λ − (3 − s)c2−s s s |x| |x| Ω Ω
(8)
From Lemma 3.3 it follows that 2
∥ωλ ∥λ =
3−s+σ (1 + o(1)) σ
∫ Ω
Vε3−s ωλ γ dx + O(∥ωλ ∥λ ), s |x|
(9)
where we used that c = 1 + o(1). Set 2
∫
(|∇f | + λf 2 )dx )2/(4−s) . |f |4−s dx s Ω |x|
Qλ (f ) := ( Ω ∫ By using (7) we have Qλ (vλ ) = Qλ (cVε )
⎧ ⎨ ⎩
2
1+
∥ωλ ∥λ 2
∥cVε ∥λ
−
2
(cVε )3−s ωλ dx |x|s Ω ∫ (cVε )4−s dx |x|s Ω
∫
2 (cVε )2−s ωλ dx ⎬ |x|s Ω (cVε )4−s ⎭ dx |x|s Ω
(3 − s) − ∫
∫
⎫ γ
+ O(∥ωλ ∥λ ).
Recalling that ε/βλ → 1 as λ → ∞, we can see lim
λ→∞
2 ∥cVε ∥λ
∫ = lim
λ→∞
Ω
(cVε )4−s 1 4−s dx = µs2−s . s 2 |x|
Hence from (8) we have Q(vλ ) = Q(cVε ) − (1 + o(1))2
2 4−s
− 2 µs 2−s
∫ Ω
Vε3−s ωλ γ dx + O(∥ωλ ∥λ ). s |x|
For Qλ (cVε ) we apply the calculations in the proof of (ii) of Lemma 2.1 (see [9]). It follows that ( ) 2−s ( ) 1 4−s 1 1 2 2 2 Qλ (cVε ) = µs − C1 H(0)ε + C2 λε log 1/2 + O λε + ε log 1/2 2 λ ε λ ε
(10)
(11)
for some positive constants C1 and C2 . In order to finish the proof we prove the following lemma in the same way as Step 3 of Section 4 in [16].
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Lemma 3.4. ∫ Ω
( ( )1/2 ) 1 Vε3−s ωλ 1/2 dx = O λ ε + ε log 1/2 ∥ωλ ∥λ . s |x| λ ε
This lemma and (9) yield that ( ∥ωλ ∥λ = O λ
1/2
( ε + ε log
)1/2 )
1 λ1/2 ε
,
and thus ∫ Ω
( ) 1 Vε3−s ωλ 2 2 dx = O λε + ε log 1/2 . s |x| λ ε
(12)
Consequently Proposition 3.1 follows from (10), (11), and (12). □
4. Proof of Theorem 1.3 In this section we prove Theorem 1.3. Note that lim
λ→λ∗
µN s,λ (Ω )
=
µN s,λ∗ (Ω )
( ) 2−s 1 N −s = µs . 2
(13)
Proof of Theorem 1.3. For any λ ∈ (0, λ∗ ) we write vλ as a least-energy solution of (1). Recall that by 2 the elliptic regularity theory vλ ∈ Cloc (Ω \ {0}) and vλ ∈ C 0,α (Ω ) (see [4,7]). Firstly, we claim that we only have to show the existence of a positive constant C such that ∥vλ ∥L∞ (Ω) < C holds uniformly for λ near λ∗ . Indeed, if ∥vλ ∥L∞ (Ω) < C then there exists v such that vλ ⇀ v weakly in H 1 (Ω ),
and vλ → v strongly in L2
∗
(s)
−s
(Ω , |x|
dx)
as λ → λ∗ . Since vλ is a least-energy solution of (1) we have v ̸≡ 0. Therefore ∫ ∫ 2 2 (|∇v| + λ∗ v 2 )dx (|∇vλ | + λvλ2 )dx N Ω Ω ≤ lim (∫ )2/2∗ (s) (∫ )2/2∗ (s) = µs,λ∗ (Ω ) ∗ (s) ∗ (s) 2 2 λ→λ ∗ |v| |v| |x|s dx |x|s dx Ω Ω and hence v is a minimizer of µN s,λ∗ (Ω ). Next, we show that there exists a positive constant C such that ∥vλ ∥L∞ (Ω) < C holds. Assume that λk is a sequence (a suitable subsequence is also written by λk ) such that λk → λ∗ as k → ∞. We suppose that ∥vλk ∥L∞ (Ω) → ∞ as k → ∞ and derive a contradiction. Here, we define αλ , βλ , and xλ ∈ Ω as αλk = ∥vλk ∥L∞ (Ω) = vλk (xλk ),
−
2
βλk = αλkN −2 .
Step 1. We obtain the following results: ∫ (i) Ω vλ2 k → 0, (ii) |xλk | = o(βλk ), as k → ∞. For any ε > 0 and r > 0 there exists a k0 such that for all k > k0 ⏐ ( ) ⏐⏐ ⏐ vλ (x) Ψ (x) ⏐ k ⏐ (iii) ⏐ α −U β ⏐ < ε in Ω ∩ Brβλk . λk λk
M. Hashizume / Nonlinear Analysis 160 (2017) 70–78
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Proof . We obtain (ii), (iii) by applying the technique of the proof of (iii), (iv) of Theorem 3.1 in [9]. We prove (i). For any R > 0, suppose that N0R denotes N0 defined in Section 2 with δ = R. From (iii) it follows that 2∗ (s) ∫ ∫ ∗ −s vλ k 1 N2−s U 2 (s) lim − γ(R), (14) s dx = s dx = µs k→∞ Ω R 2 |x| B + |x| R
βλ k
where ΩβRλ
= βλk (Ω ∩
k
∗
∫
N0R ),
γ(R) =
+ RN \BR +
U 2 (s) s dx. |x|
Since (13) we have 2∗ (s)
vλ k
∫ lim
k→∞
s
|x|
Ω\ΩβR
dx = γ(R).
(15)
λk
Clearly γ(R) → 0 as R → ∞. Here, we observe that ∫ ∫ ∫ vλ2 k dx = vλ2 k dx + ΩβR
Ω
vλ2 k dx = I1 + I2 .
Ω\ΩβR
λk
λk
For I1 from (14) we have ⎛ ∫ I1 ≤ ⎝
ΩβR
⎛ ⎞ 2∗ (s)−2 2∗ (s) ∫ 2s ∗ (s)−2 2 ⎝ ⎠ dx |x|
2∗ (s)
vλk
|x|
ΩβR
s
⎞
2 2∗ (s)
dx⎠
= o(1),
λk
λk
as k → ∞. For I2 from (15) we have ⎞ 2∗ (s)−2 ⎛ ∗
⎛ ∫ ⎝ I2 ≤
2 (s)
|x|
2s 2∗ (s)−2
Ω\ΩβR
dx⎠
2∗ (s)
vλk
∫ ⎝
Ω\ΩβR
|x|
⎞
2 2∗ (s)
dx⎠
s
λk
λk
= C(γ(R) + o(1))
2 2∗ (s)
as k → ∞. Letting R → ∞ after k → ∞, we obtain (i). □ Step 2. We have as k → ∞ µN s,λk (Ω )
( ) 1 = 2
2−s N −s
⎧ O(ε2 ) ⎪ ⎪ ⎨ ( ( ) 43 ) µs − H(0)ε + 1 2 ⎪O ε log ⎪ ⎩ ε
N ≥ 5, (16) N = 4,
where ε = εk is a positive constant such that ε → 0 as k → ∞. Proof . From Step 1 we have ∫ 1 N −s 2 lim |∇vλk | dx = µs2−s , k→∞ Ω 2
∫ lim λk
k→∞
Ω
vλ2 k dx = 0,
∫ lim
k→∞
Ω
2
|∇(vλk − Uβλ )| dx = 0. k
Therefore investigating the details of the asymptotic behavior of µN s,λk (Ω ) as k → ∞ (see [9,15–17], and Section 3 in this paper) we obtain (16). □ Consequently if the mean curvature of ∂Ω at 0 is negative (16) contradicts (ii) of Lemma 2.1, and which implies ∥vλk ∥L∞ (Ω) is bounded uniformly. □
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M. Hashizume / Nonlinear Analysis 160 (2017) 70–78
Acknowledgment The author is grateful to Prof. Chun-Hsiung Hsia for cooperation for his visit to National Center for Theoretical Sciences. A part of this work was done during his visit. This work was supported by Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant Number JP16J08945). References
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Nonlinear Analysis www.elsevier.com/locate/na
A critical problem on the Hardy–Sobolev inequality in boundary singularity case Masato Hashizume Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585, Japan
article
info
Article history: Received 26 January 2017 Accepted 8 May 2017 Communicated by Enzo Mitidieri Keywords: boundary singularity Hardy–Sobolev inequality minimization problem Neumann problem
abstract We study a Neumann problem with the Hardy–Sobolev nonlinearity. In boundary singularity case, the impact of the mean curvature at singularity on existence of least-energy solution is well known. Existence and nonexistence of least-energy solution is studied by Hashizume (2017) except for lower dimension case. In this paper, we improve this previous work. More precisely, we study four dimensional case and show existence of minimizer in critical case in some sense. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Let N ≥ 3, Ω be smooth bounded domain in RN , 0 < s < 2, 2∗ (s) = 2(N − s)/(N − 2) and λ > 0. The ∗ −s bounded embedding from H 1 (Ω ) to L2 (s) (Ω , |x| dx) leads to the Hardy–Sobolev inequality (∫ )2/2∗ (s) ∫ 2∗ (s) |u| 2 N µs,λ (Ω ) ≤ (|∇u| + λu2 )dx, s dx |x| Ω Ω where the constant µN s,λ (Ω ) is the largest possible constant defined by ∫ 2 (|∇u| + λu2 )dx N Ω µs,λ (Ω ) = inf ( )2/2∗ (s) . u∈H 1 (Ω)\{0} ∫ |u|2∗ (s) dx |x|s Ω The Dirichlet case, that is, the minimization problem of ∫ µD s (Ω )
2
|∇u| dx = inf (∫ )2/2∗ (s) ∗ (s) 1 2 u∈H0 (Ω)\{0} |u| dx s |x| Ω
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2017.05.003 0362-546X/© 2017 Elsevier Ltd. All rights reserved.
Ω
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is studied by many researchers in both interior singularity case and boundary singularity case. In interior singularity case, properties of µD s (Ω ) are similar to the best constant of the Sobolev inequality. More precisely, µD (Ω ) is independent of Ω and never achieved. In boundary singularity case, [6–8] showed that minimizer s exists when the mean curvature of ∂Ω at 0 is negative. In addition, [6] prove the nonexistence result under N the assumption T (Ω ) ⊂ RN + for some rotation T , where R+ is a half-space. After these works [11] investigated D a generalized minimization problem concerning µs (Ω ). Existence and nonexistence of minimizer of µN s,λ (Ω ) have been studied by [6,9]. In [6], they showed the existence of minimizer under the positivity of the mean curvature at 0. However the nonpositive mean curvature case was not dealt with in [6]. Recently, some part of this problem have been clarified by [9]. The result of nonpositive mean curvature case is completely different from that of positive mean curvature case. Concerning existence of minimizer of µN s,λ (Ω ) in nonpositive mean curvature case we obtained the following result; Theorem 1.1 ([9]). Assume that N ≥ 5 and the mean curvature of ∂Ω at 0 is nonpositive. Then there exists a positive constant λ∗ = λ∗ (Ω ) such that (i) If 0 < λ < λ∗ then µN s,λ (Ω ) is attained. (ii) If λ > λ∗ then µN (Ω ) is not attained. s,λ This situation is closed to the three dimensional case of the minimization problem introduced by Brezis and Nirenberg [3]: ∫ 2 (|∇u| + λu2 )dx Ω Sλ (Ω ) := inf )N/N −2 , λ ∈ R. ( 2N u∈H01 (Ω)\{0} ∫ N −2 dx |u| Ω ˜ ∗ (Ω ) < 0 such that Sλ (Ω ) is attained when λ < λ ˜ ∗ (Ω ) and Sλ (Ω ) is In [3] they proved that there exists λ ˜ not attained when λ > λ∗ (Ω ). In addition, it is also proved that if Ω is a ball, existence of a minimizer is equivalent to λ < λ∗ (Ω ). After that, by [5] this result was extended to the general bounded domain case. Our main purpose of this paper is to improve Theorem 1.1. More precisely, we investigate the case when N = 4 and the case when λ = λ∗ . What is related to the minimization problem µN s,λ (Ω ) is the following elliptic equation: ⎧ 2∗ (s)−1 ⎪ ⎨−∆u + λu = u , u > 0 in Ω , s |x| (1) ⎪ ⎩ ∂u = 0 on ∂Ω . ∂ν Least-energy solution of (1) is defined by solution of (1) attaining µN s,λ (Ω ) and thus existence of least-energy solution of (1) is equivalent to existence of minimizer of µN (Ω ). s,λ Asymptotic behavior of least-energy solution of (1) and µN s,λ (Ω ) as λ → ∞ have been studied in [9]. These studies are natural because a least-energy solution exists for any λ when the mean curvature at 0 is positive as we know. However, not only that, these studies play a crucial role in studying the minimization problem µN s,λ (Ω ). Theorem 1.1 asserts that least-energy solution of (1) does not exist for sufficiently large λ when the mean curvature at 0 is nonpositive. In order to prove this fact, we need to investigate the asymptotic behavior of least-energy solution and µN s,λ (Ω ) as λ → ∞ under the assumption of existence of least-energy solution for any λ. This technique of the asymptotic analysis in [9] is used everywhere in this paper. Our main results is as follows: Theorem 1.2. Assume N = 4 and the mean curvature of ∂Ω at 0 is nonpositive. Then there exists a positive constant λ∗ = λ∗ (Ω ) such that
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(i) If 0 < λ < λ∗ then µN s,λ (Ω ) is attained. N (ii) If λ > λ∗ then µs,λ (Ω ) is not attained. Theorem 1.3. Assume N ≥ 4, the mean curvature at 0 is negative, and λ∗ is a constant obtained by Theorem 1.1 or Theorem 1.2. Then µN s,λ∗ (Ω ) is attained. The approaches to prove these theorems are based on [1,2,9,15–17]. The blow-up analysis for semilinear Neumann problem involving the Sobolev critical exponent was started by [1,2]. After these [15–17] studied the best constant of the Sobolev inequality, that is, they studied µN s,λ (Ω ) in the case when s = 0, and [9] N studied µs,λ (Ω ) in the case when s ∈ (0, 2). In [15] the author studied the asymptotic analysis for the Neumann problem involving the mean curvature on the boundary condition. [16] studied minimization problem on exterior domain by using the techniques in [15]. In [17] they expanded some parts of the results of [15] into the four dimensional case. Finally [9] investigated the asymptotic analysis related to µN s,λ (Ω ) as we mentioned before. This paper is organized as follows. In Section 2 we introduce some useful facts to prove Theorems 1.2 and 1.3. In Section 3 we prove Theorem 1.2. In Section 4 we prove Theorem 1.3. 2. Preliminaries In this section we prepare some facts in advance to prove Theorems 1.2 and 1.3. µs denotes the best constant of the Hardy–Sobolev inequality on RN , that is, µs is defined by } {∫ ∫ 2∗ (s) ⏐ |u| ⏐ 2 1,2 N µs := inf |∇u| dx⏐u ∈ D (R ), s dx = 1 . |x| RN RN
(2)
We set ( U (x) =
2−s
|x| 1+ (N − s)(N − 2)
−2 )− N2−s
,
which is a minimizer for µs . In addition, for ε > 0 we define a function by (x) N −2 . Uε (x) = ε− 2 U ε The following lemmas are introduced in [9]. Lemma 2.1. (i) µN s,λ (Ω ) is continuous and non-decreasing with respect to λ. (2−s)/(N −s) (ii) For any λ > 0, µN . s,λ (Ω ) ≤ µs /2 N (iii) limλ→0 µs,λ (Ω ) = 0. Lemma 2.2. We have either ˜ such that for λ ≥ λ ˜ (i) there exists λ µN s,λ (Ω ) = or
( ) 2−s 1 N −s µs , 2
(3)
(4)
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(ii) for all λ the equality of (i) does not hold and lim µN s,λ (Ω ) =
λ→∞
( ) 2−s 1 N −s µs . 2
Lemma 2.3. (2−s)/(N −s) (i) If µN then µN s,λ (Ω ) < µs /2 s,λ (Ω ) is attained. ˜ such that µN (Ω ) = µs /2(2−s)/(N −s) then µN (Ω ) is not attained (ii) If there exists a positive constant λ ˜ s,λ s,λ ˜ for all λ > λ.
We recall some facts about a diffeomorphism straightening a boundary portion around a point P ∈ ∂Ω , which was introduced in [10,12–14]. Through translation and rotation of the coordinate system we may assume that P is the origin and inner normal to ∂Ω at P is pointing in the direction of the positive xN -axis. In a neighborhood N around P , there exists a smooth function ψ(x′ ), x′ = (x1 , . . . , xN −1 ) such that ∂Ω ∩ N can be represented by xN = ψ(x′ ) =
N −1 1 ∑ 2 αi x2i + o(|x′ | ) 2 i=1
where α1 , . . . , αN −1 are the principal curvatures of ∂Ω at P . For y ∈ RN with |y| sufficiently small, we define a mapping x = Φ(y) = (Φ1 (y), . . . , ΦN (y)) by ⎧ ⎪ ⎨yj − yN ∂ψ (y ′ ) j = 1, . . . , N − 1 ∂xj Φj (y) = ⎪ ⎩y + ψ(y ′ ) j = N. N
The differential map DΦ is ⎛ ∂2ψ ′ ⎜δij − ∂x ∂x (y )yN i j ⎜ DΦ(y) = ⎜ ⎝ ∂ψ ′ (y ) ∂xj
−
⎞ ∂ψ ′ (y )⎟ ∂xi ⎟ ⎟ ⎠ 1 1≤i,j≤N −1
and near y = 0 2
|JΦ(y)| = |det DΦ(y)| = 1 − (N − 1)H(P )yN + O(|y| ). We write as Ψ (x) = (Ψ1 (x), . . . , ΨN (x)) instead of the inverse map Φ −1 (x). Br (a) denotes an open ball with center a and radius r. In addition, suppose Br = Br (0) and Br+ = {y ∈ Br |yN > 0}. Define N0 as a neighborhood around 0 such that Φ(Bδ ) = N0 . 3. Proof of Theorem 1.2 In this section we prove Theorem 1.2. In order to prove Theorem 1.2 we need the following proposition: Proposition 3.1. Suppose that there exists a least-energy solution of (1) for all λ. Then there exist positive constants C1 and C2 such that as λ → ∞ ( ) 2−s ( ) 1 4−s 1 1 N 2 2 2 µs,λ (Ω ) = µs − C1 H(0)ε + C2 λε log 1/2 + O λε + ε log 1/2 , 2 λ ε λ ε where H(0) is the mean curvature at 0 and 0 < ε = o(1/λ1/2 ).
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Assuming that Proposition 3.1 holds, we obtain Theorem 1.2 immediately. Indeed, by Proposition 3.1 N (2−s)/(N −s) minimizer of µN s,λ (Ω ) does not exist for λ sufficiently large (if the minimizer exists, µs,λ (Ω ) > µs /2 and this contradicts (ii) of Lemma 2.1). This implies the existence of λ∗ = λ∗ (Ω ) such that part (i) of ˜ = λ∗ . As a consequence (i) and (ii) follow from Lemma 2.3. Lemma 2.2 holds as λ Proof of Proposition 3.1. We note that 2∗ (s) = 4 − s. Suppose that vλ is a least-energy solution of (1), that is, vλ satisfies ⎧ v 3−s ⎪ ⎨−∆vλ + λvλ = λ s , vλ > 0 in Ω , |x| (5) ⎪ ⎩ ∂vλ = 0 on ∂Ω . ∂ν We define αλ and βλ as αλ = ∥vλ ∥L∞ (Ω) ,
βλ =
From Theorem 3.1 in [9] we have ∫ 1 4−s 2 |∇vλ | dx = µs2−s , lim λ→∞ Ω 2
1 . αλ
∫ lim λ
λ→∞
vλ2 dx = 0.
(6)
Ω
We set a cut-off function ηλ (y) = ηλ (|y|) such that support of ηλ is in Bδ/λ1/2 and ηλ = 1 in Bδ/2λ1/2 . We may assume that |∇ηλ | ≤ C/λ1/2 , |D2 ηλ | ≤ C/λ. For simplicity we write η. For η, a positive constant ε, and Uε which is defined in (4) we set { η(Ψ (x))Uε (Ψ (x)) (x ∈ N0 ), Vε (x) = 0 (x ∈ RN \ N0 ). Set ∫ ⟨ϕ, ψ⟩λ =
∫ ∇ϕ∇ψdx + λ
ϕψdx,
Ω
Ω
2
∥ϕ∥λ = ⟨ϕ, ϕ⟩λ .
In addition to (6), we have lim ∥vλ − Vβλ ∥λ = 0.
λ→∞
Let { ⏐ } ⏐ M = cVε ⏐c ∈ R+ , 0 < ε ≤ 1 ,
dist(u, M ) = inf ∥u − ϕ∥λ , ϕ∈M
and { E(ε, λ) =
⟩ } ⟨ ⏐ ∂ ⏐ =0 . ϕ ∈ H 1 (Ω )⏐⟨ϕ, Vε ⟩λ = ϕ, Vε ∂ε λ
We have the following lemma corresponding to Lemma 4.2 in [9]. Lemma 3.2. If λ is sufficiently large dist(vλ , M ) is attained by cVε , where c = c(λ), ε = ε(λ). Moreover, ε →1 βλ as λ → ∞.
and
c→1
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From this lemma there exists ωλ ∈ Eε,λ such that 2
2
2
∥vλ ∥λ = ∥cVε ∥λ + ∥ωλ ∥λ ,
vλ = cVε + ωλ ,
∥ωλ ∥λ = o(1).
(7)
By applying the proof of Lemma 4.3 in [9] we have the following lemma: Lemma 3.3. We assume ε = ε(λ) is given in Lemma 3.3. Then there exists σ > 0 and λ0 such that for all ω ∈ E(ε, λ) and λ > λ0 we have ∫ Vε2−s ω 2 2 dx ≤ ∥ω∥λ . (3 − s + σ) s |x| Ω
Multiplying (5) by ωλ and integrating on Ω by parts, we have ∫ (cVε + ωλ )3−s wλ 2 ∥ωλ ∥λ = dx. s |x| Ω For the right hand side we have ∫ ∫ ∫ Vε3−s ωλ Vε2−s ωλ2 (cVε + ωλ )3−s wλ γ 3−s 2−s dx = c dx + (3 − s)c dy + O(∥ωλ ∥λ ), s s s |x| |x| |x| Ω Ω Ω where γ = min {3, 4 − s}. Thus we have ∫ ∫ Vε3−s ωλ Vε2−s ωλ2 γ 2 3−s dx = c dx + O(∥ωλ ∥λ ). ∥ωλ ∥λ − (3 − s)c2−s s s |x| |x| Ω Ω
(8)
From Lemma 3.3 it follows that 2
∥ωλ ∥λ =
3−s+σ (1 + o(1)) σ
∫ Ω
Vε3−s ωλ γ dx + O(∥ωλ ∥λ ), s |x|
(9)
where we used that c = 1 + o(1). Set 2
∫
(|∇f | + λf 2 )dx )2/(4−s) . |f |4−s dx s Ω |x|
Qλ (f ) := ( Ω ∫ By using (7) we have Qλ (vλ ) = Qλ (cVε )
⎧ ⎨ ⎩
2
1+
∥ωλ ∥λ 2
∥cVε ∥λ
−
2
(cVε )3−s ωλ dx |x|s Ω ∫ (cVε )4−s dx |x|s Ω
∫
2 (cVε )2−s ωλ dx ⎬ |x|s Ω (cVε )4−s ⎭ dx |x|s Ω
(3 − s) − ∫
∫
⎫ γ
+ O(∥ωλ ∥λ ).
Recalling that ε/βλ → 1 as λ → ∞, we can see lim
λ→∞
2 ∥cVε ∥λ
∫ = lim
λ→∞
Ω
(cVε )4−s 1 4−s dx = µs2−s . s 2 |x|
Hence from (8) we have Q(vλ ) = Q(cVε ) − (1 + o(1))2
2 4−s
− 2 µs 2−s
∫ Ω
Vε3−s ωλ γ dx + O(∥ωλ ∥λ ). s |x|
For Qλ (cVε ) we apply the calculations in the proof of (ii) of Lemma 2.1 (see [9]). It follows that ( ) 2−s ( ) 1 4−s 1 1 2 2 2 Qλ (cVε ) = µs − C1 H(0)ε + C2 λε log 1/2 + O λε + ε log 1/2 2 λ ε λ ε
(10)
(11)
for some positive constants C1 and C2 . In order to finish the proof we prove the following lemma in the same way as Step 3 of Section 4 in [16].
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Lemma 3.4. ∫ Ω
( ( )1/2 ) 1 Vε3−s ωλ 1/2 dx = O λ ε + ε log 1/2 ∥ωλ ∥λ . s |x| λ ε
This lemma and (9) yield that ( ∥ωλ ∥λ = O λ
1/2
( ε + ε log
)1/2 )
1 λ1/2 ε
,
and thus ∫ Ω
( ) 1 Vε3−s ωλ 2 2 dx = O λε + ε log 1/2 . s |x| λ ε
(12)
Consequently Proposition 3.1 follows from (10), (11), and (12). □
4. Proof of Theorem 1.3 In this section we prove Theorem 1.3. Note that lim
λ→λ∗
µN s,λ (Ω )
=
µN s,λ∗ (Ω )
( ) 2−s 1 N −s = µs . 2
(13)
Proof of Theorem 1.3. For any λ ∈ (0, λ∗ ) we write vλ as a least-energy solution of (1). Recall that by 2 the elliptic regularity theory vλ ∈ Cloc (Ω \ {0}) and vλ ∈ C 0,α (Ω ) (see [4,7]). Firstly, we claim that we only have to show the existence of a positive constant C such that ∥vλ ∥L∞ (Ω) < C holds uniformly for λ near λ∗ . Indeed, if ∥vλ ∥L∞ (Ω) < C then there exists v such that vλ ⇀ v weakly in H 1 (Ω ),
and vλ → v strongly in L2
∗
(s)
−s
(Ω , |x|
dx)
as λ → λ∗ . Since vλ is a least-energy solution of (1) we have v ̸≡ 0. Therefore ∫ ∫ 2 2 (|∇v| + λ∗ v 2 )dx (|∇vλ | + λvλ2 )dx N Ω Ω ≤ lim (∫ )2/2∗ (s) (∫ )2/2∗ (s) = µs,λ∗ (Ω ) ∗ (s) ∗ (s) 2 2 λ→λ ∗ |v| |v| |x|s dx |x|s dx Ω Ω and hence v is a minimizer of µN s,λ∗ (Ω ). Next, we show that there exists a positive constant C such that ∥vλ ∥L∞ (Ω) < C holds. Assume that λk is a sequence (a suitable subsequence is also written by λk ) such that λk → λ∗ as k → ∞. We suppose that ∥vλk ∥L∞ (Ω) → ∞ as k → ∞ and derive a contradiction. Here, we define αλ , βλ , and xλ ∈ Ω as αλk = ∥vλk ∥L∞ (Ω) = vλk (xλk ),
−
2
βλk = αλkN −2 .
Step 1. We obtain the following results: ∫ (i) Ω vλ2 k → 0, (ii) |xλk | = o(βλk ), as k → ∞. For any ε > 0 and r > 0 there exists a k0 such that for all k > k0 ⏐ ( ) ⏐⏐ ⏐ vλ (x) Ψ (x) ⏐ k ⏐ (iii) ⏐ α −U β ⏐ < ε in Ω ∩ Brβλk . λk λk
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Proof . We obtain (ii), (iii) by applying the technique of the proof of (iii), (iv) of Theorem 3.1 in [9]. We prove (i). For any R > 0, suppose that N0R denotes N0 defined in Section 2 with δ = R. From (iii) it follows that 2∗ (s) ∫ ∫ ∗ −s vλ k 1 N2−s U 2 (s) lim − γ(R), (14) s dx = s dx = µs k→∞ Ω R 2 |x| B + |x| R
βλ k
where ΩβRλ
= βλk (Ω ∩
k
∗
∫
N0R ),
γ(R) =
+ RN \BR +
U 2 (s) s dx. |x|
Since (13) we have 2∗ (s)
vλ k
∫ lim
k→∞
s
|x|
Ω\ΩβR
dx = γ(R).
(15)
λk
Clearly γ(R) → 0 as R → ∞. Here, we observe that ∫ ∫ ∫ vλ2 k dx = vλ2 k dx + ΩβR
Ω
vλ2 k dx = I1 + I2 .
Ω\ΩβR
λk
λk
For I1 from (14) we have ⎛ ∫ I1 ≤ ⎝
ΩβR
⎛ ⎞ 2∗ (s)−2 2∗ (s) ∫ 2s ∗ (s)−2 2 ⎝ ⎠ dx |x|
2∗ (s)
vλk
|x|
ΩβR
s
⎞
2 2∗ (s)
dx⎠
= o(1),
λk
λk
as k → ∞. For I2 from (15) we have ⎞ 2∗ (s)−2 ⎛ ∗
⎛ ∫ ⎝ I2 ≤
2 (s)
|x|
2s 2∗ (s)−2
Ω\ΩβR
dx⎠
2∗ (s)
vλk
∫ ⎝
Ω\ΩβR
|x|
⎞
2 2∗ (s)
dx⎠
s
λk
λk
= C(γ(R) + o(1))
2 2∗ (s)
as k → ∞. Letting R → ∞ after k → ∞, we obtain (i). □ Step 2. We have as k → ∞ µN s,λk (Ω )
( ) 1 = 2
2−s N −s
⎧ O(ε2 ) ⎪ ⎪ ⎨ ( ( ) 43 ) µs − H(0)ε + 1 2 ⎪O ε log ⎪ ⎩ ε
N ≥ 5, (16) N = 4,
where ε = εk is a positive constant such that ε → 0 as k → ∞. Proof . From Step 1 we have ∫ 1 N −s 2 lim |∇vλk | dx = µs2−s , k→∞ Ω 2
∫ lim λk
k→∞
Ω
vλ2 k dx = 0,
∫ lim
k→∞
Ω
2
|∇(vλk − Uβλ )| dx = 0. k
Therefore investigating the details of the asymptotic behavior of µN s,λk (Ω ) as k → ∞ (see [9,15–17], and Section 3 in this paper) we obtain (16). □ Consequently if the mean curvature of ∂Ω at 0 is negative (16) contradicts (ii) of Lemma 2.1, and which implies ∥vλk ∥L∞ (Ω) is bounded uniformly. □
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Acknowledgment The author is grateful to Prof. Chun-Hsiung Hsia for cooperation for his visit to National Center for Theoretical Sciences. A part of this work was done during his visit. This work was supported by Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant Number JP16J08945). References
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