Singularities of solutions to elliptic systems involving different Hardy-type terms

Accepted Manuscript Singularities of solutions to elliptic systems involving different Hardy–type terms

Dongsheng Kang, Xiaonan Liu

PII: DOI: Reference:

S0022-247X(18)30708-X https://doi.org/10.1016/j.jmaa.2018.08.044 YJMAA 22501

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Journal of Mathematical Analysis and Applications

Received date:

28 January 2018

Please cite this article in press as: D. Kang, X. Liu, Singularities of solutions to elliptic systems involving different Hardy–type terms, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.08.044

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Singularities of solutions to elliptic systems involving different Hardy–type terms Dongsheng Kang∗, Xiaonan Liu School of Mathematics and Statistics, South–Central University for Nationalities, Wuhan 430074, P. R. China

Abstract In this paper, a system of elliptic equations is studied, which involves multiple critical nonlinearities and deferent Hardy–type terms. Asymptotic properties at the singular point of positive solutions are proved by the Moser iteration method under certain conditions. It is found that the functions in solutions have different singularities at the singular point. Keywords: system of elliptic equations; solution; asymptotic property; variational method. Mathematics Subject Classification 2000: 35J47, 35J60

1

Introduction

In this paper, we study the following elliptic system: ⎧ u ηα ∗ ⎪ −Δu − μ1 2 = u2 −1 + ∗ uα−1 v β + a1 u + a2 v in Ω, ⎪ ⎪ |x| 2 ⎪ ⎨ v ηβ ∗ −Δv − μ2 2 = v 2 −1 + ∗ uα v β−1 + a2 u + a3 v in Ω, ⎪ ⎪ |x| 2 ⎪ ⎪ ⎩ u, v > 0 in Ω \ {0}, u = v = 0 on ∂Ω,

(1.1)

where Ω ⊂ RN (N ≥ 3) is an open bounded domain with smooth boundary such that 0 ∈ Ω and the parameters satisfy the following assumptions: (H1 ) N ≥ 3,

η > 0,

0 ≤ μ2 ≤ μ1 < μ ¯ := ( N 2−2 )2 , ∗

α > 1, β > 1, α + β = 2 := √ √ ¯ − μ1 > 22−β μ ¯ − μ2 . (H2 ) μ ∗ −β

ai ≥ 0, i = 1, 2, 3,

2N . N −2

∗

E-mail address: [email protected]. This work is supported by the Fundamental Research Funds for the Central Universities of China, South–Central University for Nationalities (No. CZT18008).

1

 Let H := H01 (Ω) be the completion of C0∞ (Ω) with respect to ( Ω |∇ · |2 dx)1/2 . The functional corresponding to (1.1) is defined on H × H by 

 1  μ1 u2 + μ2 v 2 2 2 J(u, v) := dx − a u + 2a uv + a v |∇u|2 + |∇v|2 − 1 2 3 2 Ω |x|2  2∗

1 ∗ − ∗ |u| + |v|2 + η|u|α |v|β dx. 2 Ω Then J ∈ C 1 (H × H, R). A pair of functions (u0 , v0 ) ∈ H × H is said to be a solution to (1.1) if u0 , v0 > 0 and J  (u0 , v0 ), (ϕ, φ) = 0 for all (ϕ, φ) ∈ H × H, where J  (u, v) is the Fr´echet derivative of J at (u, v). Since the inverse square potential 1 ∈ C 2 (Ω \ Bδ (0)) for any δ > 0 small enough, a local elliptic regularity argument |x|2 shows that the solution (u0 , v0 ) to (1.1) has the following property (e.g. [7]): ¯ \ {0}). u0 , v0 ∈ C 2 (Ω \ {0}) ∩ C 1 (Ω

(1.2)

Problem (1.1) is related to the well–known Hardy inequality ([9]):   u2 1 dx ≤ |∇u|2 dx , ∀ u ∈ D1, 2 (RN ), 2 μ ¯ RN RN |x|  where D1, 2 (RN ) is the completion of C0∞ (RN ) with respect to ( RN |∇ · |2 dx)1/2 . In recent years, much attention has been paid to elliptic problems involving critical nonlinearities and Hardy–type terms (e.g. [3], [4], [5], [8], [11], [13]). Elliptic systems involving critical nonlinearities and the Hardy inequality have been also studied (e.g. [1], [6], [10], [12]). However, it is difficult to study the problems such as (1.1), since the Hardy–type terms μ1 |x|· 2 and μ2 |x|· 2 may bring different singularities at the origin to u0 and v0 in the solution (u0 , v0 ) when μ1 = μ2 . It should be mentioned that the following elliptic system is studied in [12]: ⎧

2∗ q u q q −1 ⎪ −Δu − μ = |u| + |v| |u|q−2 u + a1 u + a2 v in Ω, ⎪ 1 ⎪ 2 ⎪ |x| ⎪ ⎨

2∗ q v q q −1 (1.3) −Δv − μ = |u| + |v| |v|q−2 v + a2 u + a3 v in Ω, 2 ⎪ 2 ⎪ |x| ⎪ ⎪ ⎪ ⎩ u = v = 0 on ∂Ω, where 0 ∈ Ω ⊂ RN (N ≥ 3) and the parameters satisfy the following assumptions: √ √ ¯, μ ¯ − μ2 − μ ¯ − μ1 < 2. (H3 ) 2 ≤ q ≤ 2∗ , 0 ≤ μ2 ≤ μ1 < μ The singularities at the origin of solutions to (1.3) were proved under (H3 ). However, the case 1 < q < 2 of (1.3) has not been studied because there are crucial difficulties when applying the Moser iteration method to (1.3). 2

In this paper, we concentrate on the asymptotic properties at the origin of solutions to (1.1). By very technique methods we obtain the estimation of singularities for solutions not only for the case β ≥ 2 but also for the case 1 < β < 2 in (1.1). The ideas can be used to study the case 1 < q < 2 of (1.3) and similar results can be obtained for (1.3). The main results of this paper are summarized in the following theorem. To the best of our knowledge, the results are new. Theorem 1.1. Suppose that (H1 ) holds and (u0 , v0 ) ∈ H × H is a solution to (1.1). (i) Then there exist the constants Ci > 0, 1 ≤ i ≤ 4, such that C1 |x|−(

√

C3 |x|−(

√ μ ¯− μ ¯−μ1 )

√

√

μ ¯− μ ¯−μ2 )

≤ u0 (x) ≤ C2 |x|−( ≤ v0 (x) ≤ C4 |x|−(

√

√

√ μ ¯− μ ¯−μ1 ) √

μ ¯− μ ¯−μ1 )

, ∀ x ∈ Ω \ {0}.

, ∀ x ∈ Ω \ {0}.

(ii) Assume furthermore that 3 ≤ N ≤ 5, β ≥ 2. Then there exists a constant C5 > 0 such that v0 (x) ≤ C5 |x|−(

√ √ μ ¯− μ ¯−μ2 )

, ∀ x ∈ Ω \ {0}.

¯ − μ2 ), (iii) Assume furthermore that β < 2 and (H2 ) holds. Then for all ε ∈ (0, μ there exists a constant C6 > 0 such that √ √ v0 (x) ≤ C6 |x|−( μ¯− μ¯−(μ2 +ε) ) , ∀ x ∈ Ω \ {0}. Remark 1.2. (Open problem) Assume that (H1 ) holds and β < 2. Suppose furthermore that (H2 ) is not satisfied, that is, √ 2−β √ μ ¯ − μ1 ≤ ∗ μ ¯ − μ2 . 2 −β Then only the vague estimates in Theorem 1.1 (i) hold for v0 in the solution (u0 , v0 ), but the accurate singularity at the origin for v0 is not clear. However, we believe it is different with that of the case when (H2 ) holds but new methods should be used. In Section 2, we prove Theorem 1.1 by the Moser iteration methods. In the following argument, we always denote positive constants as C and omit dx in integrals for convenience.

2

Asymptotic properties of solutions

Suppose that (H1 ) holds and (u0 , v0 ) ∈ H × H is a solution to (1.1). For all ε ∈ (0, μ ¯ − μ2 ), we define √ √ √ ¯− μ ¯ − μi , i = 1, 2, ν3 := μ ¯− μ ¯ − (μ2 + ε), νi := μ 3

u := |x|ν1 u0 ,

v := |x|ν1 v0 ,

y := |x|ν2 v0 ,

w := |x|ν3 v0 .

Then u, v ∈ H01 (Ω, |x|−2ν1 ), y ∈ H01 (Ω, |x|−2ν2 ), w ∈ H01 (Ω, |x|−2ν3 ) (e.g. [3]), where H01 (Ω, |x|−2νi ) is the completion of C0∞ (Ω) with respect to ( Ω |x|−2νi |∇ · |2 )1/2 . √ ¯, u ∈ C 2 (Ω \ {0}), Lemma 2.1. ([3], [12]) Suppose 0 ∈ Ω ⊂ RN (N ≥ 3), τ < μ −2τ u > 0 in Ω \ {0} and −div(|x| ∇u) ≥ 0. Then for all ρ > 0 such that the ball Bρ (0) ⊂ Ω we have that u(x) ≥ min u(x), ∀ x ∈ Bρ (0) \ {0}. |x|=ρ

Proof of Theorem 1.1 (i). Choose ρ > 0 small enough such that Bρ (0) ⊂ Ω. Let ϕ be a cut–off function in Bρ (0). For all s > 1, n ∈ N, set un := min{u, n}, vn := min{v, n}, yn := min{y, n}, wn := min{w, n}. φ1 := ϕ2 uu2(s−1) , φ2 := ϕ2 vvn2(s−1) , φ3 := ϕ2 yyn2(s−1) , φ4 := ϕ2 wwn2(s−1) . n We divide the proof into two steps. Step 1. From (1.1) it follows that (u, v) solves the problem ⎧ ∗ −2ν1 u2 −1 + ηα uα−1 v β a1 u + a2 v ⎪ 2∗ ⎪ −div |x| ∇u = + , ⎪ ∗ ⎨ |x|2 ν1 |x|2ν1 ∗ ⎪ −2ν1 v 2 −1 + ηβ ⎪ uα v β−1 a2 u + a3 v (μ1 − μ2 )v ⎪ 2∗ ⎩−div |x| ∇v = + − . |x|2∗ ν1 |x|2ν1 |x|2(ν1 +1)

(2.1)

Multiply the first equation of (2.1) by φ1 and the second one by φ2 respectively. Since μ1 − μ2 ≥ 0, integrating by parts and arguing as in [3] we have   −2ν1 2 2 2(s−1) |x| ϕ |∇u| un + 4(s − 1) |x|−2ν1 ϕ2 |∇un |2 u2(s−1) n Ω

Ω

 + Ω

|x|

 ≤ C

Ω



−2ν1

|∇v|2 vn2(s−1)

(u +

2(s−1) uα v β )ϕ2 un

Ω



2(s−1)

ϕ 2 u2 un

+ Ω

+ 4(s − 1)

|x|−2ν1 (|∇ϕ|2 + ϕ2 )u2 u2(s−1) + n 2∗

+

ϕ

2

+ (v |x|2∗ ν1 2(s−1)

+ ϕ2 uvun |x|2ν1

4

2∗

 Ω

|x|−2ν1 ϕ2 |∇vn |2 vn2(s−1)



Ω

|x|−2ν1 (|∇ϕ|2 + ϕ2 )v 2 vn2(s−1) 2(s−1)

+ uα v β )ϕ2 vn 2(s−1)

+ ϕ2 v 2 vn

.

(2.2)

We use the following Caffarelli–Kohn–Nirenberg inequality ([2]):

 Ω

|u|p¯(σ,b) |x|b˜p(σ,b)

2  p(σ,b) ¯

 ≤ C(σ, b)

|x|−2σ |∇u|2 , ∀ u ∈ H01 (|x|−2σ , Ω),

Ω

(2.3)

where C(σ, b) > 0 is a constant and √ μ ¯,

−∞ < σ <

σ ≤ b ≤ σ + 1,

By (2.3) and the fact that νi <





Ω

Ω

√

2 |ϕuus−1 n | |x|2∗ ν1

|ϕvvns−1 | |x|2∗ ν1

∗

2∗

2N . N − 2 + 2(b − σ)

p¯(σ, b) :=

μ ¯ , p¯(νi , νi ) = 2∗ , 1 ≤ i ≤ 3, we have that  22∗

 22∗

 ≤C  ≤C

2 |x|−2ν1 |∇(ϕuus−1 n )| ,

Ω

(2.4) |x|−2ν1 |∇(ϕvvns−1 )|2 .

Ω

Then from (2.2) and (2.4) it follows that



  22∗ ∗ 2 2∗ |ϕuus−1 |ϕvvns−1 |2 2∗ n | + |x|2∗ ν1 |x|2∗ ν1 Ω Ω

 ≤ Cs |x|−2ν1 (|∇ϕ|2 + ϕ2 )(u2 u2(s−1) + v 2 vn2(s−1) ) n 

Ω

2(s−1)

∗

2(s−1)

∗

(u2 + uα v β )ϕ2 un

+ (v 2 + uα v β )ϕ2 vn + |x|2∗ ν1 Ω  2 2 2(s−1) 2(s−1) 2(s−1)  ϕ u un + ϕ2 uvun + ϕ2 v 2 v n . + |x|2ν1 Ω

(2.5)

From the H¨older inequality it follows that  



2(s−1)

Ω

ϕ2 u2 un |x|2ν1

Ω

2(s−1) ϕ 2 v 2 vn |x|2ν1

∗

Ω

Ω

∗ 2(s−1) ϕ 2 v 2 vn |x|2∗ ν1



≤ Cρ2



Ω

Ω

∗ |ϕvvns−1 |2 |x|2∗ ν1 ∗

Ω

Ω

∗ |ϕvvns−1 |2 |x|2∗ ν1



5

∗

2 |ϕuus−1 n | |x|2∗ ν1

2 |ϕuus−1 n | ∗ |x|2 ν1

≤ ≤



≤ Cρ



2(s−1)

ϕ 2 u 2 un |x|2∗ ν1

2

 22∗

 22∗

 22∗ 

, (2.6)

, ∗

Bρ (0)

 22∗ 

u2 |x|2∗ ν1 2∗

Bρ (0)

v |x|2∗ ν1

 2∗2−2 ∗

 2∗2−2 ∗

, (2.7)

.

By (2.5)–(2.7) and arguing as in Theorem 1.2 of [12] we have that u, v ∈ L∞ (Bρ (0)). By (1.2) we have that there exist the constants C1 , C2 > 0, such that u0 ≤ C1 |x|−ν1 , v0 ≤ C2 |x|−ν1 , ∀ x ∈ Bρ (0) \ {0}.

(2.8)

Step 2. From (1.1) it follows that (u, y) solves the problem ⎧ ∗ u2 −1 ηα uα−1 y β a1 u a2 y ⎪ −2ν1 ⎪ ∇u) = + + + , −div(|x| ⎪ ∗ ⎨ |x|2 ν1 2∗ |x|αν1 +βν2 |x|2ν1 |x|ν1 +ν2

(2.9) 2∗ −1 α β−1 ⎪ u y u y y ηβ a a ⎪ 2 3 −2ν ⎪ ⎩−div(|x| 2 ∇y) = 2∗ ν + ∗ αν +βν + ν +ν + 2ν . |x| 2 2 |x| 1 2 |x| 1 2 |x| 2 √ ¯, for all ρ > 0 small enough such that Bρ (0) ⊂ Ω, from Lemma 2.1, Since ν1 , ν2 < μ (1.2) and (2.9) it follows that there exist the constants C3 , C4 > 0, such that u0 ≥ C3 |x|−ν1 ,

v0 ≥ C4 |x|−ν2 , ∀x ∈ Bρ (0) \ {0}.

(2.10)

Furthermore, from (2.8) and (2.10) it follows that there exists ρ > 0 small enough such that u0 v0 < uα0 v0β , ∀x ∈ Bρ (0). (2.11) Since Ω is bounded, the desired results can be concluded by (1.2), (2.8) and (2.10). Proof of Theorem 1.1 (ii). Suppose that β ≥ 2. By (H1 ) we have 2 ≤ β < 2∗ − 1 and thus 3 ≤ N ≤ 5. From (H1 ) and (2.11) it follows that uy uα y β α β = u v < u v = , ∀ x ∈ Bρ (0). 0 0 0 0 |x|ν1 +ν2 |x|αν1 +βν2 From (2.3) and the H¨older inequality it follows that

  ∗ 2 |ϕyyns−1 |2 2∗ ≤C |x|−2ν2 |∇(ϕyyns−1 )|2 , |x|2∗ ν2 Ω Ω  2∗2−2  2 2∗ 2(s−1)  ∗ 2  ∗ ∗ ϕ y yn |ϕyyns−1 |2 2∗ y2 ≤ , ∗ ∗ ∗ 2 ν2 |x|2 ν2 |x|2 ν2 Ω Ω Bρ (0) |x|

  2 2 2(s−1) ∗ 2 ϕ y yn |ϕyyns−1 |2 2∗ 2 ≤ Cρ . |x|2ν2 |x|2∗ ν2 Ω Ω

(2.12)

(2.13)

Multiply the first equation of (2.9) by φ1 and the second one by φ3 respectively. Integrating by parts and arguing as in [3] we have   |x|−2ν1 ϕ2 |∇u|2 u2(s−1) + 4(s − 1) |x|−2ν1 ϕ2 |∇un |2 u2(s−1) n n Ω

Ω

6

 |x|

+ Ω

 ≤ C

Ω

−2ν2

ϕ

2

|∇y|2 yn2(s−1)

 + 4(s − 1)

|x|−2ν1 (|∇ϕ|2 + ϕ2 )u2 u2(s−1) + n

Ω



Ω

|x|−2ν2 ϕ2 |∇yn |2 yn2(s−1) |x|−2ν3 (|∇ϕ|2 + ϕ2 )y 2 yn2(s−1)

2(s−1) u + |x|β(ν1 −ν2 ) uα y β + u2 + |x|ν1 −ν2 uy ϕ2 un + |x|2∗ ν1 Ω  2∗ 2(s−1)  y + |x|α(ν2 −ν1 ) uα y β + |x|ν2 −ν1 uy + y 2 )ϕ2 yn . + |x|2ν2 Ω 

2∗

(2.14)

From (2.12)–(2.14) it follows that



 22∗

 ∗ 2 2∗ |ϕuus−1 |ϕyyns−1 |2 2∗ n | + |x|2∗ ν1 |x|2∗ ν2 Ω Ω





|∇ϕ|2 + ϕ2 |x|−2ν1 u2 u2(s−1) ≤ Cs + |x|−2ν2 y 2 yn2(s−1) n Ω



∗

2(s−1)

∗

2(s−1)

Ω

ϕ 2 u2 un |x|2∗ ν1

Ω

ϕ 2 y 2 yn |x|2∗ ν2

+  +

2(s−1) 2(s−1)  + yn ϕ2 u α y β u n + |x|αν1 βν2 2(s−1) 2(s−1)  ϕ2 u2 u n ϕ2 y 2 y n . + + |x|2ν1 |x|2ν2

(2.15)

Note that ν1 ≥ ν2 ≥ 0. Taking ρ ∈ (0, 1) small enough we have that yn ≤ un ,

yn un y u ≤ ν1 , ≤ ν1 , ∀x ∈ Bρ (0) such that y ≤ u, ν ν 2 2 |x| |x| |x| |x|

(2.16)

un ≤ yn , ∀x ∈ Bρ (0) such that u ≤ y. Since α + β = 2∗ , β ≥ 2, from (2.15), (2.16) and the H¨older and Young inequalities it follows that  2 α β 2(s−1) ϕ u y yn |x|αν1 +βν2 Ω  β−2

  2α∗

 ∗ 2  ∗ ∗ 2∗ |ϕyyns−1 |2 2∗ y2 u2 ≤ . (2.17) 2∗ ν 2 2∗ ν 1 |x|2∗ ν2 Ω Bρ (0) |x| Bρ (0) |x| 

2(s−1)

ϕ 2 uα y β u n |x|αν1 +βν2 Ω    ∗ 2(s−1) ∗ 2(s−1) ∗ 2(s−1) ϕ 2 y 2 un ϕ 2 y 2 un α ϕ 2 u 2 un β β ≤ ∗ + + 2 Ω |x|2∗ ν1 2∗ [y≤u] |x|2∗ ν2 2∗ [u≤y] |x|2∗ ν2 7

 ≤

∗

Ω

2(s−1)

ϕ2 u2 un |x|2∗ ν1

β + ∗ 2



∗

Ω

2(s−1)

ϕ 2 y 2 yn |x|2∗ ν2

,

(2.18)

Taking ρ small enough, from (2.6), (2.7), (2.15), (2.17) and (2.18) it follows that



 22∗  ∗ 2 2∗ |ϕuus−1 |ϕyyns−1 |2 2∗ n | + |x|2∗ ν1 |x|2∗ ν2 Ω Ω   ≤ Cs |x|−2ν1 |∇ϕ|2 u2 u2(s−1) + Cs |x|−2ν2 |∇ϕ|2 y 2 yn2(s−1) . n Ω

(2.19)

Ω

By (2.19) and arguing as in Theorem 1.2 of [12] we have that u, y ∈ L∞ (Bρ (0)). Furthermore, there exists a constant C > 0, such that v0 ≤ C|x|−ν2 , ∀ x ∈ Ω \ {0}. The proof is complete. Proof of Theorem 1.1 (iii). Suppose that (H1 ) and (H2 ) hold with β < 2. Note that (u, w) solves the problem ⎧ ∗ ⎪ u2 −1 ηα uα−1 wβ a1 u a2 w −2ν ⎪ 1 ⎪ ∇u)= 2∗ ν1 + ∗ αν1 +βν3 + 2ν1 + ν1 +ν3 , −div(|x| ⎪ ⎨ |x| 2 |x| |x| |x| ∗ ⎪ ⎪ w2 −1 ηβ uα wβ−1 a2 u a3 w εw ⎪ −2ν3 ⎪ ∇w)= + + + − . −div(|x| ⎩ |x|2∗ ν3 2∗ |x|αν1 +βν3 |x|ν1 +ν3 |x|2ν3 |x|2(1+ν3 )

(2.20)

From (2.3) and the H¨older inequality it follows that

 Ω

 

|ϕwwns−1 |2 |x|2∗ ν3 2∗

∗

2(s−1)

Ω

ϕ2 w w n |x|2∗ ν3

Ω

ϕ2 w 2 w n |x|2ν3

2(s−1)

 22∗

 ≤C

Ω

|x|−2ν3 |∇(ϕwwns−1 )|2 ,

∗ ∗ 2  ∗  2 −2 2∗ |ϕwwns−1 |2 2∗ w2 ≤ , ∗ ∗ 2 ν3 |x|2 ν3 Ω Bρ (0) |x|

 ∗ 2 |ϕwwns−1 |2 2∗ 2 ≤ Cρ . |x|2∗ ν3 Ω



(2.21)

Multiply the first equation of (2.20) by φ1 and the second one by φ4 respectively. Integrating by parts and arguing as in [3] we have   |x|−2ν1 ϕ2 |∇u|2 u2(s−1) + 4(s − 1) |x|−2ν1 ϕ2 |∇un |2 u2(s−1) n n Ω

Ω



+ Ω

|x|−2ν3 ϕ2 |∇w|2 wn2(s−1) + 4(s − 1) 8

 Ω

|x|−2ν3 ϕ2 |∇wn |2 wn2(s−1)

 ≤ C

Ω

  −2ν3 2 2(s−1) (|∇ϕ|2 + ϕ2 ) |x|−2ν1 u2 u2(s−1) + |x| w w n n

  2∗ u uα w β u2 uw  2 2(s−1) + + + ϕ un + 2 ∗ ν1 |x|αν1 +βν3 |x|2ν1 |x|ν1 +ν3 Ω |x|    2∗ w uα wβ uw w2 εw2  2 2(s−1) . + + αν1 +βν3 + ν1 +ν3 + 2ν3 − 2(1+ν3 ) ϕ wn 2∗ ν 3 |x| |x| |x| |x| Ω |x|

(2.22)

From (2.11) it follows that there exists ρ > 0 small enough such that w2

=

|x|2(1+ν3 )

v02 , |x|2

uw uα w β α β = u v < u v = , ∀x ∈ Bρ (0). 0 0 0 0 |x|ν1 +ν3 |x|αν1 +βν3

(2.23)

Since (H1 ) and (H2 ) hold, a direct calculation shows that αν1 = (2∗ − β)ν1 < 2 + (2 − β)ν2 , which together with (2.8) and (2.10) implies that uα0 ≤ C|x|−αν1 < Cε|x|−(2+(2−β)ν2 ) ≤

ε 2−β v , ∀ x ∈ Bρ (0), ε > 0. |x|2 0

Consequently, uα wβ v02 w2 α β = u v < ε = ε , 0 0 |x|αν1 +βν3 |x|2 |x|2(1+ν3 )

∀ x ∈ Bρ (0), ∀ ε > 0.

(2.24)

From (2.4) and (2.21)–(2.24) it follows that  22∗

 ∗ 2 2∗ |ϕuus−1 |ϕwwns−1 |2 2∗ n | + |x|2∗ ν1 |x|2∗ ν3 Ω Ω



≤ Cs (|∇ϕ|2 + ϕ2 ) |x|−2ν1 u2 u2(s−1) + |x|−2ν3 w2 wn2(s−1) n



Ω

   2∗ ∗ u u2  2 2(s−1)  w2 w2  2 2(s−1)  ϕ un ϕ wn . + + + + |x|2∗ ν1 |x|2ν1 |x|2∗ ν3 |x|2ν3 Ω Then following the argument of Theorem 1.1 (ii) we have that u, w ∈ L∞ (Bρ (0)). In particular, there exists a constant C > 0, such that v0 ≤ C|x|−ν3 , ∀ x ∈ Ω \ {0}. The proof is thus complete. Acknowledgement The authors acknowledge the anonymous referee for carefully reading this paper and making important comments. 9

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