Asymptotic behavior and uniqueness of boundary blow-up solutions to elliptic equations

Asymptotic behavior and uniqueness of boundary blow-up solutions to elliptic equations

Qiaoyu Tian

and Yonglin Xu

School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, P.R. China

Received 10 July 2014, appeared 11 January 2015 Communicated by Gabriele Villari

Abstract. In this paper, under some structural assumptions of weight functionb(x)and nonlinear term f(u), we establish the asymptotic behavior and uniqueness of boundary blow-up solutions to semilinear elliptic equations

(∆u=b(x)f(u), x∈_Ω, u(x) =_∞, x∈∂Ω,

where Ω ⊂R^N is a bounded smooth domain. Our analysis is based on the Karamata regular variation theory and the López-Gómez localization method.

Keywords: boundary blow-up solutions, asymptotic behavior, López-Gómez localiza- tion method, Karamata regular variation theory.

2010 Mathematics Subject Classification: 35J75, 35J66.

1 Introduction and main results

In this paper, we deal with the asymptotic behavior of boundary blow-up solutions to semi- linear elliptic equations

∆u=b(x)f(u), x ∈_Ω,

u(x) =_∞, x ∈_∂Ω, ^(1.1)

whereΩ⊂_R^N (N≥3)is a bounded smooth domain, weight functionb(x)satisfies (b₁) there exists a positive nondecreasing functiona(x)∈ C([0,δ])such that

d(limx)→0

b(x)

a(d(x)) =1, (1.2)

where

1 a(r)

Z _r

0 a(s)ds∈C¹([0,δ]);

BCorresponding author. Email: tianqiaoyu2004@163.com

(b₂) for anyx₀∈∂Ω, there existsτ>0, such thatb(x)∈C¹(_Ωτ(x₀)∩_Ω)satisfies

b_x0(r)∈C¹((0,τ)), b⁰_x0(r)>0 for eachr∈ (0,τ), (1.3) and

x∈_∂Ω,limx→x0, r→0

b_x(r)

bx₀(r) =_{1, (1.4)}

whereΩτ(x0)is a ball inR^N of radiusτcentered at x0, boundary normal sectionsbx(r) defined as

bx(r) =b(x−rnx), r >0, r∼0, (1.5) wherenx stands for the outward unit normal vector atx ∈∂Ω.

The nonlinear term f(u)satisfies

(f₁) f ≥0 is locally Lipschitz continuous on[0,∞)and f(u)/uis increasing on(0,∞); (f₂) there exist some L ∈ C²([A,∞)) satisfying limu→_∞L(u) = _{∞ if} r = _{1 and} L⁰ ∈

NRV−r(0≤r ≤1), a slowly varying functionL and p≥0 such that

ulim→_∞

f(L(u))

L⁰(u)u^p+r =1, (1.6)

where p>1−rif 0≤r <1 andp ≥0 ifr=1.

The main result of this paper is the following theorem.

Theorem 1.1. Suppose that b(x)satisfies(b₁)–(b₂)and f(u)satisfies(f₁)–(f₂). Then, problem(1.1) possesses a unique positive solution u(x). Moreover, for each x₀ ∈ ∂Ω, any positive solution u(x) satisfies

limr→0

u(x₀−rn_x0) I(x₀)^−p−p1L(_Φx0(d(x)))

=

p+1 p−1

^p_p⁺₋¹₁

, (1.7)

where

Φx0(t) =

Z _∞

t

"

Z _s

0

L⁰(_Φx0)b_x0 L⁰(_Φx0)

_p+¹_r+1#−^p_p⁺₊^r_r⁺₋¹₁

ds, (1.8)

I(x₀) =lim

t→0

Φx0(t)_Φ⁰⁰_x0(t) [_Φ⁰_x

0(t)]^{2 , (1.9)}

L,Lappear in(1.6)and b_x0 is defined by(1.5).

The interest in these problems goes back to the pioneering works of López-Gómez. Pre- cisely, López-Gómez [11], used the so-called López-Gómez’s localization method, ascertained asymptotic behavior of boundary blow-up solutions to problem (1.1) with f(u) =u^pandb(x) vanishing on the boundary of the underlying domain at different rates according to the point of boundary. This results was developed by López-Gómez [12], Cano-Casanova and López- Gómez [1,2], Wei and Zhu [18], Wang and Wang [19] and Xie [20]. In particular, Huang et. al. [10] obtained asymptotic behavior of boundary blow-up solutions to problem (1.1) with nonlinear term f satisfying

(f₃) there exists a slowly varying functionH andp>1 such that

ulim→_∞

f(u)

H(u)u^p =_{1. (1.10)}

Remark 1.2. Note that (f₃)implies that f(u) ∈ RVp, see Remark 1.1 of [10]. It can easily be seen that f(L(u))∈ RV_pif(f₂)holds. Thus f is a normalized varying function at infinity with index p/(1−r)if 0≤ r < 1 and f is rapidly varying with index∞if r =1, for more details see [9]. Consequently, the main results of this paper give a unified asymptotic behavior of boundary blow-up solutions to problem (1.1).

Remark 1.3. Based on the results of López-Gómez [1,2,11,12], Ouyang and Xie [13,14], Xie [20], Xie and Zhao [21] established some similar asymptotic behavior of boundary blow-up solu- tions to problem (1.1). Recently, Huang et. al. [8], using the Karamata regular variation theory approach introduced by Cîrstea and R˘adulescu [5,6], established asymptotic behavior and uniqueness of boundary blow-up solutions to problem (1.1) with f satisfying (_f3)_{, extended} the main results of [13,14,20]. Similarly, we can obtain similar asymptotic behavior of bound- ary blow-up solutions to problem (1.1) with f satisfying(f₂).

Remark 1.4. For the existence of boundary blow-up solutions to problem (1.1), see Theorem 1.1 of [4].

Remark 1.5. One easily sees thatΦx0(t), defined by (1.8), is a decreasingC²-function on some interval(0,ς), for someς>0. Consequently, taking into account Lemma 3.1 in [3], I(x₀)≥1.

Furthermore, −_Φ⁰_x

0(t) is normalized regularly varying at zero of index I(x₀)/(1−I(x₀)) if I(x₀)>_{1 andΦx0}(t)has a representation formula if I(x₀) =_1.

Remark 1.6. By (1.8), we know that

Φ⁰_x0(t) =−

"

Z _t

0

L⁰(_Φx0)bx₀

L⁰(_Φx0)

_p+¹_r+1#−^p_p⁺₊^r_r⁺₋¹₁

, (1.11)

and

Φ⁰⁰_x0(t) = ^p+r+1 p+r−₁

"

Z _t

0

L⁰(_Φx0)b_x0 L⁰(_Φx0)

_p+¹_r+1#−²_p⁽₊^p_r⁺₋^r₁⁾

L⁰(_Φx0)b_x0 L⁰(_Φx0)

_p+¹_r+1

. (1.12) Thus, taking into account(f2)and limt→0Φx₀(t) =∞, we obtain

limt→0

Φ⁰⁰_x0(t)L⁰(_Φx0(t)) bx₀(t)f(L(_Φx0(t)))

=_lim

t→0

L⁰(_Φx0(t))_Φ^p_x⁺₀ ^r(t) f(L(_Φx0(t)))

Φx0(t)_Φ⁰⁰_x0(t) [_Φ⁰_x0(t)]²

⁻⁽p+r)

[_Φ⁰⁰_x0(t)]^p+r+1 [_Φ⁰_x

0(t)]^2(p+r)

L⁰(_Φx0(t)) L⁰(_Φx0(t))bx₀(t)

= [I(x₀)]^−(p+r)

p+r+1 p+r−1

p+r+1

,

and

limt→0

[_Φ⁰_x0(t)]²L⁰⁰(_Φx0(t)) bx0(t)f(L(_Φx0(t)))

=lim

t→0

Φx0(t)L⁰⁰(_Φx0(t)) L⁰(_Φx0(t))

L⁰(_Φx0(t))_Φ^p_x0^+r(t) f(L(_Φx0(t)))

Φx₀(t)_Φ⁰⁰_x0(t) [_Φ⁰_x0(t)]²

−(p+r+1)

×[_Φ⁰⁰_x

0(t)]^p+r+1 [_Φ⁰_x

0(t)]^2(p+r)

L⁰(_Φx0(t)) L⁰(_Φx0(t))bx₀(t)

=−r[I(x₀)]^−(p+r+1)

p+r+1 p+r−1

p+r+1

.

The structure of this paper is as follows. In Section 2, we collect some preliminary results of Karamata regular variation theory. In Section 3 we prove some auxiliary results. Theorem1.1 will be proved in Section 4.

2 Auxiliary results

The main purpose of this section is to provide some concepts from the theory of regular variation. For detailed accounts of the theory of regular variation, its extensions and many of its applications, we refer the interested reader to [7,15–17]. When the regular variation occurs at infinity and there is no possibility of confusion, we omit “at infinity”.

Definition 2.1. A positive measurable function f defined on[D,∞)for some D> 0 is called regularly varying (at infinity) with indexp∈_R(written f ∈ RV_p) if for allξ >0

ulim→_∞

f(ξu) f(u) =ξ^p.

When the index of regular variation p is zero, we say that the function is slowly varying.

The transformation f(u) =u^pL(u)reduces regular variation to slow variation.

Proposition 2.2.Assume that L is slowly varying. Then the convergence L(ξu)/L(u)→1as u→_∞ holds uniformly on each compactε-set in(0,∞).

Proposition 2.3. If L is slowly varying, then (i) lnL(u)/ lnu→0as u →_∞;

(ii) for anyα>0, u^αL(u)→_{∞, u}^−αL(u)→0as u→ _∞;

(iii) (L(u))^αvaries slowly for everyα∈_R;

(iv) if L₁varies slowly, so do L(u)L₁(u)and L(u) +L₁(u).

Now we collect some important results which will be used in the proof of Theorem1.1.

Definition 2.4. A function u ∈ C²(_Ω) is a (classical) subsolution to problem (1.1), if u= +_∞on∂Ωand

∆u≥b(x)f(u), x∈_Ω.

Similarly,uis a (classical) supersolution to problem (1.1), ifu= +_∞on∂Ωand

∆u≤b(x)f(u), x∈_Ω.

The following comparison principle which plays an important role in the proof of Theorem 1.1 will be used in later sections.

Proposition 2.5. Let f be continuous on (0,∞) such that f(u)/u is increasing for u > 0, Let b(x)∈C(_Ω)be a nonnegative function. Assume that u₁,u₂ ∈C²(_Ω)are positive such that

(∆u1−b(x)f(u₁)≤0≤_∆u2−b(x)f(u₂), x∈_Ω, lim sup_{d(x,∂Ω)→0}(u₂−u₁)(x)≤0.

Then we have u₁ ≥u2 inΩ.

3 Auxiliary results

To prove Theorem 1.1 by the López-Gómez localization method, firstly consider the corre- sponding singular problem with radial weight function b(x) in a ball or an annular domain.

Note that, in this case, (3.2) is uniformly satisfied on∂Ω.

Theorem 3.1. SupposeΩr(x₀) = {x ∈ _R^N : |x−x₀| < r}, f(u) satisfies (f₁)–(f₂), and b(x) = b(r− kx−x₀k), b∈ C([0,r]: [0,∞))satisfies(b₁). Then, problem(1.1)possesses a unique positive solution u(x). Moreover, any positive solution u(x)satisfies

d(limx)→0

u(x)

Φ(d(x)) = I⁻

p p−1

p+1 p−1

^p_p⁺₋¹₁

, (3.1)

where

I =_lim

t→0

Φ(t)_Φ⁰⁰(t)

[_Φ⁰(t)]^{2 , Φ}(t) =

Z _∞

t

"

Z _s

0

L(_Φ)a L(_Φ)

_p+¹₁#⁻

p+1 p−1

ds, (3.2)

L,Lappear in(1.6)and a appears in(b₁).

Similarly, we have the following corresponding results whenΩ= _Ωr1,r2(x0) = x ∈ _R^N : r₁< |x−x₀|<r₂ .

Theorem 3.2. Suppose Ω = _Ωr1,r2(x₀), f(u) satisfies (f₁)–(f₂), and b(x) = b(r₂− kx−x₀k), b∈C([0,r]: [0,∞))satisfies(b₁). Then problem(1.1) possesses a unique positive solution u(x)and (3.1)holds.

Note that, when the domain is an annular domain, d(x) =

(r₂− |x−x₀|, (r₁+r₂)/2≤ |x−x₀|<r₂,

|x−x0| −r₁, r₁≤ |x−x0|<(r₁+r2)/2.

In the following, the proof of Theorem 3.1 will be given. Theorem 3.2 can be proved by similar arguments, more details are omitted here.

Proof of Theorem3.1. It is interesting to note that (1.2) holds uniformly, for eachε > 0; choose δ>0 sufficiently small such that,

(₁−ε)a(d(x)−β)<b(x)<(₁+ε)a(d(x) +β)_{, 0}<β<d(x)<_δ.

For fixed β ∈ (0,δ), define u±(x) = L(ξ^±Φ(d(x)±β)), x ∈ _Ω^±

β, where u(x)is the solution to problem (1.1), Φ(t) is defined by (3.2), Ωδ = {x ∈ _{Ω, 0} < d(x) < δ}_{, Ω}⁻

β = _Ω2δ\_Ω^¯_β, Ω⁺_β =_Ω2δ−β and

ξ^±=

"

1±ε 1∓ε

p+1 p−1

p+1

I^p

#_p−¹₁ . Consequently,

∇u±(x) =ξ^±L⁰(ξ^±Φ(d(x)±β))_Φ⁰(d(x)±β)∇d(x),

∆u±(x) = (ξ^±)²L⁰⁰(ξ^±Φ(d(x)±β))[_Φ⁰(d(x)±β)]² +ξ^±L⁰(ξ^±Φ(d(x)±β))_Φ⁰⁰(d(x)±β) +ξ^±L⁰(ξ^±Φ(d(x)±β))_Φ⁰(d(x)±β)_∆d(x). Thus,

∆u+(x)−b(x)f(u+(x))

≥(ξ⁺)²L⁰⁰(ξ^±Φ(d(x) +β))[_Φ⁰(d(x) +β)]²+ξ⁺L⁰(ξ^±Φ(d(x) +β))_Φ⁰⁰(d(x) +β) +ξ⁺L⁰(ξ⁺Φ(d(x) +β))_Φ⁰(d(x) +β)_∆d(x)−(1+ε)a(d(x) +β)f(u+(x))

=_a(_d(_x) +β)_f(_u+(_x))[_A1⁺(_d(_x) +β) +_A2⁺(_d(_x) +β)_∆d(_x)−(₁+ε)]_, and

∆u−(x)−b(x)f(u(x))

≤(ξ⁻)²L⁰⁰(ξ⁻Φ(d(x)−β))[_Φ⁰(d(x)−β)]²+ξ⁻L⁰(ξ⁻Φ(d(x)−β))_Φ⁰⁰(d(x)−β) +ξ⁻L⁰(ξ⁻Φ(d(x)−β))_Φ⁰(d(x)−β)_∆d(x)−(1−ε)a(d(x)−β)f(u−(x))

=a(d(x)−β)f(u−(x))[_A1⁻(d(x)−β) +_A2⁻(d(x)−β)_∆d(x)−(1−ε)], where

A₁^±(t) = (ξ^±)²L⁰⁰(ξ^±Φ(t))[_Φ⁰(t)]² a(t)f(L(ξ^±Φ(t))) + ^ξ

±L⁰(ξ^±Φ(t))_Φ⁰⁰(t) a(t)f(L(ξ^±Φ(t))) ^, A₂^±(t) = ^ξ

±L⁰(ξ^±Φ(t))_Φ⁰(t) a(t)f(L(ξ^±Φ(t))) ^. Similar computations as in Remark1.6show that

limt→0A₁^±(t) =^h(ξ^±)²I^−(p+r)−rξ^±I^−(p+r+1)i

p+r+1 p+r−1

p+r+1

, and

limt→0A₂^±(t) =lim

t→0

ξ^±L⁰(ξ^±Φ(t))_Φ⁰⁰(t) a(t)f(L(ξ^±Φ(t)))

Φ⁰(t) Φ⁰⁰(t) =0.

Consequently, lim

d(x)±β→0

A₁^±(d(x)±β) +_A2^±(d(x)±β)_∆d(x)−(1±ε)=±ε, which implies that we can chooseδ>0 such that

(∆u⁺_β −b(x)f(u⁺_β)≥0, x∈_Ω⁺

β,

∆u⁻_β −b(x)f(u⁻_β)≤0, x∈_Ω⁻_β.

Define M(2δ) = max_d(x)≥2δu(x), where u(x)is a nonnegative solution of problem (1.1).

Obviously, u(x) ≤ M(2δ) +u⁻_β, x ∈ {x ∈ _Ω : d(x) = 2δ} and lim_d→β[M(2δ) +u⁻_β] = _∞.

Namely, u(x) ≤ M(2δ) +u⁻_β, x ∈ _∂Ω⁻_β. On the other hand, ∆(M(2δ) +u⁻_β) = _∆u⁻_β ≤ b(x)f(u⁻_β)≤b(x)f(M(2δ) +u⁻_β),x ∈_Ω⁻_β, the comparison principle of elliptic equations leads to

u(x)≤ M(2δ) +u⁻_β, x∈_Ω⁻_β. (3.3) Defineu⁺_β(x) =L(ξ⁺Φ(2δ)),x∈ {x ∈_Ω:d(x) =2δ−β}andN(2δ) =L(ξ⁺Φ(2δ)). It is easy to see that

u⁺_β(x)≤ N(2δ) +u(x), x ∈ {x ∈_Ω: d(x) =2δ−β}, lim

d→0[u⁺_β(x)−N(2δ)−u(x)] =−_∞.

That is, u⁺_β(x) ≤ N(2δ) +u(x), x ∈ ∂Ω⁺_β. Note that ∆(u⁺_β(x)−N(2δ)) = _∆u⁺_β(x) ≥ b(x)f(u⁺_β)≥b(x)f(u⁺_β(x)−N(2δ)). This fact, combined with the comparison principle shows that

u⁺_β(x)≤ N(2δ) +u(x), x∈_Ω⁺

β. (3.4)

According to (3.3) and (3.4), we find

u⁺_β(x)−N(2δ)≤u(x)≤ M(2δ) +u⁻_β, x ∈_Ω⁻

β ∩_Ω⁺

β. This yields

u⁺_β(x)−N(2δ)

L(ξ^±Φ(d(x))) ≤ ^u(x)

L(ξ^±Φ(d(x))) ≤ ^M(2δ) +u⁻_β

L(ξ^±Φ(d(x)))^{, x} ∈_Ω⁻_β ∩_Ω⁺_β. (3.5) Letting ε→ 0 andd(x)→0 in (3.5) leads to (3.1), here we use the fact thatd(x) →0 implies β→0 ifx∈ _Ω⁻_β ∩_Ω⁺_β.

4 Proof of Theorem 1.1

In this section, we will prove Theorem1.1by the localization method introduced in [11,12].

Proof. Fixedε >0, according to (1.4), there existρ = ρ(ε)∈ (0,η)andµ= µ(ε)such that for eachx ∈∂Ω∩_Ωρ(x0),r∈ (0,µ),

1−ε < ^bx(r)

bx₀(r) = ^b(x−rn_x)

b(x₀−rnx₀) <1+ε. (4.1) DefineB =x−rnx :x∈ ∂Ω∩_Ωρ(x0), r ∈[0,µ] . Note that for eachy ∈_B(ρ,µcan be shortened if necessary), there exists a uniquey₀ ∈_∂Ω∩_Ωρ(x₀), andr(y)∈[0,µ], such thaty= y₀−r(y)n_y0,r(y) =|y−y₀|=dist(y,∂Ω). Furthermore, there existsr₀∈ (0, min{ρ/2,µ/2}), such that Ωr0(x0−r0nx0) ⊂ _{Ω, and Ωr0}(x0−r0nx0)∩∂Ω = {x0}. Thus there exists σ₀ > 0 such that for σ∈ (0,σ₀], Ωr0(x₀−(r₀+σ)n_x0)⊂ _Ω∩IntB. Consequently, forσ ∈ [0,σ₀]and y∈_Ωr0(x₀−(r₀+σ)n_x0),

b(y) =b(y₀−r(y)_ny0)≥(₁−ε)b(x₀−r(y)_nx0) = (₁−ε)b_x0(r(y))

≥ (1−ε)b_x0(dist(y,∂Ωr0(x₀−(r₀+σ)n_x0))),

which shows thatb(y)≥(₁−ε)b_x0(r_σ)_{, where}r_σ=_dist(y,∂Ωr0(x₀−(r₀+σ)_nx0))_.

LetU be the unique solution to problem

(∆u= (1−ε)bx₀(rσ)f(u), x∈_Ωr0(x₀−(r₀+σ)nx₀),

u(x) = +_∞, x∈∂Ωr0(x₀−(r₀+σ)_nx0), (4.2) whereσ∈[0,σ₀]. Equation (3.1) shows that

lim

x→_∂Ωr0(x0−(r0+σ)nx0)

U(x)

K₁(x0)B₁(rσ) = (1−ε)^β, where

B₁(t) =

Z _r

0

Z _s

0

(H◦ B₁^−β(t))b_x0(t)dt ds, K₁(x₀) = [β(β+1)C_x0−β]^β, β= ¹ p−1, C_x0 =lim

t→0

[B₁⁰(t)]²

B₁(t)bx₀(t)H◦ B⁻₁^β(t)^. Thus u|_Ωr

0(x0−(r0+σ)nx0) is a bounded subsolution of (4.2), hence, for each σ ∈ [0,σ₀] and x∈ _Ωr0(x₀−(r₀+σ)n_x0),u_σ= u|_Ωr

0(x0−(r0+σ)nx0) ≤ U, and lim sup

x→_∂Ωr0(x0−(r0+σ)nx0)

u_σ

K₁(x₀)B₁(rσ) ≤(1−ε)^β. Lettingσ →_{0 gives}

limr→0

u(x₀−rn_x0)

K₁(x0)B₁(r) ≤(₁−ε)^β_. This is valid for any sufficiently smallε>0, then

limr→0

u(x₀−rn_x0)

K₁(x₀)B₁(r) ≤1. (4.3)

For anyx₀∈∂Ω, there exist 0<r₁<r₂andσ₀, such that Ω⊂ ^\

0≤σ≤σ0

Ωr1,r2(x₀+ (r₁+σ)_nx0), ∂Ω∩_Ωr1,r2(x₀+r₁nx0) ={x₀} andr₁is small enough,r2is large enough such thatΩ⊂ _Ωr1,r2/3(x0+r₁nx0).

By (4.1), we find that for each y ∈ _Ω2η(x₀)∩_{Ω, where} η ∈ min{ρ,µ} is small, b(y) = b(y₀−r(y)n_y0)≤ (1+ε)b_x0(r(y)) ≤(1+ε)b_x0(dist(y,∂Ωr1(x₀+r₁n_x0))). Define the function eb: Ωr₁,r2(x0+r₁nx0) → [0,∞) as eb(y) = ^eb(r) = (1+ε)bx0(r), where y ∈ _Ω2η(x0)∩_Ω and r = dist(y,∂Ωr1,r2(x₀+r₁n_x0)). Moreover,eb(dist(y,∂Ωr1,r2(x₀+ (r₁+σ)n_x0))≥ b(y), for each y∈_Ω,σ∈ [0,σ₀],

LetUbe the unique solution to

(∆u=eb(r)f(u), x ∈_Ωr1,r2(x₀+ (r₁+σ)n_x0), u(x) = +_∞, x ∈_∂Ωr1,r2(x₀+ (r₁+σ)nx₀), wherer=dist(y,∂Ωr1,r2(x₀+ (r₁+σ)_nx0)), and

x→_∂Ωr1,r2(limx0+(r₁+σ)nx0)

U(x)

K₂(x₀)B₂(r) = (1+ε)^β,

where

B₂(_t) =

Z _r

0

Z _s

0

(_H◦ B₂^−β(_r))_bx0(_r)_{ds dt,} K₂(x₀) = [β(β+1)C_x0−β]^β, β= ¹

p−1, C_x0 =lim

t→0

[B₂⁰(t)]²

B₂(t)bx₀(r)H◦ B₂^−β(t)^.

Moreover,U|_Ω is a subsolution of (1.1), this implies that U(x) ≤ u(x), for eachσ ∈ [0,σ₀] andx ∈_Ωr1,r2(x0+ (r₁+σ)nx0)∩Ω. This yields

limr→0

u(x₀−rn_x0)

K2(x0)B₂(r) ≥(₁+ε)^β_. Lettingσ→0, we derive that

lim inf

x→x0, x∈_Ωr1,r2(x0+r1nx0)

u(x)

K₂(x₀)B₂(r) ≥1. (4.4) It can easily be seen that B₁(r) = B₂(r) and K₁(x₀) = K₂(x₀). Using (4.3) and (4.4), we obtain (1.7).

Acknowledgements

This works was partially supported by the National Natural Science Foundation of China (No. 11401473, 11326100), Natural Science Foundation of Gansu Province (No. 145RJZA214), Fundamental Research Funds for the Central Universities (No. 31920140058, 31920130001, 31920130006), Talent Introduction Scientific Research Foundation of Northwest University for Nationalities (No. xbmuyjrc201305, xbmuyjrc201121), Research and Innovation Teams of Northwest University for Nationalities.

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