Regularity theory on A harmonic system and A Dirac system

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R E S E A R C H

Open Access

Regularity theory on

A

-harmonic system and

A

-Dirac system

Fengfeng Sun and Shuhong Chen

*

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Minnan Normal University, Fujian, Zhangzhou 363000, China

Abstract

In this paper, we show the regularity theory on anA-harmonic system and anA-Dirac system. By the method of the removability theorem, we explain how anA-harmonic system arises from anA-Dirac system and establish that anA-harmonic system is in fact the real part of the correspondingA-Dirac system.

Keywords: A-harmonic system;A-Dirac system; Caccioppoli estimate; natural growth condition; removable theorem

1 Introduction

In this paper, we consider the regularity theory on anA-Dirac system,

–DA˜(x,u,Du) =f(x,u,Du), in, (.)

and anA-harmonic system,

–divA(x,u,∇u) =f(x,u,∇u), in. (.)

Hereis a bounded domain inRn ₍_n_≥_),_A₍_x_,_u_,_∇_u_{) and}_f₍_x_,_u_,_∇_u_{) are measurable} functions deﬁned on×Rn_×_RnN_, _N_{is an integer with}_N_{> ,}_u_:_→_Rn_{is a vector} valued function. Furthermore,A(x,u,∇u) andf(x,u,∇u) satisfy the following structural conditions withm> :

(H) A(x,u,p)are diﬀerentiable functions inpand there exists a constantC> such that

∂A(_∂x,_pu,p)≤C +|p| m–

 _{for all}₍_x_,_u_,_p₎∈×_Rn_×_RnN_.

(H) A(x,u,p)are uniformly strongly elliptic, that is, for someλ> we have

∂A(x,u,p)

∂p ν

α

i

ν_jβ≥λ +|p|m–|_ν|_.

(H) There existβ∈(, )andK: [,∞)→[,∞)monotone nondecreasing such that

A(x,u,p) –A(x˜,u˜,p)≤K|u||x–x˜|m+|u–u˜|m β

m_{ +}|_p|m

for allx,x˜∈,u,u˜∈Rn_{, and}_p_∈_RnN_{. Without loss of generality, we take}_K_≥__.

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(H) There exist constantsCandCsuch that f(x,u,p)≤C|p|m+C.

(H) and (H) imply

_A₍_x_,_u_,_p_{) –}_A₍_x_,_u_,ξ)≤C +|p|+|ξ| m–

 |_p_–_ξ|_; _(.)

A(x,u,p) –A(x,u,ξ)(p–ξ)≥λ +|p|+|ξ| m–

 |_p_–_ξ| _(.)

for allx∈,u∈Rn_and_p_,_ξ_∈_RnN_{, where}_λ_{>  is a constant.}

Deﬁnition . We say that a functionu∈W_loc,m()∩L∞() is a weak solution to (.), if the equality

A(x,u,∇u)∇φdx=

f(x,u,∇u)φdx (.)

holds for allφ∈W_,m() with compact support.

In this paper, we assume that the solutions of the A-harmonic system (.) and the A-Dirac system (.) exist [] and establish the regularity result directly. In other words, the main purpose of this paper is to show the regularity theory on anA-harmonic sys-tem and the correspondingA-Dirac system. It means that we should know the properties of anA-harmonic operator and anA-Dirac operator. This main context will be stated in Section . Further discussion can be found in [–] and the references therein.

In order to prove the main result, we also need a suitable Caccioppoli estimation (see Theorem .). Then by the technique of removable singularities, we can ﬁnd that solutions to anA-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Cliﬀord valued solutions to the correspondingA-Dirac sys-tem.

The technique of removable singularities was used in [] to remove singularities for monogenic functions with modulus of continuityω(r), where the setsrn_ω₍_r_{) and} Haus-dorﬀ measure are removable. Kaufman and Wu [] used the method in the case of Hölder continuous analytic functions. In fact, under a certain geometric condition related to the Minkowski dimension, sets can be removable for A-harmonic functions in Hölder and bounded mean oscillation classes []. Even in the case of Hölder continuity, a precise removable sets condition was stated []. In [], the author showed that under a certain oscillation condition, sets satisfying a generalized Minkowski-type inequality were remov-able for solutions to theA-Dirac system. The general result can be found in [].

Motivated by these facts, one ask: Does a similar result hold for the more general case of the systems (.) and (.)? We will answer this question in this paper and obtain the following result.

Theorem . Let E be a relatively closed subset of.Suppose that u∈Lm

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under the structure conditions(H)-(H)in\E,and u is of the type of an m,k-oscillation in\E.If for each compact subset K of E

\K

d(x,K)m(k–)–k<∞, (.)

then u extends to a solution of the A-Dirac system in.

2 A-Dirac system

In this section, we would introduce theA-Dirac system. Thus the definition of theA-Dirac operator is necessary. We first present the definitions and notations as regards the Clifford algebra at first [].

We writeUnfor the real universal Cliﬀord algebra overRn. The Cliﬀord algebra is gen-erated overRby the basis of the reduced products

{e,e, . . . ,ee, . . . ,e· · ·en}, (.)

where{e,e, . . . ,en}is an orthonormal basis ofRn_{with the relation}_eiej₊_ejei_{= –}_δ ij. We writeefor the identity. The dimension ofUnis n. We have an increasing towerR⊂C⊂ H⊂U⊂ · · ·. The Cliﬀord algebra Unis a graded algebra asUn=

lUnl, whereUnl are those elements whose reduced Cliﬀord products have lengthl.

ForA∈Un,Sc(A) denotes the scalar part ofA, that is, the coeﬃcient of the elemente. Throughout this paper, ⊂ Rn _{is a connected and open set with boundary} _∂_. A Cliﬀord-valued functionu:→Uncan be written asu= αuαeα, where each uα

is real-valued andeαare reduced products. The norm used here is given by| αuαeα|=

( αuα) 

_{, which is sub-multiplicative,}|_AB| ≤_C|_A||_B|_.

The Dirac operator deﬁned here is

D= n

j= ej ∂

∂xj. (.)

AlsoD= –. Hereis Laplace operator.

Throughout, Qis a cube in with volume|Q|. We writeσQfor the cube with the same center asQand with side lengthσ times that ofQ. Forq> , we writeLq₍_,_U

n) for the space of Cliﬀord-valued functions inwhose coeﬃcients belong to the usualLq₍₎ space. Also,W,m₍_,_U

n) is the space of Clifford valued functions inwhose coefficients as well as their first distributional derivatives are inLq(). We also writeLq_loc(,Un) for

Lq₍_,_U

n), where the intersection is over allcompactly contained in . We simi-larly writeW_loc,m(,Un). Moreover, we writeM={u:→Un|Du= }for the space of monogenic functions in.

Furthermore, we deﬁne the Dirac Sobolev space

WD,m() =

u∈Un

|u|m+

|Du|m<∞

. (.)

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With those deﬁnitions and notations and also of theA-Dirac operator, we deﬁne the linear isomorphismθ:Rn_→_U

nby

θ(ω, . . . ,ωn) = n

i=

ωiei. (.)

Forx,y∈Rn_,_Du_{is deﬁned by}_θ₍_∇_φ_{) =}_D_φ_{for a real-valued function}_φ_{, and we have}

–Scθ(x)θ(y)=x,y, (.)

θ(x)=|x|. (.)

HereA˜(x,ξ,η) :×U×Un→Unis deﬁned by

˜

A(x,u,η) =θAx,u,θ–η, (.)

which means that (.) is equivalent to

ScθA(x,u,∇u)θ(∇φ)dx=

ScA˜(x,u,Du)Dφdx

=

Scf(x,u,Du)φdx. (.)

For the Clifford conjugation (ej· · ·ejl) = (–)lejl· · ·ej, we define a Clifford-valued inner product asαβ¯ . Moreover, the scalar part of this Clifford inner productSc(αβ¯ ) is the usual inner productα,βinRn_{, when}_α_and_β_{are identified as vectors.}

For convenience, we replaceA˜ withAand recast the structure systems above and deﬁne the operator:

A(x,ξ,η) :×U×Un→Un, (.)

whereApreserves the grading of the Cliﬀord algebra,x→A(x,ξ,η) is measurable for all

ξ,η, andξ→A(x,ξ,η),η→A(x,ξ,η) are continuous for a.e.x∈.

Deﬁnition . A Cliﬀord valued functionu∈W_locD,m(,Uk

n)∩L∞(,Unk), fork= , , . . . ,n, is a weak solution to system (.) under conditions (H)-(H). If for allφ∈W_,m(,Uk

n), then we have

A(x,u,Du)Dφdx=

f(x,u,Du)φdx. (.)

3 Proof of the main results

In this section, we will establish the main results. At ﬁrst, a suitable Caccioppoli estimate [, ] for solutions to (.) is necessary.

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we have

Q

 +|p| m–

 |_Du_–_p

|+|Du–p|m

dx

≤C

 (σ|Q|)/n

σQ

 +|p| m–

 |_u_–_P|_dx

+ 

(σ|Q|)m/n

σQ

|u–P|mdx+

σQ Gdx

, (.)

where P=u(x) –u+p(x–x)and

σQ

Gdx=σ|Q|K|u|+|p|

 +|p| m

–β_σ|_Q|

β n

+C_+C_|p|m

σ|Q|



n_. _(.)

Proof Denoteu(x) –u–p(x–x) byv(x) and  <|Q|



n _<_σ|_Q|n _<_min{_,_dist₍_x_,_∂₎}_for

σ > , consider a standard cut-oﬀ functionη∈C_∞(σQ(x)) satisfying ≤η≤,|∇η|< 

|Q|/n,η≡ onQ(x). Thenϕ=ηvis admissible as a test-function, and we obtain

σQ

A(x,u,Du)·(Du–p)ηdx

= –

σQ

A(x,u,Du)ηv· ∇ηdx+

σQ

f(x,u,Du)·ϕdx.

We further have

–

σQ

A(x,u,p)·(Du–p)ηdx

= 

σQ

A(x,u,p)ηv· ∇ηdx–

σQ

A(x,u,p)·Dϕdx,

and

σQ

A(x,u,p)·Dϕdx= .

Adding these equations yields

σQ

A(x,u,Du) –A(x,u,p)(Du–p)ηdx

= –

σQ

A(x,u,Du) –A(x,u,p)

(Du–p)ηv· ∇ηdx

–

σQ

A(x,u,p) –Ax,u+p(x–x),p

·Dϕdx

–

σQ

Ax,u+p(x–x),p

–A(x,u,p)·Dϕdx+

σQ

f(x,u,Du)·ϕdx

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where

I= C

σQ

 +|Du|+|p| m–

 |_Du_–_p

|η|v||∇η|dx;

II=K|u|+|p|

σQ

|v|β|Du–p|

 +|p| m

_η_dx_;

III= K|u|+|p|

σQ

|v|β+|∇η| +|p| m

_η_dx_;

IV=K|u|+|p|

σQ

|x–x|m+p(x–x) m_mβ

 +|p| m



·η|∇η||v|+η|Du–p| dx; V= σQ

C|Du|m+C

|v|ηdx,

after using (.), (H), (H).

For positiveε, to be ﬁxed later, using Young’s inequality, we have

I≤C

σQ

 + |Du–p|+ |p| m–

 |_Du_–_p

|η|v||∇η|dx

≤C

σQ

 +|p| m–

 |_Du_–_p

|η|v||∇η|dx+

σQ

|Du–p|m–η|v||∇η|dx

≤Cε

σQ

 +|p| m–

 |_Du_–_p

|ηdx+C 

ε

σQ

 +|p| m–

 |_v||∇_η|_dx

+Cε

σQ

|Du–p|mηdx+C(ε)

σQ

|v|m|∇η|mdx.

Using Young’s inequality twice inII, we have

II≤ε

σQ

|Du–p|ηdx+ 

εK

_|_u |+|p|

 +|p|

m

σQ

|v|βdx

≤ε

σQ

|Du–p|ηdx+  ε σQ  (σ|Q|)/n|v|

 dx

+

ε

K|u|+|p|

 +|p| m

–β_σ|_Q|(

β

–β+n)/n

≤ε

σQ

 +|p| m–

 |_Du_–_p

|ηdx+ 

ε

σQ

 +|p| m–



 (σ|Q|)/n



|v|dx

+

ε

K|u|+|p|

 +|p| m

–β_σ|_Q|(

β

–β+n)/n_,

and similarly we see

III≤  

σQ

|v||∇η|dx

+ K|u|+|p|

 +|p|

m

σQ

σ|Q|

β n

_|

v| (σ|Q|)/n

β

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≤

σQ

 (σ|Q|)/n



|v|dx+K|u|+|p|

 +|p| m

–β_σ|_Q|(

β

–β+n)/n

≤ +|p| m–  σQ  (σ|Q|)/n



|v|dx

+K|u|+|p|

 +|p| m

–β_σ|_Q|(

β

–β+n)/n

and

IV≤

σQ

K|u|+|p|

 +|p| m

_σ|_Q|

β n_{ +}|_p

|m β

m_η|_Du_–_p

|+ η|∇η||v|

dx

≤

σQ

K|u|+|p|

 +|p| m

_σ|_Q|

β n_{ +}|_p

| β

η|Du–p|+ η|∇η||v| dx ≤ε σQ

 +|p| m–

 |_Du_–_p

|ηdx+

σQ

 +|p| m–

 |∇_η|_|_v_|_dx

+

 +

ε

K|u|+|p|

 +|p| (m_+β)

σ|Q| n+β

n _,

and for positiveμ, to be ﬁxed later, this yields

V =

σQ

C|Du|mu–u–p(x–x)ηdx+

σQ

 (σ|Q|)/n|v|η

C

σ|Q|n_η_dx

≤

σQ C

( +μ)|Du–p|m+

 + 

μ

|p|m

u–u–p(x–x)ηdx

+ εC

 

σ|Q|n+n ₊  εσ|Q|/n

σQ| v|dx

≤C( +μ)M+p

σ|Q|n

σQ

|Du–p|mηdx+C

 + 

μ

|p|m

σQ

|v|ηdx

+ εC

 

σ|Q| n+

n ₊ 

ε

σQ  (σ|Q|)/n|v|

_dx

≤C( +μ)M+p

σ|Q|



n

σQ

|Du–p|mηdx+ 

ε

σQ  (σ|Q|)/n|v|

_dx

+ε 

C_+C_

 + 

μ 

|p|m

σ|Q|n+n _.

By (.), we obtain

σQ

A(x,u,Du) –A(x,u,p)

(Du–p)ηdx

≥λ

σQ

 +|Du|+|p| m–

 |_Du_–_p

|ηdx

≥λ

σQ

 +|Du–p|+|p| m–

 |_Du_–_p

|ηdx

≥λ

σQ

 +|p| m–

 |_Du_–_p

|ηdx+

σQ

|Du–p|mηdx

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Combining these estimates in (.) and noting thatK_≤_K–β _(as_K≥_{), (}_σ|_Q|₎

β

(–β)n ≤

(σ|Q|)nβ _for_σ_{> , [( +}|_p_|₎m_] 

–β ≥_{( +}|_p |)(

m

+β)_{, and }≤_ 

–β_{, we can estimate}

λ– Cε– ε–C( +μ)M+p

σ|Q|



n

· σQ

 +|p|

m–  |_Du_–_p

|ηdx+

σQ

|Du–p|mηdx

≤

C ε +



ε+C(ε) + 

 (σ|Q|)/n

σQ

 +|p| m–

 |_u_–_P|_dx

+ 

(σ|Q|)m/n

σQ

|u–P|mdx

+ 



ε+ 

 –β

·K|u|+|p|

 +|p| m

–β_σ|_Q| n+β

n

+ε 

C_+C_

 + 

μ 

|p|m

σ|Q| n+

n _.

Deﬁneε=ε(λ,m),μ=μ(C,M,m,λ) small enough, we obtain

σQ

 +|p| m–

 |_Du_–_p

|ηdx+

σQ|

Du–p|mηdx

≤C

 (σ|Q|)/n

σQ

 +|p| m–

 |_u_–_P|_dx

+ 

(σ|Q|)m/n

σQ

|u–P|mdx+

σQ Gdx

,

whereC=C(m,λ,β,M) and

σQ

Gdx=σ|Q|K|u|+|p|

 +|p| m

–β_σ|_Q|

β n ₊_C

+C|p|m

σ|Q|



n_.

Now let the domain of the left-hand side beQ, then we can get the right inequality

immediately.

In order to remove singularity of solutions toA-Dirac system, we also need the fact that real-valued functions satisfying various regularity properties. Thus we have the following.

Deﬁnition .[] Assume thatu∈L_loc(,Un),q> , and that –∞<k< . We say thatu is of the type of aq,k-oscillation inwhen

sup

Q⊂|Q|

–(qk+n)/qn _inf uQ∈MQ

Q

|u–uQ|q /q

<∞. (.)

Ifq=  andk= , then the inequality (.) is equivalent to the usual deﬁnition of the bounded mean oscillation; whenq=  and  <k≤, then the inequality (.) is equivalent to the usual local Lipschitz condition []. Further discussion of the inequality (.) can be found in [, ]. In these cases, the supremum is ﬁnite if we chooseuQto be the average value of the functionuover the cubeQ.

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The following lemma shows that Deﬁnition . is independent of the expansion factor of the sphere.

Lemma .[] Suppose that F∈L

loc(,R),F> a.e.,r∈R andσ,σ> .If

sup σQ⊂Q

|Q|r

Q F<∞,

then

sup σQ⊂Q

|Q|r

Q

F<∞. (.)

Then we proceed to prove the main result, Theorem ..

Proof of Theorem. LetQbe a cube in the Whitney decomposition of\E. The de-composition consists of closed dyadic cubes with disjoint interiors which satisfy

(a) \E=_Q_∈_WQ,

(b) |Q|/n_≤_d₍_Q_,_∂₎_≤__|_Q_|/n_,

(c) (/)|Q|/n≤ |Q|/n≤|Q|/nwhenQ∩Qis not empty.

Hered(Q,∂) is the Euclidean distance betweenQand the boundary of[]. IfA⊂Rn_and_r_{> , then we deﬁne the}_r_{-inﬂation of}_A_as

A(r) =B(x,r). (.)

LetQbe a cube in the Whitney decomposition of\E. Using the Caccioppoli estimate (.), we have

Q

 +|p| m–

 |_Du_–_p

|+|Du–p|m

dx

≤C

 (σQ)/n

σQ

 +|p| m–

 |_u_–_P|_dx

+ 

(σQ)m/n

σQ|

u–P|mdx+

σQ Gdx

,

with (.)

σQ

Gdx≤C|σQ|n+nβ_H_{ +}|_uQ|₊|_p |

, (.)

where

H(t) =K˜(t)( +t)m 

–β_, _K˜₍_t_{) =}_max_K₍_t_),_C ,C

,

and choose|Q|small enough such that

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By the deﬁnition of theq,k-oscillation condition, we have

Q

 +|p| m–

 |_Du_–_p

|+|Du–p|m

dx

≤C|Q|–



n|_Q|kn+n₊_C_|_Q|–mn|_Q|(mk+n)/n₊_C_|_Q|

≤C|Q|a. (.)

Herea= (n+mk–m)/n. Since the problem is local (use a partition of unity), we show that (.) holds wheneverφ∈W_,m(B(x,r)) withx∈Eandr>  suﬃciently small. Choose r= (/√n)min{,d(x,∂)}and letK=E∩ ¯B(x, r). ThenKis a compact subset ofE. Also letWbe those cubes in the Whitney decomposition of\Ewhich meetB=B(x,r). Notice that each cubeQ∈Wlies in\K. Letγ =m(k– ) –k. First, sinceγ ≥–, from [] we havem(K) =m(E) = . Also sincena–n≥γ, using (.) and (.), we obtain

B(x,r)

 +|p|

m–  |_Du_–_p

|+|Du–p|m

dx

≤C

Q∈W

|Q|a≤C Q∈W

d(Q,K)na

≤C

Q∈W

Q

d(x,K)na–ndx≤C

K()\K

d(x,K)na–ndx

≤C

K()\K

d(x,K)γ_dx_<_∞_. _(.)

Henceu∈W_locD,m().

Next letB=B(x,r) and assume thatψ∈C∞(B). Also letWj,j= , , . . . , be those cubes Q∈Wwithl(Q)≤–j.

Consider the scalar functions

φj=max

–j–d(x,K)j, . (.)

Thus eachφj,j= , , . . . , is Lipschitz, equal to  onKand as suchψ( –φj)∈W,m(B\E) with compact support. Hence

B

A(x,u,Du)Dψ–f(x,u,Du)ψdx

=

B\E

A(x,u,Du)Dψ( –φj)

–f(x,u,Du)ψ( –φj)

dx

+

B

A(x,u,Du)D(ψ φj) –f(x,u,Du)ψ φj

dx. (.)

Let

J=

B\E

A(x,u,Du)Dψ( –φj)

–f(x,u,Du)ψ( –φj)

dx,

J=

B

A(x,u,Du)D(ψ φj) –f(x,u,Du)ψ φj

dx.

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Next we estimateJas

J=

B

A(x,u,Du)ψDφjdx+

B

φjA(x,u,Du)Dψdx–

B

f(x,u,Du)ψ φjdx

=J_+J_+J_. (.)

Noting that there exists a constantCsuch that|ψ| ≤C<∞,

J_≤C Q∈Wj

B

A(x,u,Du)|Dφj|dx.

Recalling that|Q|βn_K₍_t₎≤_{, we have}

B

A(x,u,Du)|Dφj|dx

≤

B

A(x,u,Du) –A(x,u,p)|Dφj|dx

+

B

A(x,u,p) –A(x,u,p)|Dφj|dx

≤C

B

 +|Du|+|p| m–

 |_Du_–_p

||Dφj|dx

+C

B

K|u||x–x|m+|u–u|m β

m_{ +}|_p |

m

|_D_φ

j|dx

≤C

B

 +|p| m–

 ₊|_Du_–_p

|m–

|Du–p||Dφj|dx

+C

B

K|u||x–x|β+|u–u|β

 +|p| m

|_D_φ

j|dx

≤C

B

 +|p| m–

 |_Du_–_p

||Dφj|dx+C

B

|Du–p|m–|Dφj|dx

+C

B

K|u||x–x|β

 +|p| m

|_D_φ

j|dx

+C

B

K|u||u–u|β

 +|p| m

|_D_φ

j|dx

≤C

B

 +|p| m–

 |_Du_–_p

|+|Du–p|m dx +C B

 +|p| m–

 |_D_φ

j|+|Dφj|m

dx

+C

B

K|u||Q|βn_{ +}|_p_| m

|_D_φ

j|dx

+C

B

K|u||Q|βn_{ +}|_p |

m

|_D_φ

j|dx

≤C|Q|a+C

B

|Dφj|+|Dφj|m

dx+C

B

|Dφj|dx+C

B

|Dφj|dx

≤C|Q|a+C

B

j+ mjdx+C

B

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Now forx∈Q∈Wj,d(x,K) is bounded above and below by a multiple of|Q|/n_{and for} Q∈Wj,|Q|/n_≤_–j_{. Hence}

_J

≤C

Q∈Wj

|Q|a+|Q|–mn|_Q|₊_C|_Q|–n|_Q|₊|_Q|–n|_Q|n

≤C

Q∈Wj

|Q|a≤C

Wj

d(x,K)m(k–)–k. (.)

SinceWj⊂\Kand|

Wj| → asj→ ∞, it follows thatJ→ asj→ ∞. For

J_≤C Q∈Wj

B

φjA(x,u,Du)Dψdx.

Similarly, we get

B

φjA(x,u,Du)Dψdx

≤

B

A(x,u,Du) –A(x,u,p)Dψdx+

B

A(x,u,p) –A(x,u,p)Dψdx

≤C

B

 +|Du|+|p| m–

 |_Du_–_p

|dx

+C

B

K|u||x–x|m+|u–u|m β

m_{ +}|_p |

m

|_D_ψ|_dx

≤C

B

 +|p| m–

 ₊|_Du_–_p

|m–

|Du–p||Dψ|dx

+C

B

K|u||x–x|β+|u–u|β

 +|p| m

|_D_ψ|_dx

≤C

B

 +|p| m–

 |_Du_–_p

||Dψ|dx+C

B

|Du–p|m–|Dψ|dx

+C

B

K|u||x–x|β

 +|p| m

|_D_ψ|_dx

+C

B

K|u||u–u|β

 +|p| m

|_D_ψ|_dx

≤C

B

 +|p| m–

 |_Du_–_p

|+|Du–p|m

dx

+C

B

 +|p| m–

 |_D_ψ|₊_|_D_ψ_|m_dx

+C

B

|Dψ|dx+C

B

K|u||Q|βn|_D_ψ|_dx

≤C|Q|a+C

B

|Dψ|+|Dψ|mdx+C

B

|Dψ|dx

≤C|Q|a+C

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Thus,

J_≤C Q∈Wj

|Q|a+|Q|≤C Q∈Wj

|Q|a≤C

Wj

d(x,K)m(k–)–k. (.)

Sinceu∈W_loc,D() and|Wj| → asj→ ∞, we haveJ_→ asj→ ∞. In order to estimateJ_, we should use (H):

J_=

B

f(x,u,Du)ψ φjdx≤C

B

|Du–p|mdx+C

B

|p|mdx=J+J. (.)

Similar to the estimate of (.), using the Caccioppoli inequality (.) and the inequality (.), we get

J_≤C Q∈Wj

Q

|Du–p|mdx≤C

Q∈Wj

|Q|(n+mkn–m)

≤C

Wj

d(x,K)n+mk–mdx≤C

Wj

d(x,K)m(k–)–kdx.

→ (j→ ∞),

and

J_≤C Q∈Wj

Q

dx=C Q∈Wj

|Q|

≤C

Wj

d(x,K)ndx≤C

Wj

d(x,K)n+mk–mdx

≤C

Wj

d(x,K)m(k–)–kdx

→ (j→ ∞).

HenceJ→.

Combining estimatesJandJin (.), we prove Theorem ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

FS participated in design of the study and drafted the manuscript. SC participated in and conceived of the study and the amendment of the paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

Supported by National Natural Science Foundation of China (No: 11201415); Program for New Century Excellent Talents in Fujian Province University (No: JA14191).

Received: 27 February 2014 Accepted: 23 October 2014 Published:04 Nov 2014 References

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