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R E S E A R C H

Open Access

Interior regularity of obstacle problems for

nonlinear subelliptic systems with VMO

coefficients

Guangwei Du

*

and Fushan Li

*Correspondence:

guangwei87@mail.nwpu.edu.cn School of Mathematical Sciences, Qufu Normal University, Qufu, China

Abstract

This article is concerned with an obstacle problem for nonlinear subelliptic systems of second order with VMO coefficients. It is shown, based on a modification of

A-harmonic approximation argument, that the gradient of weak solution to the corresponding obstacle problem belongs to the Morrey spaceL2,X,λloc.

MSC: 35J70; 35B65; 35J50

Keywords: Subelliptic systems; Obstacle problem; VMO;L2,X,λloc-regularity

1 Introduction

In this paper we consider weak solutions to an obstacle problem for the following

nonlin-ear subelliptic system in a bounded domainof Euclidean spaceRn:

aαβij (x)Xβuj

=Bi(x,u,Xu) +Xαg α

i(x,u,Xu), i= 1, 2, . . . ,N, (1.1)

whereX= (X1, . . . ,Xm) (m≤n) is a system of smooth real vector fields satisfying the

Hör-mander’s rank condition,∗is the formal adjoint of.

If we setA(x) ={aαβij (x)},B= (Bi),g= (gαi), then (1.1) reads

–X∗A(x)Xu=B(x,u,Xu) –Xg(x,u,Xu).

Given two vector-valued functionsψ= (ψ1, . . . ,ψN) andθ= (θ1, . . . ,θN) withθ(x)≥ψ(x) a.e. on(i.e.θi(x)≥ψi(x) a.e. on,i= 1, 2, . . . ,N), we define the set

Kθψ=vS1,2X ,RN:vψa.e. in,vθS1,2

X,0

,RN.

Here the functionsψ andθ are called obstacle and boundary datum, respectively. The

functionu∈Kθψ is called a weak solution to the obstacle problem related to (1.1) if

A(x)Xu·Xϕdx

B(x,u,Xu)ϕdx+

g(x,u,Xu)·Xϕdx (1.2)

holds for allϕC0∞(,RN) withϕ+uψa.e.x.

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As we know, the uniform ellipticity requirement on coefficients is not sufficient to get the local boundedness of solutions even for one single equation in the Euclidean metric (see [1]). Therefore some additional assumptions on the coefficients is needed to ensure the regularity results. In [2–4], Campanato obtained theL2,λ-regularity and Hölder continuity

for the weak solutions of elliptic systems with continuous coefficients. See also [5–8] for related results.

Since the functions of vanishing mean oscillation (VMO) can have some kind of dis-continuities, regularity results under a VMO assumption have been established by many authors; see, for example, [9–12] for elliptic systems, and [13–17] for subelliptic systems constructed by Hörmander’s vector fields. Huang in [9] established the gradient estimates in the generalized Morrey spaces of weak solutions to the linear elliptic systems with VMO coefficients. Similar results for the nonlinear elliptic systems were obtained by Daněček and Viszus in [10] and [11]. In [15] and [16] Di Fazio and Fanciullo proved that the local gradient estimates in [9] still hold true for the subelliptic systems structured on Höman-der’s vector fields. Dong and Niu [14] established the Morrey and Campanato regularity for weak solutions to the nondiagonal subelliptic systems. The direct methods were mainly used to prove the desired results in the papers mentioned above. An important step of this kind of methods is to establish the higher integrability of gradients of weak solutions. These arguments were also used to prove the Morrey regularity and Hölder continuity for weak solutions to the obstacle problems associated with a single elliptic equation with constant coefficients or continuous coefficients; see [18–22].

Recently, another method calledA-harmonic approximation has been widely applied to

prove the optimal partial regularity for nonlinear elliptic systems or subelliptic systems in the Heisenberg group and Carnot groups; see [23–29]. This method is based on Simon’s technique of harmonic approximation ([30]) and generalized by Duzaar and Grotowski in [31] in order to deal with partial regularity for nonlinear elliptic systems. The key point is to show that a function which is “approximately harmonic”, i.e. a function closes

suf-ficiently to some harmonic function inL2. Making use of this method, one can simplify

the proof avoiding the proof of a suitable reverse Hölder inequality for the gradient of a weak solution. We also mention that Daněček-John-Stará [32] proved the Morrey space regularity for weak solutions of Stokes systems with VMO coefficients by using a mod-ifiedA-harmonic approximation lemma. Inspired by this work, Yu and Zheng [33] ob-tained optimal partial regularity for quasilinear elliptic systems with VMO coefficients by

a modification ofA-harmonic approximation argument.

In the present paper we study the interior regularity of weak solutions to the obstacle

problem related to the system (1.1) by the technique ofA-harmonic approximation, which

implies that these solutions have the same kind of regularity as the weak solutions of (1.1). Throughout this article, we make the following assumptions.

(H1) The coefficientsaαβij are bounded measurable and such that, for some suitable λ> 0and> 0,

λ|ξ|2≤aαβij (x)ξαβj|ξ|2, x∈Rn,ξ∈RmN;

(H2) The functionsBi,giα:Rn×RN×RmN→Rare both Carathéodory functions and

for almostxand all(u,ξ)∈RN×RmN, there existsL> 0such that

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i(x,u,ξ)≤fiα(x) +L|ξ|γ,

where1≤γ0<QQ+2,0≤γ < 1, and

fL2XQ/(Q+2),λQ/(Q+2),RN, f˜∈L2,Xλ,RmN, Qn<λ<Q.

HereQis the homogeneous dimension relative toandf= (fi),f˜= (f).

We are now in the position to state our main result.

Theorem 1.1 Suppose that(H1)–(H2)hold and that aαβij ∈VMO()for i,j= 1, 2, . . . ,N,

α,β= 1, 2, . . . ,m.Let u∈Kθψ be a weak solution to the obstacle problem for system(1.1) with XψL2,Xλ(,RmN),then XuLX2,λ,loc(,RmN).Moreover, if Qn<λ< 2then uCX0,(2–λ)/2(,RN).

The paper is organized as follows. In the next section we recall some concepts and facts

associated to Carnot–Carathéodory spaces and give the proof of the modifiedA-harmonic

approximation lemma for vector fields. In Sect. 3, we consider the following linear subel-liptic system with VMO coefficients:

XA(x)Xu=XA(x)Xψ,

and we prove a comparison principle and a Morrey type estimate for weak solutions of

the above system by a modification ofA-harmonic approximation argument. Section 4 is

devoted to the proofs of Theorem 1.1. On the basis of the Morrey type estimate established for linear subelliptic system, we can first prove theL2,Xλ,loc-regularity for weak solutions of the obstacle problems and then interior Hölder continuity is obtained by virtue of the equivalence between the Campanato space and the Hölder continuity function space (see [34, 35]).

In what follows, we usecto denote a positive constant that may vary from line to line.

2 Some notations and preliminaries

Let

=

n

k=1 bαk

∂xk

, bαkC∞,α= 1, 2, . . . ,m

be a family of vector fields inRnsatisfying Hörmander’s condition ([36]):

rankLie{X1, . . . ,Xm}

=n.

We consideras a first order differential operator acting onu∈Lip(Rn) defined as

Xαu(x) = (x),∇u(x)

, α= 1, 2, . . . ,m.

We denote byXu= (X1u, . . . ,Xmu) the gradient ofuand hence|Xu(x)|= (

m

α=1|Xαu(x)|2)

1 2.

An absolutely continuous curveγ : [a,b]→Rnis said to be admissible if

γ (t) =

m

α=1 (t)Xα

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for some functions(t) satisfying

m

α=1(t)2≤1. The Carnot–Carathéodory distance

d(x,y) generated byXis defined by

d(x,y) =infT> 0 : there is an admissible curveγ,γ(0) =x,γ(T) =y.

Forx∈RnandR> 0 we let

BR(x) =B(x,R) =

y∈Rn:d(y,x) <R.

In what follows, ifσ> 0 andB=B(x,R) we writeσBto indicateB(x,σR). Furthermore, if E⊂Rnis a Lebesgue measurable set with Lebesgue measure|E|, we setuE=–

Eu dxthe

integral average ofuonE.

In [37], it was proved that for every connectedKthere exist constantsC1,C2> 0 and 0 <λ< 1 such that

C1|xy| ≤d(x,y)C2|xy|λ, x,yK.

Moreover, there areRd> 0 andCd≥1 such that, for anyxKandRRd,

B(x, 2R)CdB(x,R). (2.1)

Property (2.1) is the so-called “doubling condition” which is assumed to hold on the spaces

of homogeneous type. The best constantCdin (2.1) is called the doubling constant. We

call thatQ=log2Cdis the homogeneous dimension relative to. As a consequence of

(2.1), we have

|BtR| ≥C–2d tQ|BR|, ∀RRd,t∈(0, 1). (2.2)

We now introduce the relevant Sobolev spaces. Given 1≤p<∞, the Sobolev space

S1,Xp(,RN) is the Banach space

S1,Xp,RN=uLp,RN:X

αuLp

,RN,α= 1, 2, . . . ,m

endowed with the norm

uS1,p

X (,RN)=uL p(,RN)+

m

α=1

XαuLp(,RN).

Here,Xαuis the distributional derivative ofuL1loc(,RN) defined by

Xαu·φdx=

u·Xαφdx,φC0,RN,

where

∗= –

n

k=1

∂xk

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is the formal adjoint of, not necessarily a vector field in general. The spaceSX1,p,0(,RN)

is defined as the completion ofC0 (,RN) under the norm ·

S1,Xp(,RN).

In addition, we also need the following Sobolev inequalities for vector fields.

Theorem 2.1([38, 39]) For every compact set K,there exist constants C> 0andR¯> 0

such that,for any metric ball B=B(x0,R)with x0K and0 <R≤ ¯R,for any fS1,Xp(BR),

BR

|ffR|κpdx

1 κp

CR

BR

|Xf|pdx 1

p

,

where1≤κQ/(Qp),if1≤p<Q; 1κ<∞,if pQ.Moreover,

BR

|f|κpdx 1

κp

CR

BR

|Xf|pdx 1

p

,

whenever fS1,X,0p(BR).

Now we define the Morrey spaces, the Campanato spaces, VMO and the Hölder spaces with respect to the Carnot–Carathéodory metric. To simplify our description, we intro-duce the following notation:

(x,R) =B(x,R), fx,R=

1

|(x,R)|

(x,R) f(y)dy,

and

d0=min{diam,Rd}.

Definition 2.2 For 1 <p<∞andλQ, we say thatfLploc(,RN) belongs to the Morrey

spaceLpX,λ(,RN) if

fLp

X (,RN)= sup

x,0<ρ<d0

ρλ |(x,ρ)|

(x,ρ) f(y)p

dy 1

p

<∞;

fLploc(,RN) belongs to the Campanato spaceLp,λ

X (,RN) if

fLp

X (,RN)= sup

x,0<ρ<d0

ρλ

|(x,ρ)|

(x,ρ)

f(y) –fx,ρpdy 1

p

<∞.

Definition 2.3 Forα∈(0, 1), the Hölder spaceC0,Xα(¯,RN) is the collection of functions

f :¯ →RNsatisfying

fC0,α

X (¯,RN)=sup |f|+sup¯

|f(x) –f(y)| d(x,y)α <∞.

We say thatf is locally Hölder continuous, i.e.fCX0,α(,RN), iff C0,α

X (K,RN) for every

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Definition 2.4 We say thatfL1

loc(,RN) belongs to BMO(,RN) if

f∗= sup

x,0<ρ<d0

1

|(x,ρ)|

(x,ρ)

f(y) –fx,ρdy<∞;

f belongs to VMO(,RN) iff ∈BMO(,RN) and

ηr(f) = sup x,0<ρ<r

1

|(x,ρ)|

(x,ρ)

f(y) –fx,ρdy→0, r→0.

The integral characterization for a Hölder continuous function was shown in [35] and [34].

Lemma 2.5 If–p<λ< 0,thenLXp,λ(,RN)C0,α

X (,RN),α= –

λ

p.

3 Morrey type estimate for a subelliptic system

In this section we will prove by the modifiedA-harmonic approximation technique a

Mor-rey type estimate for the subelliptic system

XA(x)Xu=XA(x)Xψ. (3.1)

Let us first recall that a functionhSX1,2(,RN) is calledA-harmonic forABil(RmN) if

hsatisfies

A(Xh,Xϕ)dx= 0, ∀ϕC01,RN.

We cite theA-harmonic approximation lemma for vector fields as follows ([24, 31]).

Lemma 3.1 Consider fixed positiveλand,and m,N∈Nwith m≥2.Then for any

givenε> 0there existsδ=δ(m,N,λ,,ε)with the following property:for any A∈Bil(RmN)

satisfying

A(ξ,ξ)≥λ|ξ|2, for allξ∈RmN, (3.2)

and

A(ξ,ξ˜)≤|ξ||˜ξ|, for allξ,ξ˜∈RmN, (3.3)

for any gS1,2X (Bρ(x0),RN) (for someρ> 0,x0∈Rn)satisfying

(x0)

|Xg|2dx≤1 (3.4)

and

(x0)

A(Xg,Xϕ)dxδ sup

(x0)

||, ∀ϕC0(x0),RN

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there exists an A-harmonic function hS1,2X (Bρ(x0),RN)such that

(x0)

|Xh|2dx≤1 and 1

ρ2–

(x0)

|gh|2dxε.

Similarly to [32] and [33], we can prove the following modification of theA-harmonic

approximation lemma.

Lemma 3.2 Let0 <λ<and m∈Nwith m≥2be fixed.Then,for anyε> 0,there exists

a constant k=k(m,N,λ,,ε)such that the following holds:for any A∈Bil(RmN)satisfying conditions(3.2), (3.3)and for any uS1,2X (Bρ(x0),RN),there exists an A-harmonic function

hS1,2X (Bρ(x0),RN)such that

(x0)

|Xh|2dx

(x0)

|Xu|2dx (3.6)

and,moreover,there existsϕC0∞(Bρ(x0),RN)such that

L(Bρ(x0),RN)

1

ρ (3.7)

and

(x0)

|uh|2dx

ερ2

(x0)

|Xu|2dx+k(ε)

ρ4

|(x0)|

(x0)

A(Xu,Xϕ)dx 2

. (3.8)

Proof For any givenε> 0 anduS1,2X (Bρ(x0),RN), we takeδ(ε) as in the above Lemma 3.1

and set

g=

(x0)

|Xu|2dx –12

u.

Then (3.4) holds. Assume that forgthe inequality (3.5) is true. From Lemma 3.1, there is anA-harmonic functionwsatisfying

(x0)

|Xw|2dx≤1, 1

ρ2–

(x0)

|wg|2dxε

and thus the functionh= (–

(x0)|Xu|

2dx)12wsatisfies (3.6). Moreover, we have

|uh|2= –

(x0)

|Xu|2dx· |gw|2,

which implies

(x0)

|uh|2dx

(x0)

|Xu|2dx·–

(x0)

|gw|2dxερ2

(x0)

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If, vice versa, there is a nonconstant functionϕ˜∈C0∞(Bρ(x0),RN) such that

(x0)

A(Xg,Xϕ˜)dx>δ(ε) sup

(x0) |˜|.

Settingϕ=ρsup ϕ˜

Bρ(x0)|˜|, it follows that ρ

δ(ε) –

(x0)

A(Xg,Xϕ)dx> 1.

We now take h = . Using the Poincaré inequality and the fact that Xg =

(–

(x0)|Xu|

2dx)–12Xuwe deduce

(x0)

|uh|2dx=

(x0)

|u|2dx

2

(x0)

|Xu|2dx

4 δ2(ε)

(x0)

A(Xg,Xϕ)dx 2

(x0)

|Xu|2dx

4|(x0)|

δ2(ε)

(x0)

A(Xu,Xϕ)dx 2

4 δ2(ε)|B

ρ(x0)|

(x0)

A(Xu,Xϕ)dx 2

. (3.10)

Combining (3.9) and (3.10) and takingk(ε) = c

δ2(ε) complete the proof.

Now we are in a position to establish the Morrey type estimate for gradient of weak solution to (3.1) based on Lemma 3.2.

Lemma 3.3 Suppose that A(x)satisfies(H1)and uSX1,2,loc(,RN)is a weak solution to the

system(3.1),i.e.,

A(x)Xu·Xϕdx=

A(x)Xψ·Xϕdx,ϕC0,RN.

Then for any x0there exists a constant c> 0such that,for all Bρ(x0)⊂BR(x0)⊂,

R<Rd,

(x0)

|Xu|2dxc

ρ

R Q

+ε+ηR(A) BR(x0)

|Xu|2dx+c

BR(x0)

||2dx. (3.11)

Proof For fixedx0and 0 <R<Rd, denoteBR:=BR(x0). Letηbe a cut-off function on

BRrelative to, i.e.ηC∞0 (BR,RN) and satisfies

0≤η(x)≤1, η(x) = 1 in, (x)≤

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Taking the functionϕ=η2(uu

R) as a test function, it follows that

BR

η2A(x)Xu·Xu dx

= –2

BR

A(x)η(u–uR)Xu·Xηdx+

BR

η2A(x)Xψ·Xu dx

+ 2

BR

A(x)η(u–uR)Xψ·Xηdx.

From (H1) and Young’s inequality

λ

BR

η2|Xu|2dx≤2

BR

|ηXu||uuR|||dx

+

BR

|ηXψ||ηXu|dx+ 2

BR

|ηXψ||uuR|||dx

ε

BR

η2|Xu|2dx+ (R–ρ)2

BR

|uuR|2dx+

BR

||2dx.

Choosingε<λ, it follows that

|Xu|2dxc (R–ρ)2

BR

|uuR|2dx+c

BR

||2dx. (3.12)

Next we defineAR=–

BRA(x)dx. By Lemma 3.2, there exists anAR-harmonic function

hS1,2X (BR,RN) such that (3.6)–(3.8) hold. Moreover, by standard results of the subelliptic

system with constant coefficients (see for example [34]), we have

|Xh|2dxc

ρ

R Q

BR

|Xh|2dx, ∀0 <ρR.

Therefore, from (3.12) and (3.6) it follows that for any 0 <ρ<R/2

|Xu|2dx

c ρ2

B

|uu2ρ|2dx+c

B

||2dx

c ρ2

B

uu2ρ– (h–h2ρ)

2 dx+

B

|hh2ρ|2dx

+c

B

||2dx

c ρ2

B

|uh|2dx+c

B

|Xh|2dx+c

B

||2dx

c ρ2

B

|uh|2dx+c

ρ

R Q

BR

|Xh|2dx+c

BR

||2dx

c ρ2

B

|uh|2dx+c

ρ

R Q

BR

|Xu|2dx+c

BR

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For the first term in the right-hand side, we have from Lemma 3.2

c

ρ2

B

|uh|2dx

B

|Xu|2dx+ck(ε) ρ 2

|B2ρ|

B

ARXu·Xϕdx

2

BR

|Xu|2dx+

ρ2 |BR|

BR

ARXu·Xϕdx

2

, (3.14)

whereϕC0∞(B2ρ,RN) satisfiesXϕL∞(B2ρ,RN)≤

1

R <

1

2ρ. Sinceuis a weak solution to

(3.1), it follows that

BR

ARXu·Xϕdx

2 =

BR

(ARA)Xu·Xϕdx+

BR

AXu·Xϕdx 2

≤2

BR

(ARA)Xu·Xϕdx

2 + 2

BR

AXu·Xϕdx 2

:=I1+I2.

From Hölder’s inequality, using (H1), we have

I1≤ 1 2ρ2

BR

|Xu|2dx·

BR

|ARA|2dx

|BR|

ρ2 1

|BR|

BR

|ARA|dx·

BR

|Xu|2dx

|BR|

ρ2 ηR(A)

BR

|Xu|2dx

and

I2≤

2

2ρ2

BR

||dx 2

2|BR|

2ρ2

BR

||2dx.

Hence

BR

ARXu·Xϕdx

2

c|BR|

ρ2

ηR(A)

BR

|Xu|2dx+

BR

||2dx

. (3.15)

Combining (3.15), (3.14) and (3.13), we have, for any 0 <ρ<R/2,

|Xu|2dxc

ρ

R Q

+ε+ηR(A) BR

|Xu|2dx+c

BR

||2dx.

ForR/2ρR, obviously

|Xu|2dx

BR

|Xu|2dx≤2Q

ρ

R Q

BR

|Xu|2dx.

A combination of these two cases leads to (3.11) for 0 <ρR.

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Lemma 3.4 Suppose that u,ψSX1,2(BR,RN)satisfy

XA(x)Xu=XA(x)Xψ

where A(x)satisfies(H1).Ifψu on∂BR,thenψu a.e.in BR.

Proof For anyϕC0∞(BR,RN) we have

BR

A(x)Xu·Xϕdx=

BR

A(x)Xψ·Xϕdx. (3.16)

Setu+=max{u, 0}. Sinceψuon∂BR, we conclude that (see [40, Lemma 6]) (ψu)+

S1,2X,0(BR,RN) and

X(ψu)+= ⎧ ⎨ ⎩

X(ψu), ψ>u,

0, ψu.

Choosingϕ= (ψu)+in (3.16) gives

BR

A(x)X(ψu)·X(ψu)+dx= 0,

which implies

BR∩{ψ>u}A(x)X(ψu)·X(ψu)dx= 0.

From (H1) we have

BR

X(ψu)+ 2

dx=

BR∩{ψ>u}

X(ψu)2dx

≤1 λ

BR∩{ψ>u}A(x)X(

ψu)·X(ψu)dx= 0.

Thus from Poincaré inequality, we obtain

BR

(ψu)+2dxcR2

BR

X(ψu)+2dx= 0,

which implies (ψu)+= 0, orψua.e. inBR. The proof is complete.

4 Proof of main result

In this section we are going to prove our main result. To this end, we need a generalized iteration lemma, which can be found in [9, Proposition 2.1].

Lemma 4.1 Let H be a nonnegative almost increasing function on the interval[0,T]and

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(1) for any0 <ρRT,there existA,B,εanda> 0such that

H(ρ)≤

A

ρ

R a

+ε

H(R) +BF(R);

(2) there existsτ∈(0,a)such thatFρ(τρ) is almost increasing in(0,T].Then there exist positive constantsε0andCsuch that,for any0≤εε0,

H(ρ)≤CF(ρ)

F(R)H(R) +CB·F(ρ), 0 <ρRT,

whereε0depends only onA,aandτ.

Proof of Theorem1.1 LetBR=B(x0,R)⊂⊂be an arbitrary ball aroundx0of radiusR

and letu∈Kθψ be a weak solution to the obstacle problem related to (1.1). InBRwe splitu

asu=w+ (u–w), wherewSX1,2(BR,RN) is the weak solution to the following system:

⎧ ⎨ ⎩

X∗(A(x)Xw) =X∗(A(x)Xψ) inBR,

w=u on∂BR.

(4.1)

Sincew=uψa.e. on∂BR, it follows from Lemma 3.4 thatwψ a.e. inBR.

By the definition of weak solutions, we have

BR

A(x)Xw·X(wu)dx=

BR

A(x)Xψ·X(wu)dx.

From (H1) and Young’s inequality one gets

λ

BR

|Xw|2dx

BR

A(x)Xw·Xw dx

BR

|Xw||Xu|dx+

BR

|||XwXu|dx

ε

BR

|Xw|2dx+

BR

|Xu|2dx+

BR

||2dx.

Choosingε<λleads to

BR

|Xw|2dxc

BR

|Xu|2dx+c

BR

||2dx. (4.2)

On the basis of (4.2), it follows from Lemma 3.3 that for any 0 <ρR

|Xu|2dx

≤2

|Xw|2dx+ 2

X(uw)2dx

c

ρ

R Q

+ε+ηR(A) BR

|Xw|2dx+c

BR

||2dx+ 2

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c

ρ

R Q

+ε+ηR(A) BR

|Xu|2dx

+c

BR

||2dx+ 2

BR

X(uw)2dx. (4.3)

Note thatwuis admissible as a test function in the definition of weak solutions to the obstacle problem due towuS1,2X,0(BR,RN) andwψa.e. inBR. Applyingwuto (1.2)

leads to

A(x)Xu·X(uw)dx

B(x,u,Xu)(uw)dx+

g(x,u,Xu)·X(uw)dx.

From (H1)–(H2) and Hölder’s inequality, we have

λ

BR

|XuXw|2dx

BR

A(x)X(uw)·X(uw)dx

≤–

BR

A(x)Xw·X(uw)dx+

BR

B(x,u,Xu)|uw|dx

+

BR

g(x,u,Xu)|XuXw|dx

≤–

BR

A(x)Xψ·X(uw)dx+

BR

|f|+L|Xu|γ0|uw|dx

+

BR

f|+L|Xu|γ|XuXw|dx

c

BR

|XuXw|2dx 1

2

BR

||2dx 1 2 + BR

|f|+|Xu|γ02Q/(Q+2)dx

Q+2 2Q

+c

BR

|XuXw|2dx 1

2

BR

f|+|Xu|γ2dx 1 2 , which means BR

|XuXw|2dxc

BR

||2dx+c

BR

f|+|Xu|γ2dx

+c

BR

|f|+|Xu|γ02Q/(Q+2)dx

Q+2

Q

. (4.4)

In view of 1≤γ0<QQ+2, 0≤γ < 1, it follows by Hölder’s inequality that

BR

|Xu|2γ0Q/(Q+2)dx

Q+2

Q

=

BR

|Xu|2γ0Q/(Q+2)dx

Q+2 2γ0Q2γ0

|BR|

Q+2 2γ0Q–12

BR

|Xu|2dx 1

22γ0

=|BR|

Q+2

Qγ0

BR

|Xu|2dx γ0

(14)

and

BR

|Xu|2γdx

|BR|

1 2γ–12

BR

|Xu|2dx 1

22γ

≤ |BR|1–γ

BR

|Xu|2dx γ

ε

BR

|Xu|2dx+|BR|. (4.6)

Combine (4.4)–(4.6) to deduce

BR

|XuXw|2dx(R) +ε

BR

|Xu|2dx+c

BR

||2dx

+c

BR

|f|2Q/(Q+2)dx Q+2

Q

+c

BR

f|2dx+|BR|, (4.7)

whereω(R) =|BR|

Q+2

Qγ0(

BR|Xu|

2dx)γ0–1.

From (4.7) and (4.3), we find, for any 0 <ρR(we may supposeR< 1),

|Xu|2dxc

ρ

R Q

+ε+ηR(A) +ω(R) BR

|Xu|2dx+c

BR

||2dx

+c

BR

|f|2Q/(Q+2)dx (Q+2)/Q

+c

BR

f|2dx+c|BR|

c

ρ

R Q

+ϑ(R)

BR

|Xu|2dx+c˜|BR| ,

whereϑ(R) =ε+ηR(A) +ω(R),c˜=˜c(f2L2Q/(Q+2),λQ/(Q+2)+˜f2L2,λ+2Lp,λ). By the absolute

continuity of Lebesgue integral, we know thatω(R)→0 asR→0. Finally, we can takeR< R0such thatηR(A) is small enough due to the VMO property ofA(x). If we takeF(ρ) =|Bρρ|λ,

0 <Qλ<τ <Q, we claim that Fρ(τρ) =ρ|Bτ+λρ| is almost increasing. In fact, it follows from (2.2) that, for anyt∈(0, 1),

(tρ)τ

F(tρ)/

ρτ

F(ρ)= (tρ)τ+λ

|Btρ|

/ρ

τ+λ

||

=t

τ+λ|B ρ|

|Btρ| ≤

C2d+λQCd2.

By Lemma 4.1, we obtain, for 0 <ρR,

|Xu|2dxc||

ρλ , (4.8)

which shows thatXuL2,Xλ,loc(,RmN).

On the other hand, from Poincaré inequality and (4.8) we see that

|u|2dx2

|Xu|2dxc||

(15)

which impliesuL2,Xλ,loc–2(,RN) and souC0,(2–λ)/2

X (,RN) according to Lemma 2.5. The

proof is finished.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11201258), National Science Foundation of Shandong Province of China (ZR2011AM008, ZR2011AQ006, ZR2012AM010).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 13 December 2017 Accepted: 23 February 2018

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