R E S E A R C H
Open Access
Global higher integrability for very weak
solutions to nonlinear subelliptic equations
Guangwei Du and Junqiang Han
**Correspondence:
[email protected] Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, P.R. China
Abstract
In this paper we consider the following nonlinear subelliptic Dirichlet problem:
X∗A(x,u,Xu) +B(x,u,Xu) = 0, x∈
, u–u0∈WX1,,0r(
),
whereX={X1,. . .,Xm}is a system of smooth vector fields defined inRnwith globally
Lipschitz coefficients satisfying Hörmander’s condition, and we prove the global higher integrability for the very weak solutions.
MSC: Primary 35H20; secondary 35J60
Keywords: nonlinear subelliptic equations; very weak solutions; global higher integrability
1 Introduction and main result
The theory of very weak solutions was introduced in the work of Iwaniec and Sbordone []. Iwaniec and Sbordone realized that the usual Sobolev assumption for weak solutions top-harmonic equation can be relaxed to a slightly weaker Sobolev space and proved that very weak solutions are actually classical weak solutions by using the nonlinear Hodge decomposition to construct suitable test functions. Based on Whitney’s extension theo-rem and theory ofApweights, Lewis [] showed a completely different proof and obtained the same result to certain elliptic systems. After [] and [], many authors have devoted their energy to the study of the regularity of such solutions; see for example [–] and the references therein. We mention here that Xie and Fang [] obtained the global higher integrability for very weak solutions to a class of nonlinear elliptic systems with Lipschitz boundary condition by using Hodge decomposition to construct a suitable test function. Recently the authors in [] proved the global regularity result for a second-order degen-erate elliptic systems ofp-Laplacian type in the Euclidean setting.
In , Zatorska-Goldstein [] showed the local higher integrability of very weak so-lutions to the nonlinear subelliptic equations
X∗A(x,u,Xu) +B(x,u,Xu) = , x∈, (.)
where⊂Rnis a bounded domain andX={X
, . . . ,Xm}(m≤n) is a system of smooth vector fields in Rnwith globally Lipschitz coefficients satisfying the Hörmander’s condi-tion andX∗= (X∗, . . . ,Xm∗) is a family of operators formal adjoint toXjinL.
In this work we are concerned with the boundary value problem for (.) with the bound-ary conditionu–u∈WX,,r(), i.e.,
⎧ ⎨ ⎩
X∗A(x,u,Xu) +B(x,u,Xu) = , x∈,
u–u∈WX,,r(),
(.)
and establish the global higher integrability for very weak solutions. We assume that the functions A= (A, . . . ,Am) : Rn×R×Rm →Rm and B: Rn×R×Rm →R are both Carathéodory functions satisfying
A(x,u,ξ)≤α|u|p–+|ξ|p–, (.)
B(x,u,ξ)≤α|u|p–+|ξ|p–, (.)
A(x,u,ξ) –A(x,v,ζ),ξ–ζ≥β|ξ–ζ||ξ|+|ζ|p–, (.) for a.e.x∈Rn,u∈Randξ∈Rm. Herep≥,α,βare positive constants.
A functionu∈WX,r() (r<p) is called avery weak solutionto (.) if
A(x,u,Xu)·Xϕdx+
B(x,u,Xu)ϕdx= (.)
holds for allϕ∈C∞().
In the above definition, the very weak means the integrable exponent is strictly lower than the natural exponentpand ifr=p, this is the classical definition of weak solution to (.).
To get our result, some regularity assumption introduced in [] should be imposed on. Let us first recall the notion of uniform (X,p)-fatness which can be found in []: A set
E⊂Rnis calleduniformly(X,p)-fatif there exist constantsC,R> such that cappE∩ ¯B(x,R),B(x, R)≥Ccapp
¯
B(x,R),B(x, R)
for allx∈∂Eand <R<R, wherecappis the variationalp-capacity defined in Section . We consider the following hypotheses on:
(H) there exists a constantC≥such that, for allx∈, |Bρ(x)| ≤CBρ(x)∩
Rn\,
whereρ(x) = dist(x, Rn\);
(H) the complementRn\ofis uniformly(X,p)-fat. Under the hypotheses stated above, we prove the following.
Theorem . Assume that u∈WX,s(),s>p.Then there exists aδ> such that if u∈
The key technical tool in proving Theorem . is a Sobolev type inequality with a ca-pacity term. With it we can prove a reverse Hölder inequality for the generalized gradient
Xuof a very weak solution, which allows us to get the global higher integrability ofXu. This paper is organized as follows. In Section we collect some known results on Carnot-Carathéodory spaces and prove a Sobolev type inequality characterized by capacity. Sec-tion is devoted to the proof of Theorem ..
2 Some known results and a Sobolev type inequality
Let{X, . . . ,Xm}be a system ofC∞-smooth vector fields in Rn(n≥) satisfying Hörman-der’s condition (see []):
rankLie{X, . . . ,Xm}
=n.
The generalized gradient is denoted byXu= (Xu, . . . ,Xmu) and its length is given by
Xu(x)=
m
j=
Xju(x)
.
An absolutely continuous curveγ : [a,b]→Rnis said to beadmissiblewith respect to the system{X, . . . ,Xm}, if there exist functionsci(t),a≤t≤b, satisfying
m
i=
ci(t)≤ and γ(t) = m
i=
ci(t)Xi
γ(t).
The Carnot-Carathéodory distanced(x,y) generated by{X, . . . ,Xm}is defined as the infi-mum of thoseT> for which there exists an admissible pathγ : [,T]→Rnwithγ() =x, γ(T) =y.
By the accessibility theorem of Chow [], the distancedis a metric and therefore (Rn,d) is a metric space which is called theCarnot-Carathéodoryspace associated with the sys-tem{X, . . . ,Xm}. The ball is denoted by
B(x,R) =
x∈Rn:d(x,x) <R
.
Forσ> andB=B(x,R), we will writeσBto indicateB(x,σR) anddiamthe diameter
ofwith respect tod.
It was proved in [] that the identity map is a homeomorphism of (Rn,d) into Rnwith the usual Euclidean metric, and every set which is bounded with respect to the Euclidean metric is also bounded with respect tod. Moreover, by a result of Garofalo and Nhieu [], Proposition ., if the given vector fields have globally Lipschitz coefficients in addition, then a subset of Rnis bounded with respect todif and only if it is bounded with respect to the Euclidean metric.
Hereafter we assume that the vector fieldsX, . . . ,Xmsatisfy the Hörmander condition and have globally Lipschitz coefficients.
Lemma .([, ]) For every bounded open set⊂Rnthere exists Cd≥such that
B(x, R)≤CdB(x,R) (.)
Here,|B(x,R)|denotes the Lebesgue measure ofB(x,R). The best constantCdin (.) is called the doubling constant, the measure such that (.) holds is called a doubling measure and the homogeneous dimension relative toisQ=logCd.
Given ≤p<∞, we define the Sobolev spaceWX,p() by
WX,p() =u∈Lp() :Xju∈Lp(),j= , , . . . ,m
,
endowed with the norm
uW,p
X ()=uL
p()+XuLp().
Here,Xjuis the distributional derivative ofu∈Lloc() given by the identity
Xju,ϕ=
uXj∗ϕdx, ϕ∈C∞().
The spaceWX,p() is a Banach space which admitsC∞()∩WX,p() as its dense subset. The completion ofC∞() under the norm·
WX,p()is denoted byW
,p
X,(). The following
Sobolev-Poincaré inequalities can be found in [] and []:
Lemma . Let Q be the homogeneous dimension relative to,B=B(x,R)⊂, <R< diam, ≤p<∞.There exists a constant C> such that,for every u∈WX,p(B),
–
B
|u–uB|κpdx
κp
≤CR
–
B
|Xu|pdx
p ,
where uB=–
Bu dx=
|B|
Bu dx,and≤κ≤Q/(Q–p),if≤p<Q; ≤κ<∞,if p≥Q.
Moreover,for any u∈WX,,p(B),
–
B
|u|κpdx
κp
≤CR
–
B
|Xu|pdx
p .
Next we recall a Gehring lemma on the metric measure space (Y,d,μ), whered is a metric andμis a doubling measure.
Lemma .([]) Let q∈[q, Q],q> is fixed.Assume that functions f,g are nonnegative
and g∈Lqloc(Y,μ),f ∈Lr
loc(Y,μ),for some r>q.If there exist constants b> andθ such
that for every ball B⊂σB⊂Y the following inequality holds:
–
B
gqdμ≤b
–
σB
g dμ
q + –
σB
fqdμ
+θ–
σB
gqdμ,
then there exist nonnegative constantsθ=θ(q,Q,Cd,σ)andε=ε(b,q,Q,Cd,σ)such
that if <θ<θthen g∈Lploc(Y,μ)for p∈[q,q+ε).
For the Hardy-Littlewood maximal functions
Mf(x) =sup
R>
|B(x,R)|
and
Mf(x) =sup R>
|B(x,R)|
B(x,R)∩
f(y)dy,
we will use the following properties proved in [] and [].
Lemma . If f ∈Lp(), <p≤ ∞,then M
f ∈Lp()and there exists a constant C=
C(Cd,p) > such that
MfLp()≤CfLp().
Lemma . If u∈WX,,locp (), <p<∞,then there exists C> such that,for a.e.x,y∈,
u(x) –u(y)≤Cd(x,y)M|Xu|(x) +M|Xu|(y)
.
Moreover,for any B=B(x,R)⊂and u∈WX,p(B),we have
u(x) –uB≤CRMB|Xu|(x), a.e. x∈B. (.)
It is worth noting that from Lemma . and Lemma . we can infer that, for a.e.x∈B
andu∈WX,,p(B),
u(x)≤CRMB|Xu|(x). (.)
Letω(x)≥ be a locally integrable function, we say thatω∈Ap, <p<∞, if there exists some positive constantAsuch that
sup
B⊂Rn
–
B
ωdx
–
B
ω–pdx
p–
≤A<∞.
Lemma . Assumeω∈L
loc(Rn)is nonnegative and <p<∞.Thenω∈Apif and only if
there exists a constant C> such that
Rn
|Mf|pωdx≤C
Rn
|f|pωdx,
for all f ∈Lp(ω(x)dx).
The (X,p)-capacityof a compact setK⊂inis defined by
capp(K,) =inf
|Xu|pdx:u∈C∞(),u= onK
and for an arbitrary setE⊂, the (X,p)-capacity ofEis
capp(E,) = inf
G⊂open E⊂G
sup
K⊂G Kcompact
We will use the following two-sided estimate of (X,p)-capacity in []: Forx∈and <R<diam, there existC,C> such that
C|
B(x,R)|
Rp ≤capp
¯
B(x,R),B(x, R)≤C|
B(x,R)|
Rp . (.)
Lemma .([]) IfRn\is uniformly(X,p)-fat,then there exists <q<p such thatRn\
is also uniformly(X,q)-fat.
The uniform (X,q)-fatness also implies uniform (X,p)-fatness for allp≥q, which is a simple consequence of Hölder’s and Young’s inequality.
At the end of this section we prove a Sobolev type inequality characterized by capacity. A similar inequality in the Euclidean setting can be found in [].
Lemma . Let⊂Rnbe a bounded open set with the homogeneous dimension Q, <
q<∞and <R<diam.For any x∈,denote B=B(x,R)and N(ϕ) ={y∈ ¯B:ϕ(y) = }.
Then there exists a constant C=C(Q,q) > such that,for allϕ∈C∞(B)∩WX,q(B),
–
B
|ϕ|κqdx
κq
≤C
capq(N(ϕ), B)
B
|Xϕ|qdx
q
, (.)
where≤κ≤Q/(Q–q)if≤q<Q and≤κ<∞if q≥Q.
Proof We always assumeϕB= ; otherwise, (.) follows immediately from Lemma . and (.). Let η∈C∞(B), ≤η≤ such thatη= onB¯ and|Xη| ≤ Rc. Denotingv=
η(ϕB–ϕ)/ϕB, thenv∈C∞(B) andv= inN(ϕ). It follows from Lemma . that
capqN(ϕ), B≤
B
|Xv|qdx
≤ |ϕB|–q
B
|Xη|q|ϕ–ϕB|qdx+|ϕB|–q
B
|Xϕ|qdx
≤C|ϕB|–q
B
|Xϕ|qdx,
and then
|ϕB| ≤C
capq(N(ϕ), B)
B
|Xϕ|qdx
q
. (.)
Then Lemma . and (.) lead to
–
B
|ϕ|κqdx
κq
≤
–
B
|ϕ–ϕB|κqdx
κq
+|ϕB|
≤CR
–
B
|Xϕ|qdx
q
+C
capq(N(ϕ), B)
B
|Xϕ|qdx
q
≤C
capq(N(ϕ), B)
B
|Xϕ|qdx
where in the last step we used the estimate
capqN(ϕ), B≤capq(B¯, B)≤C|B|R–q.
The proof is complete.
3 Proof of Theorem 1.1
Assume that the functionu∈WX,p–δ()(δ< ) is a very weak solution to the Dirichlet problem (.). Choose a ballBsuch that⊂ Band letBbe a ball of radiusRwith
B⊂Bfor fixed <R< . There are two cases: (i) B⊂or (ii) B\=∅. In the case
(i), the following estimate has been proved in []:
–
B
|Xu|p–δdx≤θ–
B|
Xu|p–δdx+b
–
B|
u|p–δ+
–
B|
Xu|tdx
p–δ
t
, (.)
whereθ small enough,b> ,max{, (p–δ)∗}<t<p–δ.
When B\=∅, a similar inequality (see (.) below) will be achieved.
Step. Letηbe a smooth cut-off function on B, i.e.η∈C∞(B) such that
≤η≤, η= onB and |Xη| ≤c/R.
Defineuˆ=η(u–u) and
Eμ=
x∈Rn:M|Xuˆ|(x)≤μ, forμ> . (.)
We conclude from Lemma . and the assumption (H) thatuˆis Lipschitz continuous on
Eμ∪(Rn\).
Indeed, ifx,y∈Eμ∩, then Lemma . implies|ˆu(x) –uˆ(y)| ≤cμd(x,y); ifx,y∈Rn\, thenuˆ(x) =uˆ(y) = . We setBρx=B(x,ρx) withρx= dist(x, Rn\) for the casex∈Eμ∩ andy∈Rn\. Sinceuˆis zero on Rn\, it follows that
Bρx∩(Rn\)
|ˆu–uˆBρx|dz=
Bρx∩(Rn\)
|ˆuBρx|dz
=Bρx∩
Rn\|ˆuBρx|
and then, from assumption (H) and Lemma .,
|ˆuBρx| ≤C
|Bρx∩(Rn\)|
|Bρx|
|ˆuBρx|
= C
|Bρx|
Bρx∩(Rn\)
|ˆu–uˆBρx|dz
≤C–
Bρx
|ˆu–uˆBρx|dz≤cCρx–
Bρx
|Xuˆ|dz
Therefore, we have by (.) and (.)
uˆ(x) –uˆ(y)=uˆ(x)
≤uˆ(x) –uˆBρx+|ˆuBρx|
≤cρxM|Xuˆ|(x) +cCμρx
≤cCμρx
≤cCμd(x,y).
It follows thatuˆis a Lipschitz function onEμ∪(Rn\) with the Lipschitz constantcCμ.
As in [], we can use the Kirszbraun theorem (see e.g. []) to extenduˆ to a Lipschitz functionvμdefined on Rnwith the same Lipschitz constant. Moreover, there existsμ
such that, for everyμ≥μ,suppvμ⊂B∩.
In fact, letD= B∩andx∈Rn\(B∩), we have by Lemma . that
M|Xuˆ|(x) = sup
Bx,B∩B=∅ –
B
|Xuˆ|(y)dy≤ Cd
|B|
D
|Xuˆ|(y)dy,
where|B|>|B|,Cdis the doubling constant. Setting
μ=
Cd
|B|
D
|Xuˆ|(y)dy,
thenM|Xuˆ|(x)≤μ,μ≥μ, which impliesvμ(x) =uˆ(x) = forx∈Rn\(B∩). So we can
take the functionvμas a test function in (.).
Letμ≥μand takevμas a test function in (.) to have
B∩
A(x,u,Xu)·Xvμdx+
B∩
B(x,u,Xu)vμdx= .
Noting thatvμ=uˆon (B∩)∩Eμand thatsuppuˆ⊂D, we have by the structure condi-tions onA(x,u,ξ) andB(x,u,ξ)
D∩Eμ
A(x,u,Xu)·Xu dxˆ +
D∩Eμ
B(x,u,Xu)u dxˆ
≤ (B∩)\Eμ
A(x,u,Xu)|Xvμ|dx+
(B∩)\Eμ
B(x,u,Xu)|vμ|dx
≤cμ
(B∩)\Eμ
|u|p–+|Xu|p–dx, (.)
where in the last inequality we use the fact that|Xvμ| ≤cμ,|vμ| ≤cRμ(see []). Multiplying both sides of (.) byμ–(+δ)and integrating over (μ,∞), we get
L:=
∞
μ
D∩Eμ
μ–(+δ)A(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆdx dμ
≤c
∞
μ
(B∩)\Eμ
Interchanging the order of integration and applying (.), we have
P=c
B
M|Xuˆ|
μ
μ–δ|u|p–+|Xu|p–dμdx
≤ c
–δ
(B∩)\Eμ
M|Xuˆ|–δ|u|p–+|Xu|p–dx
≤c
B∩
|u|p–δ+|Xu|p–δdx+c
B∩
M|Xuˆ|p–δdx. (.)
Using Lemma . and Lemma ., we have
c
B∩
M|Xuˆ|p–δdx
≤c
D
|Xuˆ|p–δdx
≤c
D
|Xu–Xu|p–δdx+
c Rp–δ
B
|u–u|p–δdx
≤c
D
|Xu–Xu|p–δdx+
c|B| Rp–δ
capp–δ(N(u–u), B)
B
|Xu–Xu|p–δdx
,
whereN(u–u) ={x∈ ¯B:u(x) =u(x)}. Sinceu–uvanishes outside, we have Rn\⊂ {u–u= }. On the other hand, by Lemma . and assumption (H), there existsδsuch
that if <δ<δ, Rn\is uniformly (X,p–δ)-fat, and hence
capp–δN(u–u), B
≥capp–δB¯∩Rn\, B
≥ccapp–δ(B¯, B)≥c|B|R–(p–δ). (.)
From (.) and the doubling condition, we derive
c
B∩
M|Xuˆ|p–δdx≤c
D|
Xuˆ|p–δdx
≤c
D
|Xu|p–δdx+c
D
|Xu|p–δdx, (.)
and then (.) becomes
P≤c
B∩
|u|p–δdx+c
B∩
|Xu|p–δdx+c
B∩
|Xu|p–δdx. (.)
As regards the estimation ofL, by changing the order of integration, we have
L=
∞
μ
D
μ–(+δ)A(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆχ{M|Xuˆ|(x)≤μ}dx dμ
=
D\Eμ
∞
M|Xuˆ|
μ–(+δ)A(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆdx dμ
+
D∩Eμ
∞
μ
= δ
D\Eμ
M|Xuˆ|–δA(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆdx
+ δ
D∩Eμ
μ–δA(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆdx.
SinceD\Eμ=D\(D∩Eμ), (.) and (.) imply
L= δ
D
M|Xuˆ|–δA(x,u,Xu)·Xu dxˆ – δ
D∩Eμ
M|Xuˆ|–δA(x,u,Xu)·Xu dxˆ
+ δ
D\Eμ
M|Xuˆ|–δB(x,u,Xu)u dxˆ
+ δ
D∩Eμ
μ–δA(x,u,Xu)·Xuˆ+B(x,u,Xu)uˆdx
≥
δ
D
M|Xuˆ|–δA(x,u,Xu)·Xu dxˆ
–α δ
D∩Eμ
M|Xuˆ|–δ|u|p–+|Xu|p–|Xuˆ|dx
–α δ
D
M|Xuˆ|–δ|u|p–+|Xu|p–|ˆu|dx
:=
δ(I– αI–αI). (.)
Step. Next, we will estimateIi(i= , , ) one by one. Now for estimation ofI. To this end, define the sets
D=
x∈D\B:M|Xuˆ| ≤δMD|Xu–Xu|
,
D=
x∈D\B:M|Xuˆ|>δMD|Xu–Xu|
andB=B∩. Thus
I=
B∪D
M|Xuˆ|–δA(x,u,Xu) –A(x,u,Xu)
·ηX(u–u)dx
+
B∪D
M|Xuˆ|–δA(x,u,Xu)·η(Xu–Xu)dx
+
D
M|Xuˆ|–δA(x,u,Xu)·Xη(u–u)dx
+
D
M|Xuˆ|–δA(x,u,Xu)·Xu dxˆ .
Since (M|Xuˆ|)–δ≤ |Xuˆ|–δa.e., it follows from (.) and (.) that
I≥β
B
M|Xuˆ|–δ|Xu–Xu|pdx
–α
B
|Xuˆ|–δ|u|p–+|Xu|p–
× |Xu–Xu|dx+
D
M|Xuˆ|–δ|u|p–+|Xu|p–
+
D
M|Xuˆ|–δ|u|p–+|Xu|p–Xη(u–u)dx
+
D
M|Xuˆ|–δ|u|p–+|Xu|p–|Xuˆ|dx
:=I–α(I+I+I+I). (.)
Since the function (M|Xuˆ|)–δis anA
p-weight, we obtain from Lemma . that
I≥cβ
B
M|Xuˆ|–δMB|Xu–Xu|
p
dx.
By the doubling condition and Lemma . we see that, forx∈ B∩,
M|Xuˆ|(x)≤ sup
Bx,B⊂B –
B
|Xuˆ|dy+ sup
Bx,B∩∂B=∅ –
B
|Xuˆ|dy
≤MBX(u–u)+
c R – B
|u–u|s dx s +c
|B|
D
|Xu–Xu|s
dx
s
≤MBX(u–u)+
c R
caps(N(u–u), B)
B
X(u–u)s
dx s +c
|B|
D
|Xu–Xu|s
dx
s
≤MBX(u–u)+c
|B|
D
|Xu–Xu|s
dx
s
, (.)
wheremax{, (p–δ)∗}<s<p–δis such that Rn\is uniformly (X,s)-fat and the last inequality comes from an argument similar to (.).
To continue, we define
G=
x∈B
∩:MBX(u–u)≥c
|B|
D
|Xu–Xu|s
dx
s
.
So from (.) we see thatM|Xuˆ| ≤cMB|X(u–u)|onG, and then
I≥c
G
MB|Xu–Xu|
–δ
MB|Xu–Xu|
p dx ≥c B ∩
|Xu–Xu|p–δdx–c|B|
|B|
D
|Xu–Xu|s
dx
p–δ
s
≥c
B ∩
|Xu|p–δdx–c
D
|Xu|p–δdx–c|B|
|B|
D
|Xu|sdx
p–δ
s
. (.)
Using the factXuˆ=X(u–u) onBand Young’s inequality, we have
I≤c
D
|u|p–δ+|Xu|p–δ
dx+cε
D
|Xu–Xu|p–δdx
≤c
D
|u|p–δdx+c
D
|Xu|p–δdx+cε
D
Next from the definition ofDand Lemma ., we see
I≤cδ–δ
D
MD|Xu–Xu| –δ
|u|p–+|Xu|p–
dx
≤c
D
|u|p–δ+|Xu|p–δ
dx+cε
D
|Xu–Xu|p–δdx
≤c
D
|u|p–δdx+c
D
|Xu|p–δdx+cε
D
|Xu|p–δdx. (.)
ForI, we have by using|Xη(u–u)| ≤ |Xuˆ|+|Xu–Xu|
I≤c
D
M|Xuˆ|–δ|u|p–+|Xu–Xu|p–+|Xu|p–Xη(u–u)dx ≤c
D
M|Xuˆ|–δ|u|p–+|Xu|p–
|Xuˆ|+|Xu–Xu|
dx
+c
D
M|Xuˆ|–δ|Xu–Xu|p–|Xη||u–u|dx ≤c
D
M|Xuˆ|–δ|u|p–+|Xu|p–
|Xuˆ|dx
+c
D
M|Xuˆ|–δ|u|p–+|Xu|p–
|Xu–Xu|dx
+ c
R
D
M|Xuˆ|–δ|Xu–Xu|p–|u–u|dx
:=K+K+K.
Using Young’s inequality and (.), we get
K≤c
D
|u|p–+|Xu|p–
|Xuˆ|–δdx
≤c
D
|u|p–δdx+c
D
|Xu|p–δdx+cε
D
|Xu|p–δdx. (.)
By the definition ofDand noting that|X(u–u)| ≤MD|X(u–u)|a.e.D,
K≤cδ–δ
D
|Xu–Xu|–δ
|u|p–+|Xu|p–
dx
≤c
D
|u|p–δdx+c
D
|Xu|p–δdx+cε
D
|Xu|p–δdx. (.)
Finally, by Young’s inequality,
K≤
cδ–δ
R
D
MD|Xu–Xu| –δ
|Xu–Xu|p–|u–u|dx
≤cδ–δ
R
D
|Xu–Xu|p––δ|u–u|dx
≤cε
D
|Xu–Xu|p–δdx+c
D
u–u
R
In order to estimate the second component of the right-hand side, we lets= (p–δ)( –ϑ), where <ϑ<p–p–δ+δQifp–δ≤Qand <ϑ<min{p–pδ––δQ,
}ifp–δ>Q. Denote
κ=
⎧ ⎨ ⎩
Q
Q–s, s<Q, , s>Q,
thenκs≥p–δ. Using Lemma . and Lemma ., we derive
–
B
u–u
R
p–δdx
p–δ
≤cR–
–
B
|u–u|κs
dx
κs
≤cR–
caps(N(u–u), B)
B
X(u–u)s dx s ≤c
|B|
D
|Xu–Xu|s
s ,
where the proof of the last inequality is similar to (.). Therefore,
c
B
u–u
R
p–δdx≤c|B|
|B|
D
|Xu–Xu|s
dx
p–δ
s
. (.)
Inserting (.) into (.), we have
K≤cε
D
|Xu|p–δdx+c
D
|Xu|p–δdx
+c|B|
|B|
D
|Xu|sdx
p–δ
s
. (.)
A combination of (.), (.) and (.) implies
I≤c
D
|u|p–δ+|Xu|p–δ
dx
+cε
D
|Xu|p–δdx+c|B|
|B|
D
|Xu|sdx
p–δ
s
. (.)
The definition ofDand Lemma . give
I≤c
D
M|Xuˆ|–δ|u|p–+|Xu|p–dx
≤cδ–δ
D
MD|Xu–Xu| –δ
|u|p–+|Xu|p–dx
≤cδ–δ
D
|Xu–Xu|p–δdx+
D
|u|p–δ+|Xu|p–δdx
≤c
D
|u|p–δ+|Xu|p–δ
dx+cδ
D
The previous estimates show that
I≥c
B ∩
|Xu|p–δdx–c
D
|u|p–δ+|u
|p–δ+|Xu|p–δ
dx
–c(ε+δ)
D
|Xu|p–δdx–c|B|
|B|
D
|Xu|tdx
p–δ
t
, (.)
wheret=max{s,s}<p–δ.
Now we address the estimation ofI. Using (.), we have
I≤
D|
u|p–|Xuˆ|–δdx+
D∩Eμ
M|Xuˆ|–δ|Xu|p–|Xuˆ|dx
≤c
D
|u|p–δdx+cε
D
|Xuˆ|p–δdx+
D∩Eμ
|Xu|p–M|Xuˆ|–δdx
≤c
D
|u|p–δ+|Xu
|p–δ
dx
+cε
D
|Xu|p–δdx+
D∩Eμ
|Xu|p–M|Xuˆ|–δdx. (.)
To estimate the last integral in (.), let <τ <andx∈D∩Eμ. If|Xu| ≥τ–μ, then
M|Xuˆ| ≤μ≤τ|Xu|and
|Xu|p–M|Xuˆ|–δ≤ |Xu|p–τ|Xu|–δ=τ–δ|Xu|p–δ; (.)
if|Xu|<τ–μ, then
|Xu|p–M|Xuˆ|–δ≤τ–μ p–
μ–δ≤τ–pμp–δ. (.)
By (.) and (.), we deduce that, for anyx∈D∩Eμ,
|Xu|p–M|Xuˆ|–δ≤cτ–δ|Xu|p–δ+τ–pμp–δ. (.)
For the second term in (.), we first observe from the proof of (.) that
R
–
B
|u–u|s
dx
s
≤c
|B|
D
|Xu–Xu|s
dx
s .
Noticingμ=|cB|
D|Xuˆ|dx, we have from Hölder’s inequality
τ–pμp–δ≤cτ–p
|B|
D
Xη(u–u) +ηX(u–u)dx p–δ
≤cτ–p
R
–
B
|u–u|s
dx
sp–δ
+cτ–p
|B|
D
X(u–u)
s
dx
p–δ
≤cτ–p
|B|
D
X(u–u)
s
dx
p–δ
s
≤cτ–p
|B|
D
|Xu|p–δdx
+cτ–p
|B|
D
|Xu|sdx
p–δ
s
. (.)
By (.) and (.), it follows that
D∩Eμ
|Xu|p–M|Xuˆ|–δdx
≤cτ–δ
D
|Xu|p–δdx+c
D
|Xu|p–δdx+cτ–p|B|
|B|
D
|Xu|sdx
p–δ
s
. (.)
Taking (.) into (.), we have
I≤c
D|
u|p–δdx+c
D|
Xu|p–δdx
+cε+τ–δ
D|
Xu|p–δdx+cτ–p|B|
|B|
D|
Xu|sdx
p–δ
s
. (.)
For the estimation ofI: From (.), Lemma . and a similar process to the proof of
(.), we have
I≤c
D
|u|p–+|Xu|p–|ˆu|–δdx
≤
D
|u|p–δ+cε
D
|Xu|p–δdx+c
D
|u–u|p–δdx
≤c
D
|u|p–δ+|Xu
|p–δ
dx
+cε
D|
Xu|p–δdx+c|B|
|B|
D|
Xu|tdx
p–δ
t
, (.)
wheret=max{s,s}<p–δ.
Step. Taking into account (.), (.), substituting (.), (.) and (.) into (.), and lettingε=τ–δ, it follows that
B ∩
|Xu|p–δdx
≤c
B∩
|u|p–δ+|u|p–δ+|Xu|p–δ
dx
+cδ+τ–δ
B∩
|Xu|p–δdx+cτ–p|B|
|B|
B∩
|Xu|tdx
p–δ
t
. (.)
To sum up the cases B⊂and B\=∅, we let
g(x) =
⎧ ⎨ ⎩
and
f(x) =
⎧ ⎨ ⎩
(|u–u|+|u|+|Xu|)t, x∈,
, x∈Rn\.
Thus we have from (.) and (.)
–
B
gqdx≤b
–
B
g dx
q + –
B
fqdx
+θ–
B
gqdx,
where q= p–tδ, θ =c(δ +τ–δ) and b=cτ–p. Choosing τ, δ small enough, we see by Lemma . that there existst=p–δ+ε, for someε> , such that|Xu| ∈Lt().
Furthermore, we will show that there existst>r=p–δ such thatu∈Lt(). Since
u–u∈WX,,r(), we obtain from Lemma . that, forr<Q,r∗=Qr/(Q–r),
|u–u|r ∗
dx
r∗
≤C()
X(u–u)rdx
r <∞.
Takingt=min{s,r∗}>r, we have
|u|tdx
t
≤
|u–u|tdx
t +
|u|tdx
t
≤C
|u–u|r ∗
dx
r∗
+
|u|tdx
t
and thenu∈Lt() byu
∈Ls(). Ifr≥Qthen we can apply the above reasoning for any
r∗<∞to obtainu∈Lt().
We setp˜=min{t,t}>p–δandu∈WX,˜p(). Repeating the preceding reasoning, we know that there existsδ˜> such thatu∈WX,p+δ˜() and the proof is complete.
4 Conclusions
In this paper, we obtained the global higher integrability for very weak solutions to the Dirichlet problem for a nonlinear subelliptic equation on Carnot-Carathéodory spaces which implies that such solutions are classical weak solutions. It is a generalization of the corresponding result in the classical Euclidean setting.
Acknowledgements
The authors are grateful to anonymous reviewers for their careful reading of this paper and their insightful comments and suggestions, which improved the paper a lot. The current work is supported by the National Natural Science Foundation of China (No. 11271299).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All 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.
References
1. Iwaniec, T, Sbordone, C: Weak minima of variational integrals. J. Reine Angew. Math.454, 143-161 (1994) 2. Lewis, JL: On very weak solutions of certain elliptic systems. Commun. Partial Differ. Equ.18(9-10), 1515-1537 (1993) 3. Giannetti, F, Passarelli di Napoli, A: On very weak solutions of degeneratep-harmonic equations. Nonlinear Differ. Equ.
Appl.14(5-6), 739-751 (2007)
4. Kinnunen, J, Lewis, JL: Very weak solutions of parabolic systems ofp-Laplacian type. Ark. Mat.40(1), 105-132 (2002) 5. Xie, S, Fang, A: Global higher integrability for the gradient of very weak solutions of a class of nonlinear elliptic
systems. Nonlinear Anal.53(7-8), 1127-1147 (2003)
6. Fattorusso, L, Molica Bisci, G, Tarsia, A: A global regularity result for some degenerate elliptic systems. Nonlinear Anal.
125, 54-66 (2015)
7. Zatorska-Goldstein, A: Very weak solutions of nonlinear subelliptic equations. Ann. Acad. Sci. Fenn., Math.30(2), 407-436 (2005)
8. Kilpeläinen, T, Koskela, P: Global integrability of the gradients of solutions to partial differential equations. Nonlinear Anal.23(7), 899-909 (1994)
9. Danielli, D, Garofalo, N, Phuc, NC: Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces. In: Sobolev Spaces in Mathematics. I: Sobolev Type Inequalities, pp. 117-151. Springer, New York (2009)
10. Hörmander, L: Hypoelliptic second order differential equations. Acta Math.119, 147-171 (1967)
11. Chow, WL: Über systeme von linearen partiellen differentialgleichungen erster ordnung. Math. Ann.117, 98-105 (1939)
12. Nagel, A, Stein, EM, Wainger, S: Balls and metrics defined by vector fields. I: basic properties. Acta Math.155, 103-147 (1985)
13. Garofalo, N, Nhieu, DM: Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. J. Anal. Math.74, 67-97 (1998)
14. Hajłasz, P, Koskela, P: Sobolev met Poincaré. Mem. Am. Math. Soc.688, 1-101 (2000)
15. Lu, G: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Rev. Mat. Iberoam.8(3), 367-439 (1992)
16. Danielli, D: Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ. Math. J.44(1), 269-286 (1995)
