Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups

HoÈlder Continuity for Sub-Elliptic Systems Under the Sub-Quadratic Controllable Growth in Carnot Groups (*)

JIALINWANG(**)(***) - DONGNILIAO(***) - ZEFENGYU(***)

ABSTRACT- This paper is devoted to optimal partial regularity of weak solutions to nonlinear sub-elliptic systems for the case15m52under the controllable growth condition in Carnot groups. We begin with establishing a Sobolev-Poincare type inequality for the functionu2HW^1;m(V;R^N) withm2(1;2), and then partial reg- ularity with optimal local HoÈlder exponent for horizontal gradients of weak solu- tions to the systems is established by usingA-harmonic approximation technique.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 35H20, 35B65.

KEYWORDS. Carnot group; Nonlinear sub-elliptic system; Sub-quadratic con- trollable growth condition; Optimal partial regularity;A-harmonic approxima- tion technique

1. Introduction and statements of main results

In this paper, we consider nonlinear sub-elliptic systems of second or- der in divergence form under the sub-quadratic (15m52) controllable growth condition in Carnot groups, and settle optimal partial regularity for horizontal gradients of weak solutions.

More precisely, let VGbe a bounded domain in a Carnot groupG with general step, and consider the following system

ÿX^k

iˆ1

X_iA^a_i(j;u;Xu)ˆB^a(j;u;Xu); j2V; u2R^N; Xu2R^kN (1:1)

(*) Research supported by National Natural Science Foundation of China (No.

11126294 and No. 11201081), supported by Natural Science Foundation of Jiangxi, China (No. 2010GQS0021), and supported by Science and Technology Planning Project of Jiangxi Province, China (No. GJJ13657).

(**) Indirizzo dell'A.: Key Laboratory of Numerical Simulation Technology of Jiangxi Province, Gannan Normal University, Ganzhou, 341000, Jiangxi, P. R. China.

E-mail: jialinwang1025@hotmail.com

(***) Indirizzo degli Autori: School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, P. R. China.

with sub-quadratic controllable structure conditions (H1)-(H3) and (C), where

(H1): A^a_i(j;u;p) is differentiable with respect top, with bounded and continuous derivatives, that is, there exists a constantCsuch that

A^a_i;pj

b(j;u;p)C(1‡j jp²)^mÿ22 ; (j;u;p)2VR^NR^kN; 15m52;

(1:2)

where we denote byA^a_i;pj

b()ˆ@A^a_i()

@p^j_b ;

(H2):A^a_i(j;u;p) satisfies the following ellipticity condition A^a_i;pj

b(j;u;p)h^a_ih^b_j l(1‡j jp²)^mÿ22 j jh²; 8h2R^kN; (1:3)

wherelis a positive constant;

(H3): A^a_i(j;u;p) is HoÈlder continuous with exponent g2(0;1) in the first and second variables, i.e.,

A^a_i(j;u;p)ÿA^a_i(~j;u;~ p)K(j j)(du ^m(j;~j)‡uÿu~^m)^mg(1‡j jp²)^mÿ12 ; (1:4)

where K(): [0;1)![0;1) is monotonously nondecreasing. Without loss of generality, it is convenient to takeK()1;

(C) (Controllable growth condition): Denoterˆ mQ

Qÿmand require B^a(j;u;p)

j j4a pj j^m(1ÿ1r)‡b uj j^rÿ1‡c;

(1:5)

where Q3 is the homogeneous dimension in Carnot groups (see (2.3) below), anda;bandcare positive constants.

Furthermore, (H1) infers that there exists a continuously nonnegative and bounded functionv(s;t): [0;1)[0;1)![0;1), wherev(s;0)ˆ0 for alls, andv(s;t) is monotonously nondecreasing insfor fixedt;v(s;t) is concave and monotonously nondecreasing intfor fixeds, such that for all (j;u;p);(~j;u;~ ~p)2VR^NR^kN,

A^a_i;pj

b(j;u;p)ÿA^a_i;pj

b(j;u;p)~ C(1‡j jp²‡j jp~²)^mÿ22 v(j j;p jpÿ~pj):

(1:6)

As is well known, even under reasonable assumptions onA^a_i; B^ain the systems, people cannot in general expect that weak solutions of nonlinear elliptic systems of equations will be classical (i.e.C²-solutions) like elliptic scalar equations. This was first shown by De Giorgi [1]. Then the goal is to establish partial regularity of weak solutions for systems. Such regularity

means that for any weak solutionuof a system, there exists an open subset V0Vsuch thatVnV0has a zero Lebesgue measure anduor its gradient Du is locally regular in V₀. We refer the reader to monographs of Gia- quinta [2, 3] and Chen-Wu [4].

There are different methods to prove partial regularity: the direct approach was first carried out by Giaquinta-Modica [5]; Guisti-Miranda [6]

was earlier to employ the blow-up method; furthermore, Duzaar and Grotowski in [7] generalized so called A-harmonic approximation techni- que in [8] and gave the remarkable proof of partial regularity for systems with quadratic growth conditions(mˆ2). This method has two major ad- vantages: the first is that we only need to establish a Caccioppoli type inequality; this avoids having to prove an inverse HoÈlder inequality. The other is that one can obtain the optimal HoÈlder exponent in partial reg- ularity. Then, Duzaar et al. [9] studied partial regularity of almost mini- mizers of quasi-convex variational integrals with sub-quadratic growth, and we also note that the degenerate p-Laplacian version of the method has been obtained in [10], and applied to the partial regularity in [11]. In the paper [9], Duzaar et al. actually provide a partially new proof of the original regularity result of Carozza et al. [12]. Later, Chen and Tan in [13, 14] extended Duzaar and Grotowski's results [7] to more general nonlinear elliptic systems under the super-quadratic growth (m>2) and sub- quadratic growth (15m52), respectively.

Several regularity results were focused on systems constructed by basic vector fields in Carnot groups. Capogna and Garofalo in [15] showed the partial HoÈlder regularity for quasi-linear sub-elliptic systems under the quadratic structure conditions in Carnot groups of step two. Shores in [16] considered a homogeneous quasi-linear system under the quadratic growth condition on the Carnot group with general step. She first estab- lished higher differentiability and smoothness for weak solutions of the system with constant coefficients, and then deduced the partial regularity.

Their methods depend mainly on generalization of classical direct method in the Euclidean space. Later, by the method ofA-harmonic approxima- tion, FoÈglein in [17] treated the homogeneous nonlinear system

ÿX²ⁿ

iˆ1

XiA^a_i(j;Xu)ˆ0; aˆ1; ;N

on the Heisenberg group under super-quadratic structure conditions. She got partial regularity for the horizontal gradient of weak solutions to the initial system. Then the first author and Niu in [18] considered more gen-

eral nonlinear sub-elliptic systems (see (1.1)) in Carnot groups under super- quadratic growth conditions, and established the optimal partial regularity of the horizontal gradient of weak solutions.

In this paper, we will apply the method ofA-harmonic approximation adapted to the setting of Carnot groups to study partial regularity for the system (1.1) under the controllable growth conditions with sub-quadratic case (15m52). The key point is to establish a certain excess-decay esti- mate for the excess functionF. In the casem2, this function is given by

F(j₀;r;p₀)ˆB_r(j₀)^ÿ1

G

Z

Br(j0)

jXuÿp₀j²‡jXuÿp₀j^m dj;

whereas in the case 15m52, one uses F(j₀;r;p0)ˆBr(j₀)^ÿ1

G

Z

Br(j0)

V(Xu)ÿV(p0) j j²dj;

(1:7)

where V(A)ˆ(1‡ jAj²)^mÿ24 for A2Hom(R^k;R^N). It is shown that if F(j₀;r;p₀) is small enough on a ballB_r(j₀)V, then for some fixedu2(0;1) one has the excess improvementF(j₀;ur;p₀)Cu^2gF(j₀;r;p₀). Iteration of this result yields the excess-decay estimate which implies the regularity result. Although the underlying philosophy in this paper is encouraged by that in [14, 9], some different treatment are necessary. Since basic vector fields (see (2.1) below) of Lie algebras corresponding to the Carnot group are more complicated than gradient vector fields in the Euclidean space, we have to find a different auxiliary function in proving Caccioppoli type in- equality. Inspired by [17], we choose horizontal variables to construct such a suitable function (see Remark 1 below); Besides, the non-horizontal deri- vatives of weak solutions will happen in the Taylor type formula on the Carnot group and cannot been effectively controlled in the present hy- potheses. So the method employing Taylor's formula in [14] is not appro- priate in our setting. In order to obtain the desired decay estimate, we need establish and use the Sobolev-Poincare type inequality (3.1) instead.

The main result in this paper is as follows.

THEOREM 1. Assume that coefficients A^a_i and B^a satisfy conditions (H1)-(H3)and(C), and u2HW^1;m(V;R^N)be a weak solution to the sys- tem(1.1)withVG, i.e.,

Z

V

A^a_i(j;u;Xu)X_iW^adjˆ Z

V

B^a(j;u;Xu)W^adj 8W2C¹₀ (V;R^N):

(1:8)

Then there exists an open subsetV0 V;such that u2G^1;g(V0;R^N), withg in(1.4).Furthermore,VnV0ˆS1[S2and meas(VnV0)ˆ0, where

S1ˆn

j₀2V: lim

r!0^‡sup (Xu)ÿ _j0;r

ˆ 1o

; S2ˆ

(

j₀2V: lim

r!0^‡infjBr(j₀)j^ÿ1 Z

Br(j0)

Xuÿ(Xu)_j0;r

^mdj>0 )

:

To the best of our konwledge, in the Heisenberg group,G^0;gregularity of weak solutions to sub-ellipticp-Laplacian equations is valid for 15p51 and was proved by several people during the 90's in [19, 20, 21], but when turning toG^1;gregularity, it is worthy of pointing out that the remarkable contribution of HoÈlder continuity for the gradient of weak solutions to sub- ellipticp-Laplacian equations is due to Capogna [22, 23], Marchi [24, 25], Domokos [26, 27], Manfredi and Mingione et al. [28, 29], but the exponent p should be near 2, and the limitation 2p54 appears in the most recent of the cited works. Recently, Garofalo in [30] obtained theG^1;gregularity of weak solution which possess some special symmetries for 2p. When turning to partial continuity of weak solutions, our result shows that the G^1;g continuity is also valid for exponent 15p52.

The plan of this paper is organized as follows: In Section 2, we in- troduce two functions and their some useful properties, and collect some basic notions and facts associated to Carnot groups. Since in the sub- quadratic case we are dealing with functions belonging to the horizontal Sobolev spaceHW^1;m with 15m52, in the proof of the main results, a Sobolev-Poincare type inequality (see (3.1) below) will be used. So we will prove this inequality and a prior estimate for constant systems in Section 3. In Section 4, we show a Caccioppoli type inequality for weak solutions to the system (1.8). Section 5 is devoted to the proof of The- orem 1, and the process of the proof is split into four steps. An estimate on weak solutions is established in the first step. Second step begins with proving three ``small conditions'' (see (5.13)-(5.15) below), and then obtain the excess-improvement for the functionFin (1.7). Third step is to show that we can iterate the statement in the second step. A decay estimate

F(j₀;r;p0)Cr^2g

is obtained in the last step, wheregis the same as (1.4). Then we can infer the conclusion of Theorem 1 employing Lemma 2 and Lemma 3 below.

2. Preliminaries

A Carnot groupGof stepris a simply connected, nilpotent Lie group whose Lie algebra eg admits a stratification, i.e., egˆL^r

jˆ1V^j such that [V¹;V^j]ˆV^j‡1; jˆ1; ;rÿ1 and [V¹;V^r]ˆ f0g:Let X^l_idenote a left- invariant basis vector field ofV^lwith 1lrand 1im_l, wherem_lis the dimension ofV^l. For the sake of simplicity, we letX_iˆX_i¹,kˆm₁, and denote by Xˆ(X₁; ;X_k) the sub-elliptic gradient. We will say that Xi(iˆ1; ;k) are the horizontal vector fields with the form

X_iˆ@_i‡ Xⁿ

jˆi‡1

a_ij(j)@_j; X_i(0)ˆ@_i; (2:1)

where a_ij(j) is a polynomial. For a vector valued function uˆ(u¹; ;u^N):G7!R^N, we let Xiu^a(iˆ1; ;k;aˆ1; ;N) be a horizontal direction derivative, and say thatXuis the horizontal Jacobian with entriesX_iu^a.

Denoting

jˆ(j¹;j²; ;j^r)ˆx¹₁;x¹₂; ;x¹_m1;x²₁; ;x²_m2; ;x^r₁; ;x^r_ml 2G;

the distance from origin defined by

d(j)ˆ X^r

lˆ1

X^ml

iˆ1

x^l_i²

!^r!_l 2

4

3 5

2r!1

: (2:2)

For any j;h2G, we set d(j;h)ˆd(h^ÿ1j), where h^ÿ1ˆ ÿhˆ (ÿh¹; ;ÿh^r) is the reverse of h, and is the multiplication rule in G defined by jejˆj‡ej‡P(j;ej); j;ej2G; where P:GG7!G has polynomial components.

Following [31], we introduce the gauge ball and sphere B_r(j)ˆ fh2G d(j;j h)5rg; @Br(j)ˆ fh2G d(j;j h)ˆrginG, respectively. In what follows, we denote by v_GˆjB(0;1)j_G the volume of unit ball. Then

B(j;r)

j j_Gˆv_Gr^Q, where

QˆX^r

lˆ1

l ml

(2:3)

is the homogeneous dimension ofG.

DEFINITION1. Let 1m51, andVGbe an open set. Ifu2L^m(V) satisfies

u

k k_HW1;m(V)k ku _Lm(V)‡X^k

iˆ1

X_iu

k k_Lm(V)51;

(2:4)

we say thatubelongs to the horizontal Sobolev space. The spaceHW₀^1;m(V) is the completion ofC¹₀ (V) under the norm (2.4).

DEFINITION2. LetVG,g2(0;1), and denote

G^1;g(V)ˆfu2L¹(V)jX_iu2G^g(V);iˆ1; ;kg;

whereG^g(V)ˆn

u2L¹(V)juj_gˆ sup

j;~j2V;j6ˆ~j

u(j)ÿu(~j) d(j;~j)^g 51o

. We say that G^1;g(V) is a Folland-Stein space with the norm

u

k k_G^1;g_(V)ˆk ku _L1(V)‡kXuk_L1(V)‡juj_g‡ jXuj_g: (2:5)

DEFINITION 3. Let VG; 1m51 and m0. We say that u2L^m(V) belongs to the Campanato spaceL^m;m(V) if

u

k k_L^m;m_(V)ˆk ku _Lm(V)‡ sup

j2V;r5diamVr^ÿm Z

Br(j)\V

u(z)ÿu_j;r ^mdz 0

B@

1 CA

m1

51:

(2:6)

Capogna in [23] proved a Campanato type lemma in Carnot groups.

LEMMA2. Letj2VG;m1, r>0, and Q5mQ‡m, then L^m;m(B_r(j))G^g(B_r(j)); gˆmÿQ

m : (2:7)

Throughout the paper, we shall use the functions V ˆV_m, WˆW_m:Rⁿ !Rⁿdefined by

V(Q)ˆQ=(1‡j jQ²)^2ÿm4 ; W(Q)ˆQ=(1‡j jQ^2ÿm)¹² (2:8)

for eachQ2Rⁿ andm>1. By the elementary inequalityk kx ²

2ÿmk kx ₁ 2^m2k kx 2

2ÿmapplied to the vectorxˆ(1;j jQ^2ÿm)2R² we conclude that 1‡j jQ²

^2ÿm₂

1‡j jQ^2ÿm2^m21‡j jQ^22ÿm₂

; (2:9)

which immediately yields W(Q)

j j jV(Q)j 2^m4jW(Q)j:

(2:10)

The purpose of introducing W is the fact that in contrast to j jV ^m2, the functionj jW^m2 is convex (see [9]).

The following lemma includes some useful properties of the functionV, which can be found in Lemma 2.1 of [12].

LEMMA3. Let m2(1;2)and V; W:Rⁿ !Rⁿbe the functions defined in(2.8 ).Then there holds for anyQ₁;Q₂2Rⁿand t>0:

(1). 1

p2minj j;Q₁ j jQ₁^m2

jV… †Q₁ j minj j;Q₁ j jQ₁^m2

; (2). jV(tQ₁)j maxÿt;t^m2

V(Q₁) j j;

(3). jV(Q₁‡Q₂)j C(m)(jV(Q₁)j ‡jV(Q₂)j);

(4). m

2 jQ₁ÿQ₂jjV(Q₁)ÿV(Q₂)j=(1‡j jQ₁²‡j jQ₂²)^mÿ24 C(m;n)jQ₁ÿQ₂j;

(5). jV(Q₁)ÿV(Q₂)j C(m;n)jV(Q₁ÿQ₂)j;

(6). jV(Q₁ÿQ₂)j C(m;M)jV(Q₁)ÿV(Q₂)jfor allQ₁ withjQ₂j M.

The inequalities(1)-(3)also hold if V is replaced by W.

For later purpose we state the following two simple estimates which can be easily deduced from Lemma 3 (1) and (6). LetQ₁;Q₂2RⁿwithjQ₂j Mit follows that

Q₁ÿQ₂

j j²C(m;M)jV(Q₁)ÿV(Q₂)j²; (2:11)

forjQ₁ÿQ₂j 1;while forjQ₁ÿQ₂j>1 Q₁ÿQ₂

j j^mC(m;M)jV(Q₁)ÿV(Q₂)j²: (2:12)

The next lemma due to [9] is a more general version of the Lemma 2.7 in [12].

LEMMA 4. Let0n51;a;b0; v2L^p(Br(j₀))and g be a non-nega- tive bounded function satisfying

g(t)ng(s)‡a Z

Br(j0)

V v sÿt

²dj‡b;

for all r

2t5sr. Then there exists a constant CˆC(n)such that g r

2 C(n) a Z

Br(j0)

V v r

²dj‡b 0

B@

1 CA:

We conclude this section with an algebraic fact from [32]

LEMMA5. For every t2 ÿ1 2;0

andm0, it holds

1 R¹

0m²‡jp‡s(~pÿp)j²_t ds m²‡j jp²‡j j~p²

_t 8

2t‡1; (2:13)

for any p;p~2R^kN, not both zero ifmˆ0.

3. Sobolev-Poincare type inequality and a prior estimate in the sub- quadratic case

LEMMA 6 (Sobolev-Poincare type inequality). Let m2(1;2) and u2HW^1;m(Br(j₀);R^N)with Br(j₀)V, then

(3:1) Br(j₀)^ÿ1

G

Z

Br(j0)

W uÿu_j0;r r

Qÿm2Q

dj 0

B@

1 CA

Qÿm2Q

C_P B_r(j₀)^ÿ1

G

Z

Br(j0)

W Xu… † j j²dj 0

B@

1 CA

12

:

Furthermore, the analogous inequality is valid with W replaced by V defined in (2.8), and in particular, the inequality also holds if we substitute2 for 2Q

Qÿm.

PROOF. Similarly to the proof of Theorem 2 in [9]. Following [33, Lemma 3.2], there existc>1 andC>1 such that forj2B(j₀;r)

u(j)ÿuj0;r

C

Z

Bcr(j0)

Xu(h) j j d(j;h)^Qÿ1dh:

(3:2)

SinceW^2=m(t) is non-decreasing and convex, we applyW^2=m(t) to both sides of the last inequality and have by Jensen's inequality and recalling W(0)ˆ0:

W^2=m u(j)ÿu_j0;r r

C r Z

G

d^1ÿQ(j;h)We^2=m…jXu(h)j†dh;

(3:3)

where the numberQis the homogeneous dimension inG, and W Xu(h)e…j j† ˆ 0; h2=Bcr(j₀);

W Xu(h)…j j†; h2B_cr(j₀):

(

Since W Xu(h)…j j† 2L^mÿBr(j₀)

by Lemma 3(1), it implies W Xu(h)e…j j† 2 L^m… †. Then Theorem 2.7 in [19] applied withG We^2=m…jXuj†; aˆ1 yields

Z

Br(j0)

W u(j)ÿuj0;r

r

Qÿm2Q

dj 2

64

3 75

QÿmmQ

ˆ Z

Br(j0)

W^2=m u(j)ÿu_j0;r r

_Qÿm^mQ dj 2

64

3 75

QÿmmQ

C r

Z

G

Z

G

d^1ÿQ(j;h)We^2=m…jXu(h)j†dh 0

@

1 A

QÿmmQ

dj 2

64

3 75

QÿmmQ

C r

Z

G

We^2=m…jXu(j)j†^mdj 2

4

3 5

m1

ˆC r

Z

Br(j0)

W²…jXu(j)j†dj 2

64

3 75

m1

;

or

Z

Br(j0)

W u(j)ÿu_j0;r 2r

Qÿm2Q

dj 2

64

3 75

Qÿm2Q

Cr^ÿm2 Z

Br(j0)

W²…jXu(j)j†dj 2

64

3 75

12

: (3:4)

Note thatW(Q=2)W(Q)=2, we conclude the result (3.1) from (3.4).

LEMMA7. Let u2HW^1;1(V;R^N)be a weak solution of Z

V

A^ab_ijX_ju^bX_if^adjˆ0 (3:5)

for anyf2C₀¹(V;R^N), where A^ab_ij is a constant matrix satisfying the strong Legendre-Hadamard condition:

A^ab_ijh^ah^bm_im_j >cj jh²j jm²; h2R^N; m2R^k:

Then u is smooth and there exists C01such that for any Br(j₀)V (3:6) sup

Br=2(j0)uÿuj0;r²‡r²jXuj²‡r⁴X²u²

C0Br(j₀)^ÿ1

G r² Z

Br(j0)

Xu j j²dj:

PROOF. Shores [16] has shown that a weak solutionv2HW^1;2(V;R^N) of (3.5) belongs toC¹in the subsetV₀V, and Wang and Niu [18] have established the following estimate:

Bsupr=2(j0)vÿv_j0;r²‡r²j jXv²‡r⁴ X²v²

C0Br(j₀)^ÿ1

G r² Z

Br(j0)

Xv j j²dj:

(3:7)

Then we can argue as in the proof of Proposition 2.10 in [12] to obtain the conclusion.

4. Caccioppoli type inequality

Denote by Bil (R^kN) the collection of bi-linear forms defined inR^kN, and suppose A 2Bil(R^kN). We say that a function h2HW^1;m(V;R^N) is A-harmonic, if hsatisfies

Z

V

A(Xh;XW)djˆ0; 8W2C₀¹(V;R^N):

(4:1)

Similarly to [9], one can establish the following A-harmonic approx- imation lemma for the case 15m52 in Carnot groups.

LEMMA 8. Let l and L be fixed positive numbers, 15m52, and k;N2N with k2. If for any given e>0, there exists dˆ d(k;N;l;L;e)2(0;1]with the following properties:

(I) for anyA 2Bil(R^kN)satisfying

A(n;n)l nj j² andA(n;n)Lj jn j j;n n;n2R^kN; (4:2)

(II) for any g2HW^1;m(B_r(j₀);R^N)satisfying Br(j₀)

^ÿ1

G

Z

Br(j0)

W(Xg)

j j² dj²1and (4:3)

(4:4)

Br(j₀)^ÿ1

G

Z

Br(j0)

A(Xg;XW)dj

dsup

Br(j0)jXWj; 8W2C₀¹(Br(j₀);fR^N);

then there exists anA-harmonic function h h2Hˆn

h2HW^1;m(B_r(j₀);fR^N)B_r(j₀)^ÿ1

G

Z

Br(j0)

W(Xh)

j j² dj1o

such that

B_r(j₀) ^ÿ1

G

Z

Br(j0)

W gÿh r

² dj²e:

(4:5)

To establish the main result, our first aim is to establish a suitable Caccioppoli type inequality.

LEMMA9 (Caccioppoli Type Inequality). Let u2HW^1;m(V;fR^N)be a weak solution to the system(1.1)under the conditions(H1)-(H3)and(C).

Then for everyj₀ˆ(j¹₀; ;j^r₀)2V; u02 R^N; p02 fR^kN and anyrsat- isfying05r5minf1; dist(j₀; @V)g, it holds

B_r=2(j₀) ^ÿ1

G

Z

Br=2(j0)

V(Xuÿp₀) j j² dj

C_c B_r(j₀)^ÿ1

G

Z

Br(j0)

V uÿu0ÿp0(j¹ÿj¹₀) r

!

2

dj 8>

<

>:

‡ Kÿ

ÿ

1‡j jp0^m=2_s

r^2g‡Br(j₀)^ÿ1

G

Z

Br(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

mÿ1m… †1ÿ¹_r 9

>>

=

>>

;; (4:6)

where we denote K()ˆK u…j j ‡0 j jp0†,j¹ˆ(j¹₁;j¹₂; ;j¹_k)is the horizontal component ofjˆ(j¹; ;j^r)2G,sˆmaxf2m=(mÿ2g);m=(mÿ1ÿg)g>2 and the constant CcˆCc(a;b;c;n;N;m;L;l;M).

PROOF. Let B_r(j₀)V, r=2t5sr, and a standard cut off functionh2C₀¹(B_s(j₀)) satisfy 0h1; j j5rh C

sÿtandh1 onB_t(j₀).

Inspired by the way of [17], we let vˆu(j)ÿu₀ÿp₀(j¹ÿj¹₀), and then define two functions

Wˆhv; fˆ(1ÿh)v;

(4:7) one has

XW‡XfˆXuÿp₀ (4:8)

and

XW

j j^m;jXfj^mC(m) j jXv^m‡ v sÿt ^m

: (4:9)

Using hypothesis (H2), from Lemma 5, and as the elementary inequality 1‡j ja²‡jbÿaj²3 1 ‡j ja²‡j jb²

; (4:10)

we have Z

Bs(j0)

A^a_i(j;u;p0‡XW)ÿA^a_i(j;u;p0)

XiW^adj

ˆ Z

Bs(j0)

Z¹

0

dA^a_i(j;u;p0‡uXW)

du du

2 4

3 5X_iW^adj

ˆ Z

Bs(j0)

Z¹

0

@A^a_i(j;u;p₀‡uXW)

@p^b_j duX_jW^bX_iW^adj

l Z

Bs(j0)

Z¹

0

1‡jp0‡u……XW‡p0† ÿp0†j²

h i_(mÿ2)=2

dujXWj²dj

l Z

Bs(j0)

1‡j jp₀²‡jXWÿp₀j²

_(mÿ2)=2

XW j j²dj

3^(mÿ2)=2l Z

Bs(j0)

1‡j jp₀²‡jXWj²

_(mÿ2)=2

jXWj²dj:

(4:11)

From (4.11), it follows that 3^(mÿ2)=2l

Z

Bs(j0)

1‡j jp0²‡jXWj²

_(mÿ2)=2

XW j j²dj

Z

Bs(j0)

A^a_i(j;u;p0‡XW)ÿA^a_i(j;u;p0)

XiW^adj

ˆ Z

Bs(j0)

A^a_i(j;u;Xu)XiW^adjÿ Z

Bs(j0)

A^a_i(j;u;p0)XiW^adj

‡ Z

Bs(j0)

A^a_i(j;u;p₀‡XW)ÿA^a_i(j;u;Xu)

X_iW^adj

ÿ Z

Bs(j0)

A^a_i(j;u;p₀)ÿA^a_ij;u₀‡p₀(j¹ÿj¹₀);p₀

h i

X_iW^adj

ÿ Z

Bs(j0)

A^a_ij;u₀‡p₀(j¹ÿj¹₀);p₀

ÿA^a_i(j₀;u₀;p₀)

h i

X_iW^adj

ÿ Z

Bs(j0)

Z¹

0

@A^a_i…j;u;Xuÿu(XuÿXWÿp₀)†

@p^b_j duXjf^bXiW^adj

‡ Z

Bs(j0)

B_i(j;u;Xu)W^adj :ˆI‡II‡III‡IV;

(4:12)

where we have used (4.8), (1.8) and the fact that Z

BR(j0)

A^a_i(j₀;u0;p0)XW^a djˆ0:

Noting thatWˆvonB_t(j₀), and (4.10), the left-hand side in (4.12) can be estimated by

(4:13) 3^(mÿ2)=2l Z

Bt(j0)

1‡j jp₀²‡jXWj²

_(mÿ2)=2

jXWj²dj

ˆ3^(mÿ2)=2l Z

Bt(j0)

1‡j jp₀²‡j jXv²

_(mÿ2)=2

Xv j j²dj

C(m)l Z

Bt(j0)

1‡j jp₀²‡jXuj²

_(mÿ2)=4

Xuÿp₀

j j

₂

dj

C(k;N;m;l) Z

Bt(j0)

V(Xu)ÿV(p₀)

j j²dj

C(k;N;m;l;M) Z

Bt(j0)

V(Xv) j j²dj;

where we have applied Lemma 3(4) in the second inequality and Lemma 3(6) for the final inequality.

The structure condition (H3) yields I

Z

Bs(j0)

K() 1… ‡j jp₀†^m=2j jv^gjXWjdj:

(4:14)

To obtaina suitable estimate forI, weneed take the domainB_s(j₀) into four parts:B_s(j₀)\fjv=sj>1g \fjXWj 1g,B_s(j₀)\fjv=sj>1g \fjXWj>1g, Bs(j₀)\fjv=sj 1g \fjXWj>1g, Bs(j₀)\fjv=sj 1g \fjXWj 1g, and will use Young's inequality, (2.11) and (2.12), repeatedly.

CASE1. On the partBs(j₀)\fjv=sj>1g \fjXWj 1g, we see K() 1… ‡j jp0†^m=2 v

s

^gs^gjXWj C(e)jXWj²‡C(e)v

s

^m‡C(e)hK() 1… ‡j jp₀†^m=2s^gi_2m=(mÿ2g) C(e)jV(Xv)j²‡C(e)V v

s

²‡C(e)hK() 1… ‡j jp0†^m=2i_2m=(mÿ2g) s^2g; (4:15)

where we have used the facts^g51 and 2m=(mÿ2g)>2:

CASE2. On the setB_s(j₀)\fjv=sj>1g \fjXWj>1g, it holds K() 1… ‡j jp0†^m=2 v

s

^gs^gjXWj C(e)jXWj^m‡C(e)v

s

^m‡C(e)hK() 1… ‡j jp₀†^m=2s^gi_m=(mÿgÿ1) C(e)jV(Xv)j²‡C(e)V v

s

²‡C(e)hK() 1… ‡j jp₀†^m=2i_m=(mÿgÿ1) s^2g; (4:16)

where we have usedm=(mÿgÿ1)>2:

CASE 3. On the set Bs(j₀)\fjv=sj 1g \fjXWj>1g, observing m=(mÿ1)>2;one has

K() 1… ‡j jp0†^m=2j jv^gjXWj K() 1… ‡j jp₀†^m=2s^gjXWj ejXWj^m‡C(e)

K() 1… ‡j jp0†^m=2s^g_m=(mÿ1) C(e)jV(Xv)j²‡C(e)

K() 1… ‡j jp₀†^m=2_m=(mÿ1) s^2g: (4:17)

CASE4. ForB_s(j₀)\fjv=sj 1g \fjXWj 1g, there is K() 1… ‡j jp0†^m=2j jv^gjXWj

K() 1… ‡j jp0†^m=2s^gjXWj ejXWj²‡C(e)

K() 1… ‡j jp₀†^m=2s^g₂ C(e)jV(Xv)j²‡C(e)

K() 1… ‡j jp₀†^m=2₂ s^2g: (4:18)

Combining these estimations with (4.14), we get IC(e)

Z

Bs(j0)

V(Xv)

j j²dj‡C(e) Z

Bs(j0)

V v s ²dj

‡C(e)hK() 1… ‡j jp0†^m=2i_s Bs(j₀) j j_Gs^2g; (4:19)

wheresˆmax 2m=(mf ÿ2g);m=(mÿ1ÿg)g>2.

Similarly toI, it follows that IIC(e)

Z

Bs(j0)

V(Xv)

j j²dj‡C(e) Z

Bs(j0)

V v s

²dj

‡C(e)hK() 1… ‡j jp₀†^m=2‡gi_m=(mÿ1) B_s(j₀) j j_Gs^2g: (4:20)

By (H1), Lemma 5 and (4.10), it holds

IIIC Z

Bs(j₀)

Z¹

0

1‡jXu‡u‰…XuÿXf†ÿXuŠj²

_(mÿ2)=2

du 2

4

3

5jXfjjXWjdj

C Z

B_s(j₀)

1‡jXuj²‡jXuÿXfj²

_(mÿ2)=2

Xf j jjXWjdj

C Z

Bs(j0)

1‡jXuj²‡jXfj²

(mÿ2)=2

Xf j jjXWjdj:

(4:21)

Noting thatXfBsnBt andÿ1=25(mÿ2)=250, we split the domain Bs(j0) into four parts:Bs(j0)\fjXfj>1g\fjXWj>1g,Bs(j0)\fjXfj 1g\

XW j j 1

f g, Bs(j0)\fjXfj>1g \fjXWj 1g, and Bs(j0)\fjXfj 1g \ XW

j j>1

f g. Similarly toI, it follows that

IIIC Z

Bs(j0)nBt(j0)

V(Xuÿp0) j j²dj‡

Z

Bs(j0)

V v sÿt

²dj 0

B@

1 CA:

(4:22)

Using HoÈlder inequality, one has IV C

Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †^1ÿ1rj jdjW

C Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

Z

Bs(j0)

W j j^rdj 0

B@

1 CA

1r

: (4:23)

Analogously asI, we take the domainBs(j₀) into two parts.

CASE1. ForBs(j₀)\fj jXv >1g, by Sobolev type inequality, Young's inequality and (2.12) it follows that

Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

Z

Bs(j0)

W j j^rdj 0

B@

1 CA

1r

C Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

Z

Bs(j0)

Xv j j^mdj 0

B@

1 CA

m1

C(e) Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

… †mÿ1^m

‡Ce Z

Bs(j0)

Xv j j^mdj 0

B@

1 CA

C(e) Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

… †mÿ1^m

‡Ce Z

Bs(j0)

V Xv… † j j²dj 0

B@

1 CA:

(4:24)

CASE2. On the setB_s(j₀)\fj j Xv 1g, we have Xv

j j^mj jXv²‡1;

and then it follows that Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

Z

Bs(j0)

W j j^rdj 0

B@

1 CA

1r

C(e) Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

… †mÿ1^m

‡Ce Z

Bs(j0)

V Xv… † j j²dj 0

B@

1

CA‡CejBs(j₀)j_G; (4:25)

where we have used (2.11).

Combining these estimates inIV, we have

(4:26) IV Ce Z

Bs(j0)

V Xv… † j j²dj 0

B@

1 CA

‡C(e) Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

… †mÿ1^m

‡CejBs(j₀)j_G:

Substituting (4.13), (4.19), (4.20), (4.22) and (4.26) into (4.12), we finally arrive at

C1

Z

Bt(j0)

V(Xv) j j²dj

C2

Z

Bs(j0)nBt(j0)

V(Xv)

j j²‡V v sÿt

²

dj

‡C₃ Z

Bs(j0)

V(Xv)

j j²‡V v sÿt

²

dj‡C₄hK() 1 ‡j jp₀^m=2i_s B_s(j₀) j j_Gs^2g

‡C₅ Z

Bs(j0)

Xu

j j^m‡j ju^r‡1

… †dj

2 64

3 75

1ÿ¹_r

… †mÿ1^m

:

The proof is now completed by applying Lemma 4.

REMARK1. The functionvtaken in the proof is different from that in [14], essentially.

5. Proof of the main result

In this section, we will complete the proof of Theorem 1 via four steps, and hence consideru2HW^1;m(V;R^N) (15m52) to be a weak solution of the system (1.1).

FIRST STEP. We claim that if r1 and W2C¹₀ (B_r(j₀);fR^N) with

Bsupr(j0)jXWj 1, then there exist some constants C₆ˆC₆(m;M;C_P)>1 such that

B_r(j₀) ^ÿ1

G

Z

Br(j0)

A^a_i;pj

b(j₀;u₀;p₀)(Xuÿp₀)XW^a dj C₆ sup

Br(j0)jXWj v¹²j j;p₀ F¹²(j₀;r;p₀)

F¹²(j₀;r;p₀)‡ h

‡F(j₀;r;p₀)‡r^gF(j j ‡u₀ j j)p₀ i

; (5:1)

where we denoteF(t)ˆ[K(t)(1‡t)^s0]^mÿg2m withs⁰ˆmaxn

m;rÿ1;m 2 ‡go

. PROOF. Noting the fact

Z

Br(j0)

" Z¹

0

A^a_i;pj

b(j₀;u0;uXu‡(1ÿu)p0)(Xuÿp0)du

# XW^adj

ˆ Z

Br(j0)

" Z¹

0

d

duA^a_i(j₀;u0;uXu‡(1ÿu)p0)du

# XW^adj

ˆ Z

Br(j0)

[A^a_i(j₀;u₀;Xu)ÿA^a_i(j₀;u₀;p₀)]XW^adj

ˆ Z

Br(j0)

hA^a_i(j₀;u0;Xu)ÿA^a_i(j;u;Xu)i

XW^adj‡ Z

Br(j0)

B^a(j;u;Xu)W^adj;

(5:2)

we have (5:3)

Z

Br(j₀)

A^a_i;pj

b(j0;u0;p0)(Xuÿp0)XW^a dj

ˆ Z

Br(j0)

Z¹

0

A^a_i;pj

b(j0;u0;p0)du(Xuÿp0) 2

4

3 5XW^adj

Z

Br(j₀)

Z¹

0

A^a_i;pj

b(j0;u0;p0)ÿA^a_i;pj

b(j0;u0;uXu‡(1ÿu)p0)jXuÿp0jduo

Bsup_r(j₀)jXWjdj

‡ Z

Br(j0)

A^a_i(j0;u0;Xu)ÿA^a_i(j;u0‡p0(j¹ÿj¹₀);Xu)

sup

Br(j₀)jXWjdj

‡ Z

B_r(j₀)

A^a_i(j;u0‡p0(j¹ÿj¹₀);Xu)ÿA^a_i(j;u;Xu)

sup

Br(j₀)jXWjdj

‡ Z

Br(j₀)

(a pj j^m(1ÿ1r)‡b uj j^rÿ1‡c)j jW dj :ˆI⁰‡II⁰‡III⁰‡IV⁰:

Using (H1) and the estimate (1.6) yields (Note thatmÿ250) (5:4) I⁰

Z

B_r(j₀)

Z¹

0

A^a_i;pj

b(j0;u0;p0)ÿA^a_i;pj

b(j0;u0;p0‡u(Xuÿp0))

^12‡12dt 2

4

3

5jXuÿp0jdj

C Z

Br(j₀)

Z¹

0

1‡j jp0²

^mÿ2₂

‡1‡jp0‡u(Xuÿp0)j^2mÿ2₂

¹₂

8<

:

1‡j jp0²‡jp0‡u(Xuÿp0)j^2mÿ2₂

v…j j;p0 ju(Xuÿp0)j†

¹₂

dt )

Xuÿp0

j jdj C

Z

Br(j₀)

1‡j jp0²‡jXuj^{2 mÿ2}₄

v¹²…j j;p0 jXuÿp0j†jXuÿp0jdj

C Z

Br(j0)

1‡jXuÿp0j^mÿ22

Xuÿp0

j jv¹²…j j;p0 jXuÿp0j†dj:

Let

B₁ˆ:B_r(j₀)\fjXuÿp₀j 1g; B₂ˆ:B_r(j₀)\fjXuÿp₀j>1g:

(5:5)

Then it follows that by first HoÈlder's inequality and then Jensen's in- equality

(5:6) I⁰ C Z

B1

Xuÿp₀

j jv¹²…j j;p₀ jXuÿp₀j†dj

‡C Z

B2

Xuÿp0

j j^m2v¹²…j j;p0 jXuÿp0j†dj

C Z

B1

Xuÿp₀ j j²dj 0

B@

1 CA

12 Z

Br(j0)

v…j j;p₀ jXuÿp₀j†dj 0

B@

1 CA

12

‡C Z

B2

Xuÿp0

j j^mdj 0

B@

1 CA

12 Z

Br(j0)

v…j j;p0 jXuÿp0j†dj 0

B@

1 CA

12

C Z

Br(j0)

V(Xu)ÿV(p₀)

j j²dj

0 B@

1 CA

12 Z

Br(j0)

v…j j;p₀ jXuÿp₀j†dj 0

B@

1 CA

12

‡C Z

Br(j0)

V(Xu)ÿV(p0)

j j²dj

0 B@

1 CA

12 Z

Br(j0)

v…j j;p0 jXuÿp0j†dj 0

B@

1 CA

12

2C B r(j₀)

G Br(j₀)^ÿ1

G

Z

Br(j0)

V(Xu)ÿV(p0)

j j²dj

0 B@

1 CA

12

‡ B_r(j₀)^ÿ1

G

Z

Br(j0)

v…j j;p₀ jXuÿp₀j†dj 0

B@

1 CA

12

C B _r(j₀)

GF¹²(j₀;r;p₀)v¹² j j;p₀ B_r(j₀)^ÿ1

G

Z

B1

Xuÿp₀ j j²dj

!¹₂

‡ B_r(j₀)^ÿ1

G

Z

B2

Xuÿp₀ j j^mdj

!_m¹!