Regularity results for degenerate elliptic systems

www.elsevier.com/locate/anihpc

Regularity results for degenerate elliptic systems

Michael Pingen

Fachbereich Mathematik, University of Duisburg-Essen, Lotharstr. 65, 47048 Duisburg, Germany Received 22 December 2006; accepted 12 February 2007

Available online 1 September 2007

Abstract

We prove regularity results for certain degenerate quasilinear elliptic systems with coefficients which depend on two different weights. By using Sobolev- and Poincaré inequalities due to Chanillo and Wheeden [S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985) 1191–1226; S. Chanillo, R.L. Wheeden, Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986) 1111–1134] we derive a new weak Harnack inequality and adapt an idea due to L. Caffarelli [L.A. Caffarelli, Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35 (1982) 833–838]

to prove a priori estimates for bounded weak solutions. For example we show that every bounded weak solution of the system

−D_α(a^αβ(x, u,∇u)D_βuⁱ)=0 with|x|²|ξ|²a^αβξ_αξ_β |x|^τ|ξ|²,|x|<1,τ∈(1,2)is Hölder continuous. Furthermore we derive a Liouville theorem for entire solutions of the above systems.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

Nous prouvons des résultats de régularité pour certains systèmes elliptiques quasi linéaires dégénérés avec des coefficients dépendant de deux poids différents. En employant des inégalités de Sobolev- et Poincaré dues à Chanillo et Wheeden [S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math.

107 (1985) 1191–1226 ; S. Chanillo, R.L. Wheeden, Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986) 1111–1134] nous déduisons une nouvelle inégalité de Harnack et adaptons une idée due à L. Caffarelli [L.A. Caffarelli, Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35 (1982) 833–838] pour prouver des évaluations a priori pour des solutions limitées et faibles. Par exemple, chaque solution limitée et faible du système−D_α(a^αβ(x, u,∇u)D_βuⁱ)=0 avec|x|²|ξ|²a^αβξ_αξ_β|x|^τ|ξ|²,|x|<1, τ∈(1,2)est continue selon Hölder. De plus, nous déduisons un théorème de Liouville pour les solutions entières des systèmes ci-dessus.

©2007 Elsevier Masson SAS. All rights reserved.

MSC:35D10; 35J70; 35J45

Keywords:Regularity of weak solutions; Degenerate elliptic systems; Harnack inequality; Liouville theorems

✩ This research was supported by the Deutsche Forschungsgemeinschaft.

E-mail address:michael.pingen@uni-duisburg-essen.de.

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.02.008

1. Introduction

We consider weak solutions of degenerate elliptic systems of the form

−D_α

A^αβ(x, u,∇u)D_βuⁱ

=fⁱ(x, u,∇u) (i=1, . . . , m) (1)

in a domain Ω ⊂Rⁿ, where a^αβ(x):=A^αβ(x, u(x),∇u(x)) are measurable and symmetric coefficients and f (x, u,∇u)is a measurable function. Here and in the sequel, we use the summation convention: repeated Greek indices are to be summed from 1 ton, repeated Latin indices from 1 tom. We assume there exist measurable weights v(x), w(x) >0 a.e. inΩ with the property

w(x)|ξ|²a^αβ(x)ξ_αξ_βv(x)|ξ|² ∀ξ∈Rⁿ. (2)

Furthermore we require the following structure conditions:

1. sup_Ω|u|M <∞.

2. |f (x, u, p)|aQ(x, p)andu(x)·f (x, u, p)a^∗Q(x, p)for a.e.x∈Ωand for allp∈R^n×mwith somea0, a^∗∈R, whereQ(x, p):=a^αβ(x)pⁱ_αpⁱ_β.

The notion of a weak solution of (1) will be defined in Section 4; to prove regularity for weak solutions of (1) the weightsvandwhave to satisfy three further conditions, which we will state exactly in Section 2. Roughly speakingw andz:=^v_w² have to be doubling weights and have to fulfill a weighted Poincaré- and a weighted Sobolev inequality. We will show that the weightsv(x)= |x|andw(x)= |x|^τ withτ∈ [1,2)inB₁(0)⊂Rⁿ,n3 satisfy these conditions.

Optimal regularity results for weak solutions of uniformly elliptic systems of type (1) are well known and due to Hildebrandt and Widman [9], Wiegner [17,18] and Caffarelli [3]. For the case of equal weights, i.e.v=w, which be- long to the Muckenhoupt classA₂(see Section 2 for explicit definitions), Fabes, Kenig and Serapioni [7] have proven Hölder continuity for weak solutions of an elliptic equation. For certain different weights, Chanillo and Wheeden [5] proved regularity for weak solutions of elliptic equations, while for degenerate elliptic systems only very little is known. Baldes [1] and Baoyao [2] proved some results for equal weights, e.g. weak solutions of systems with bounded weightsv=w∈A₂are Hölder continuous provided the smallness conditiona^∗+aM <1 holds. The results in this paper are of much more general nature than in [1] or [2], and, in fact, are the first regularity results for singular systems with different weights.

Our proof uses an idea of L. Caffarelli [3] who proved a priori estimates for weak solutions of certain uniformly el- liptic systems. His main tool was a weak Harnack inequality for supersolutions of a uniformly elliptic linear equation;

we will prove such a Harnack inequality for solutions of degenerate (in the above sense) elliptic equations in Sec- tion 3. The proof of this Harnack inequality is based upon a method of Trudinger [16] in which a Harnack inequality for solutions of some mildly degenerate elliptic equations was shown. Our regularity result reads as follows:

Theorem 1.1.Letube a bounded, weak solution of (1)inΩ⊂Rⁿ. The coefficientsa^αβare required to fulfill(2)with admissible weightswandv(see Section2). Under the assumptiona^∗+aM <2uis Hölder continuous and for every ΩΩthere exist constantsC=C(n, a, a^∗, M, Ω, Ω) >0andα=α(n, a, a^∗, M) >0, such that

[u]α,ΩC. (3)

In the last section we also show a Liouville theorem for entire solutions of elliptic systems, whose coefficients are degenerate in an arbitrary large compact subset ofRⁿand uniformly elliptic outside this compact set, more precisely:

Theorem 1.2.Letube a bounded, weak solution of (1)inRⁿ. The coefficientsa^αβare assumed to be of type(2)in a ballB_R(0)⊂Rⁿwith admissible weightswandvand to be uniformly elliptic outside this ball. Ifa^∗+aM <2, then u=const. a.e. inRⁿ.

This result extends a Liouville theorem for uniformly elliptic systems due to Hildebrandt and Widman [10] and Meier [11].

2. The Muckenhoupt classesAp and conditions for the weights

The Muckenhoupt classes are defined in the paper [12] by Muckenhoupt in connection with Hardy functions. Let w∈L¹_loc(Rⁿ)be a nonnegative function.

Definition 2.1.Let 1< p <∞. The weightwis an element ofA_p, if sup

B_R⊂Rⁿ

1

|B_R|

BR

w(x) dx 1

|B_R|

BR

w(x)^p−−11dx p−1

=:C_p<∞, (4)

wis to be said of classA_∞, if for every >0 there exists aδ >0 with the property that for every measurableE⊂B_R with|E|< δ|B_R|the inequalityw(E)w(B_R)holds, wherew(E)=

Ew(x) dx.

From [13] and [6] we inferA_∞=

p>1Ap. A result due to Muckenhoupt and Wheeden [14], p. 223 implies the doubling propertyfor anyw∈A_∞:

w(B_2R)Kw(B_R) with someK >0. (5)

We require the following conditions for the weightswandz=^v_w² (cf. [5]):

(1) w, z∈D_∞, i.e. the doubling property holds:w(B_2R)Cw(B_R)andz(B_2R)Cz(B_R)with a constantC >0 independent ofR.

(2) The following Poincaré inequality holds: There exists ak >1 such that for allB_R⊂Ω and allf ∈C¹(B_R)the inequality

1 z(B_R)

B_R

f − 1 z(B_R)

B_R

f z dx ^2kz dx

_2k¹

CR

1 w(B_R)

B_R

|∇f|²w dx ¹₂

(6) holds with a constantCindependent off.

(3) The following Sobolev inequality holds: There exists ak >1 such that for allB_R⊂Ω and allf ∈C₀¹(B_R)the inequality

1 z(B_R)

B_R

|f|^2kz dx ¹

2k CR

1 w(B_R)

B_R

|∇f|²w dx ¹

2

(7) holds with a constantCindependent off.

Fabes, Kenig and Serapioni [7] showed that in the casev=w∈A2conditions (2) and (3) are satisfied. In the case of different weights, Chanillo and Wheeden [4] proved that condition (2) and (3) hold, ifw∈A2,z∈D_∞and if there is aq >2 such that for all ballsBR, whose centers are inB2R, thebalance condition

s

z(B_sR) z(B_R)

¹

q C

w(B_sR) w(B_R)

¹

2

(8) holds for alls∈(0,1).

3. A weak Harnack inequality

To give a definition of a weak solution of a degenerate elliptic equation D_α

a^αβ(x)D_βu

=0 (9)

with coefficientsa^αβ(x)which satisfy (2) we first need to define the spaceH₂¹(Ω, v, w), wherevandware weights with the properties (1)–(3) of Section 2.

Definition 3.1.H₂¹(Ω, v, w)is defined as completion ofC¹(Ω)with respect to the norm u _1,2,Ω=

Ω

a^αβ(x)D_αuⁱD_βuⁱdx+

Ω

u²v dx.

H˚₂¹(v, w, Ω)denotes the completion ofC¹_c(Ω)with respect to the norm u 1,2,0,Ω=

Ω

a^αβ(x)DαuⁱDβuⁱdx.

Remark.It is possible to estimate · 1,2,Ωas follows:

Ω

|∇u|²w dx+

Ω

u²v dx u ²_1,2,Ω

Ω

|∇u|²v dx+

Ω

u²v dx <∞.

If uk∈C¹(Ω)is a sequence withuk→uinH₂¹(Ω, v, w), then uk and∇uk converge inL2(Ω, v)andL2(Ω, w) resp. If limk→∞∇u_k=v, define∇u:=v;∇uis well defined (cf. [5], §2).

Definition 3.2.u∈H₂¹(Ω, v, w)is a weak subsolution of (9), if

Ω

a^αβ(x)D_βuD_αφ dx0 (10)

holds for everyφ∈H˚₂¹(Ω, v, w), φ0.uis called a weak supersolution, if−uis a weak subsolution anduis called a weak solution, ifuis a weak subsolution and a weak supersolution.

The main result of this section is

Theorem 3.3. Let u be a nonnegative weak supersolution of (9) in Ω ⊂Rⁿ. Then for any ball B_R ⊂Ω with

z(BR)

w(BR)C₁and anyα, β, γ satisfying0< α < β <1,0< γ < kthe estimate 1

z(B_βR)

B_βR

|u|^γz dx ¹

γ C(n, α, β, γ , C1)inf

BαR

u (11)

holds, wherek >1is the constant from the Sobolev- and Poincaré inequalities.

The proof of Theorem 3.3 is divided into three lemmatas, extended proofs of these lemmatas can be found in [15].

All these lemmatas are based on a method developed by Trudinger [16].

Lemma 3.4.Letube a weak subsolution of (9)inΩ⊂Rⁿ. Then for everyB_R⊂Ωwith_w(B^z(BR)

R) C₁we have for any 0< α < β <1the estimate

sup

B_αR

uC(n, α, β, C₁) 1

z(B_βR)

BβR

|u⁺|²z dx ¹

2

. (12)

Proof. Forδ1 and 0< N <∞we define F (u)=F_δ^N(u)=

(u⁺)^δ, uN, δN^δ−1u−(δ−1)N^δ, u > N.

Useφ(x)=η²(x)F (u),η0, η∈C_c¹(B_R)as test function in (10). We arrive at

Ω

η²F(u)|∇u|²w dx2

Ω

η|η_x|F|∇u|v dx. (13)

The inequalityF (u)u⁺F(u)is easily derived; by using this relation, the Hölder inequality yields

Ω

η²(x)F(u)|∇u|²w dxC

Ω

η_x²(u⁺)²Fz dx. (14)

Define G(u):=

u 0

F(t ) ¹²dt= √

δ_δ+²₁|u⁺|^δ+12 , uN,

√δN^δ−21|u|, u > N.

With (14) we infer

Ω

η²|∇G|²w dxC

Ω

η²_x(u⁺G)²z dx.

In connection with the Sobolev inequality andGu⁺Gthis estimate implies 1

z(BR)

BR

|ηG|^2kz dx ¹

2k CR

z(B_R) w(BR)

√

C₁

1 z(BR)

BR

η_x²(u⁺G)²z dx ¹

2

. (15)

Setq:=^δ+₂¹ and take theqth root of (15). Then chooseandσ in a way thatα < σβ andηin a way that suppη⊂B_{σ R},η≡1 inB_R,|η_x|_(σ−²_)R. IfN= ∞we seeG(u)=^√_q^δ(u⁺)^q; by using the doubling property for zwe obtain

1 z(B_R)

B_R

(u⁺)^2kqz dx ¹

2kq

Cq σ−

¹

q 1 z(B_βR)

B_{σ R}

(u⁺)^2qz dx ¹

2q

. (16)

Iteration of (16):

Defineq0:=1,qi :=kqi−1=kⁱ, furthermore seti=α+(β−α)¹⁺ⁱ,σi=i−1. With this choice ofqi andi

we infer sup

B_αR

u^∞

l=0

Cql

_l−_l+1

¹

ql 1 z(B_βR)

BβR

(u⁺)²z dx ¹

2

. (17)

We can estimate the infinite product in (17) by using the geometric sum. Thus, we have sup

BαR

uC(n, α, β, C1) 1

z(B_βR)

B_βR

|u⁺|²z dx ¹₂

.

This completes the proof of Lemma 3.4. 2

Lemma 3.5.Under the hypotheses of Theorem3.3andα < β, we have 1

infBαRuexp

C− 1

z(B_βR)

BβR

loguz dx

. (18)

Proof. W.l.o.g. we assumeu >0 (in caseu0 we use Levi’s Theorem to derive the assertion). Testing (10) with the functionφ(x)=η(x)u⁻¹(x),η∈C_c¹(Ω), η0 yields the estimate

Ω

a^αβ(x)D_βuD_αηu⁻¹dx−

Ω

a^αβ(x)D_βuD_αuηu⁻²dx0.

Setv:=log(_u^t), wheretdenotes a positive constant which will be specified later. We see thatvis a weak subsolution of (9) and with Lemma 3.4 we infer

sup

B_αR

vC 1

z(B_βR)

BβR

|v⁺|²z dx ¹

2

. (19)

To estimate the right-hand side of (19) we test (10) withφ(x)=η²(x)u⁻¹(x),η∈C_c¹(Ω). With (2) and the Hölder inequality we arrive at

Ω

η²u⁻²|∇u|²w dxC

Ω

η|η_x ∇u|u⁻¹v dxC

Ω

η_x²z dx ¹

2

Ω

η²|∇u|²u⁻²w dx ¹

2

.

It follows

Ωη²u⁻²|∇u|²w dxC

Ωη²_xz dx.

Chooseηin a way thatη≡1 inBβR, suppη⊂BR,|ηx|₍₁₋²_β)R. From the last inequality we conclude together with the doubling property and the fact|∇v|²=u⁻²|∇u|²the estimate

BβR

|∇v|²w dxC 1

R²

BβR

z dx

.

We definet by means of logt=_z(B¹_βR)

B_βRloguz dx, then the weighted mean value ofv is zero and the Poincaré inequality in connection with the above inequality yields

1 z(BβR)

BβR

|v|^2kz dx ¹

2k C

z(B_βR) w(BβR)

1 z(BβR)

BβR

z dx ¹

2 C(n, β, C₁).

Combining this estimate with (19) we infer sup

B_αR

v=logt+log 1

infBαRu

C.

By considering the definition oft we finally arrive at

Binf_αRu ₋1

exp

C− 1

z(B_βR)

B_βR

loguz dx

. 2

Lemma 3.6.Under the hypotheses of Theorem3.3andα < β, we have 1

z(B_αR)

BαR

|u|^γz dx ¹

γ exp

C+ 1

z(B_βR)

BβR

loguz dx

. (20)

Proof. We may again assumeu >0. Setf=v⁻=log(^u_t)⁺(for the definition oft see the proof of Lemma 3.5) and test the weak formulation withφ(x)=η²(x)u⁻¹(x)(f^δ(x)+(2δ)^δ), whereδ1,η∈C_c¹(B_R),η0. By using the ellipticity condition we conclude

Ω

η²u⁻²

f^δ+(2δ)^δ−δf^δ−1

|∇u|²w dxC

Ω

η|η_x|u⁻¹

f^δ+(2δ)^δ

|∇u|v dx.

Now we use the inequalityδf^δ−11₂(f^δ+(2δ)^δ)in connection with|∇f|²=u⁻²|∇u|²and the Hölder inequality to infer

Ω

η²

f^δ+(2δ)^δ

|∇f|²w dxC

Ω

η²_x

f^δ+(2δ)^δ z dx.

By using once againδf^δ−11₂(f^δ+(2δ)^δ)and taking the elementary inequalityf^δ+(2δ)^δ2(f^δ+1+(2δ)^δ)into account we obtain

δ

Ω

η²f^δ−1|∇f|²w dxC

Ω

η²_x

f^δ+1+(2δ)^δ

z dx. (21)

Letq:=^δ+₂¹>1, by applying the Sobolev inequality toηf^q∈H˚₂¹(B_R, v, w)we find under consideration of (21) 1

z(B_R)

B_R

|ηf^q|^2kz dx ¹

2k CR

1 w(B_R)

B_R

η²_xf^δ+1+η²(δ+1)²f^δ−1|∇f|² w dx

¹

2

C√

qR

z(B_R) w(B_R)

√

C₁

1 z(B_R)

B_R

(ηxf^q)²z dx+(2δ)^δsup

BR

|ηx|^{2 1}

2

.

Chooseandσ in a way thatα < σβ andηin a way that suppη⊂Bσ R,η≡1 inBR and|ηx|_(σ−²_)R. With this choice of, σ andη, taking theqth root in the last estimate yields

1 z(B_R)

BR

f^2qkz dx _2kq¹

Cq^1q(σ−)^−1q

Cq+ 1

z(B_{σ R})

Bσ R

f^2qz dx _2q¹

. (22)

Now setq_i=kⁱ1,_i=α+2⁻ⁱ(β−α),σ_i=_i+2⁻ⁱ(β−α), we obtain 1

z(B_iR)

B_iR

f^2ki+1z dx ¹

2ki+1

(C2ⁱkⁱ)

1 ki

Ckⁱ+

1 z(Bσ_iR)

B_{σi R}

f^2kiz dx ¹

2ki .

In the next step we iterate this inequality; afteri−1 iteration steps we arrive at 1

z(B_iR)

B_{i R}

f^2ki+1z dx ¹

2ki+1

i j=1

Ck^j i l=j

(Ck^l2^l)

1 kl +

i j=1

(Ck^j2^j)

1 kj

1 z(BβR)

BβR

f^2kz dx ¹

2k

. (23)

We estimate the series and products in (23) and then we find with the doubling property and the Hölder inequality the following estimate for allp >2k:

1 z(B_αR)

B_αR

f^pz dx ¹

pC

p+ 1

z(B_βR)

B_βR

f^2kz dx _2k¹

. (24)

By considering the power series expansion of e^p0f forp0∈(0,e⁻¹)we infer by using (24) and the Stirling approxi- mationn! ≈√

2π n(ⁿ_e)ⁿfor largenthe estimate 1

z(B_αR)

B_αR

e^p0fz dxCe⁽

1 z(BβR )

BβRf^2kz dx)^2k1

. (25)

Sincef =v⁻the right-hand side of (25) is bounded by the proof of Lemma 3.5. Thus 1

z(B_αR)

BαR

e^p0fz dx ¹

p0

C. (26)

In the remainder of the proof we have to estimate ^u_t Lγ(z,BαR)by ^u_t Lp0(z,B_αR)(α < α< β). For this, we remark that−uis a subsolution of (9); by modifying the functionF (u)appearing in the proof of Lemma 3.4 in the sense that δ∈(−1,0)we see that the estimate (16) holds also for q∈(0,¹₂). For the iteration process we set q₀:=_2k^γ ,

q_i := ^qi_k⁻¹ →0, i→ ∞. After finitely many iteration steps we achieve 2qi < p₀<e⁻¹. From (26) and (16) (for q∈(0,1/2)) we infer with the definition oft

1 z(B_αR)

B_αR

u^γz dx ¹

γ exp

C+ 1

z(B_βR)

B_βR

loguz dx

. 2

Proof of Theorem 3.3. Multiply (18) and (20). 2

4. Results for weak solutions of degenerate elliptic systems

Now we define what we will understand under aweak solutionof a system of type (1):

Definition 4.1.u∈H₂¹(Ω, v, w,R^m)is called a weak solution of (1), if

Ω

a^αβ(x)D_βuD_αφ dx=

Ω

f (x, u,∇u)φ dx (27)

holds for allφ∈H˚₂¹(Ω, v, w,R^m).

For the proof of Theorem 1.1 we now can use an idea of L. Caffarelli [3]. In fact we only have to replace the weak Harnack inequality for weak supersolutions of uniformly elliptic equations by the weak Harnack inequality proven in Section 3 (Theorem 3.3).

Examples.

1) v(x)=w(x)= |x|^α, x∈B_R(0)⊂Rⁿ andα >−n. Ifα∈(−n, n) it is easy to show thatv=w∈A₂ and if α >−n+2 we can interpret|x|^α as a weight which arises from a quasiconformal mapping (cf. [7], pp. 105–

112). This weight has also the properties which were needed in the proof of Theorem 3.3 (cf. [7]) and so it is an admissible weight for the system (1).

2) v(x)=w(x)=(log|x|)^k,x∈B1/2(0)⊂Rⁿ,k∈2N.

3) v(x)=w(x)= |x|^α(log|x|)²,x∈B1/2(0)⊂Rⁿ,α∈(−n, n).

4) v(x)= |x|,w(x)= |x|^τ,τ ∈(1,2),x∈B1(0)⊂Rⁿ, n3. It is obvious thatw∈A2,z= |x|^2−τ∈D_∞. In view of a result due to Chanillo and Wheeden [4] it is enough to show that the balance condition (8) holds. We remark that forα >0 anda∈B_R(0)there are positive constantsc₁andc₂with the property

c1Rⁿ

R+ |a|α

BR(a)

|x|^αdxc2Rⁿ

R+ |a|α

. (28)

From (28) we infer _w(B^z(BR)

R)C₁; forq∈(2,_n+²ⁿ_τ−2]we have for anys∈(0,1)the estimates s

z(BsR(a)) z(BR(a))

¹

q Css^nq and

w(BsR(a)) w(BR(a))

¹

2 sⁿ²s^τ2.

Sincess^nq Cs^n+2τ the validity of (8) is shown.

Liouville theorem for entire solutions.Here, we assume the coefficientsa^αβ(x)satisfy the estimate 1

Cs(x)|ξ|²a^αβ(x)ξαξβCt (x)|ξ|² (29)

withC1 and s(x)=

w(x), |x|< L,

1, |x|L, t (x)=

v(x), |x|< L, 1, |x|L, wherewandvare weights which satisfy the conditions of Section 2.

The proof of the Liouville Theorem uses an idea of Meier [11], who proved the corresponding Liouville theorem for uniformly elliptic systems. First we have to consider some lemmatas:

Lemma 4.2.Letube a bounded, weak solution of (1)in a domainΩ⊂Rⁿ. Ifa^∗<1andξ ∈R^m is a vector with

|ξ|¹⁻_a^a∗, then−Dα(a^αβ(x)Dβ|u−ξ|²)0inΩ.

Proof. We useφ=η(u−ξ ),η∈C_c^∞(Ω),η0 as a test function in the weak formulation (27) and take the structure conditions of the introduction into account. 2

With the notationz1(x):=^t_s(x)^2(x)we can formulate the next lemmatas.

Lemma 4.3. Let B_4L(0)⊂Ω and u be a bounded, weak, nonnegative supersolution of D_α(a^αβ(x)D_βu)=0 in Ω⊂Rⁿ with coefficientsa^αβ(x)of the form(29), furthermore let ^z_s(B^1(BL)

L) C1. Then we have for anyR >0 with B4R(0)⊂Ωthe estimate

1 z₁(B_2R)

B_2R(0)

uz1dxC(n, C1) inf

BR(0)u.

Proof. If B_4R(0)⊂B_L(0), the lemma is a direct consequence of Theorem 3.3 with γ =1 and suitable α, β. If B_L(0)⊂B_4R(0)we can prove similarly to [8], pp. 195–198 a Harnack inequality for supersolutions ofLu=0 with uniformly elliptic coefficients on annular regionsB_4S−B_S(SL), i.e.

1 z₁(B_β1S−B_β2S)

B_β

1S−B_β

2S

uz1dxC inf

B_α

1S−B_α

2S

u (30)

with 1< β₂< α₂< α₁< β₁<4.

The main difference in the proof of (30) compared with [8] is to construct suitable test functions on the correspond- ing annular regions.

Chooseα, α₁, β, β₁in a way that 1< α < α₁<2,α₁< β₁< β <4 andB_α1L⊂B_β1R. We conclude 1

z₁(B_βR)

B_βR

uz₁dx= 1

z₁(B_βR−B_αL)+z₁(B_αL)

B_βR−B_αL

uz₁dx+

B_αL

uz₁dx

1

z₁(B_βR−B_αL)

B_βR−B_αL

uz₁dx+C 1 z₁(B_2L)

B_2L

uz₁dx

C inf

Bβ1R−Bα1L

u+C inf

Bα1L

uCinf

BR

u.

Here, we used (30) and Theorem 3.3.

IfB_L(0)⊂B_2R(0)we chooseβ=2, β1=3/2, α1=5/4, α=9/8 to arrive at the assertion. IfB_L(0)⊂B_2R(0)we choose someβ∈(2,4)withB_L(0)⊂B_βR(0); the doubling property ofz₁yields the desired estimate. 2

Lemma 4.4.Letu be a weak solution of−D_α(a^αβD_βu)0inB_4R(0)⊂Rⁿ with coefficients of the form(29). If

z₁(B_R)

s(B_R) C1, then there is a constantδ(n, C1)∈(0,1)with the property sup

BR(0)

u(1−δ) sup

B4R(0)

u+δ 1 z₁(B_R)

B_R(0)

uz₁dx.

Proof. From Lemma 4.3 we infer for the nonnegative supersolution sup_B4R(0)u−uthe estimate 1

z1(B2R)

B2R

sup

B_4R

u−u

z₁dxCinf

BR

sup

B_4R

u−u

.

With the help of the doubling property we can estimate the left-hand side from below by C˜ 1

z1(BR)

BR

sup

B_4R

u−u

z₁dx

and we infer C˜ Csup

B_4R

u−C˜ C

1 z₁(B_R)

BR

uz₁dxsup

B_4R

u−sup

B_R

u. 2

Lemma 4.5.Letube a bounded, weak solution of(1)inRⁿwith coefficientsa^αβ(x)of the form(29). If^z_s(B^1(BR)

R) C₁ for someR^L₂ anda^∗<1, then we have

(i) limR→∞ 1 z1(BR)

BR(0)u(x)z₁dx=: ¯u_∞exists and| ¯u_∞| =sup_Rn|u| =M.

(ii) limR→∞ 1 z₁(B_R)

B_R(0)|u− ¯u_∞|²z1dx=0.

(iii) sup_Rⁿ|u−ξ| = | ¯u_∞−ξ| ∀ξ∈R^mwith|ξ|¹⁻_a^a∗.

Proof. (i) In view of Lemma 4.2 we have−D_α(a^αβ(x)D_β|u−ξ|²)0∀ξ∈R^mwith|ξ|¹⁻_a^a∗. From Lemma 4.4 we infer by letting R→ ∞the estimate sup_Rⁿ|u−ξ|²limR→∞ 1

z₁(B_R)

B_R(0)|u−ξ|²z1dx. It’s obvious that the reverse inequality is also true. Thus,

Rlim→∞

1 z₁(B_R)

B_R(0)

|u−ξ|²z₁dx=sup

Rⁿ |u−ξ|². (31)

Since 1 z1(BR)

BR(0)

|u−ξ|²z₁dx= 1 z1(BR)

BR(0)

|u|²z₁dx−2ξ· 1 z1(BR)

BR(0)

uz₁dx+ |ξ|²

we see in view of (31) that limR→∞ξ·_z1(B¹_R)

B_R(0)uz1dxexists and we infer sup

Rⁿ |u−ξ|²=M²+ |ξ|²−2ξ· lim

R→∞

1 z₁(B_R)

B_R(0)

uz1dx. (32)

Setτ :=¹_aM^−a∗ and chooseu¯_∞∈R^min a way that| ¯u_∞| =Mand sup_Rⁿ|u+τu¯_∞| =(1+τ )M. Withξ:= −τu¯_∞we observe from (32)

M²= lim

R→∞u¯_∞ 1 z₁(B_R)

BR(0)

uz1dx.

Since| ¯u_∞|,|_z1(B¹_R)

B_R(0)uz₁dx|Mwe conclude assertion (i).

(ii) We have 1 z₁(B_R)

B_R(0)

|u− ¯u_∞|²z₁dx= | ¯u_∞|²+ 1 z₁(B_R)

B_R(0)

|u|²z₁dx−2u¯_∞· 1 z₁(B_R)

B_R(0)

uz₁dx.

By lettingR→ ∞we infer from the proof of (i)

Rlim→∞

1 z₁(B_R)

BR

|u− ¯u_∞|²z₁dx=M²+M²−2M²=0.

(iii) (32) and (i) yield for everyξ∈R^mwith|ξ|¹⁻_a^a∗ the equation sup

Rⁿ |u−ξ|²= | ¯u_∞|²+ |ξ|²−2ξ· ¯u_∞= | ¯u_∞−ξ|². 2