On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with L

Boundary Value Problems

Volume 2007, Article ID 65825,15pages doi:10.1155/2007/65825

Research Article

On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with L

Data

S. Bonafede and F. Nicolosi

Received 24 January 2007; Accepted 29 January 2007 Recommended by V. Lakshmikantham

We establish H¨older continuity of generalized solutions of the Dirichlet problem, asso- ciated to a degenerate nonlinear fourth-order equation in an open bounded setΩ⊂Rⁿ, withL¹data, on the subsets ofΩwhere the behavior of weights and of the data is regular enough.

Copyright © 2007 S. Bonafede and F. Nicolosi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we will deal with equations involving an operator A:W^{◦ 1,q}_2,p(ν,μ,Ω)→ (W^{◦ 1,q}_2,p(ν,μ,Ω))of the form

Au=

|α|=1,2

(−1)^|α|D^αAα

x,∇2u, (1.1)

whereΩis a bounded open set ofRⁿ,n >4, 2< p < n/2, max(2p,^√n)< q < n,νandμare positive functions inΩwith properties precised later,W^{◦ 1,q}_2,p(ν,μ,Ω) is the Banach space of all functions u:Ω→Rwith the properties|u|^q,ν|D^αu|^q,μ|D^βu|^p∈L¹(Ω),|α| =1,

|β| =2, and “zero” boundary values;∇2u= {D^αu:|α| ≤2}.

The functionsAαsatisfy growth and monotonicity conditions, and in particular, the following strengthened ellipticity condition (for a.e.x∈Ωandξ= {ξα:|α| =1, 2}):

|α|=1,2

Aα(x,ξ)ξα≥c2

|α|=1

ν(x)ξα^q+

|α|=2

μ(x)ξα^p

−g2(x), (1.2) wherec2>0,g2(x)∈L¹(Ω).

We will assume that the right-hand sides of our equations, depending on unknown function, belong toL¹(Ω).

A model representative of the given class of equations is the following:

−

|α|=1

D^α

ν

|β|=1

D^βu² (q−2)/2

D^αu

+

|α|=2

D^α

μ

|β|=2

D^βu² (p−2)/2

D^αu

= −|u|^σ−1u+f inΩ, (1.3) whereσ >1 and f ∈L¹(Ω).

The assumed conditions and known results of the theory of monotone operators allow us to prove existence of generalized solutions of the Dirichlet problem associated to our operator (see, e.g., [1]), bounded on the setsG⊂Ωwhere the behavior of weights and of the data of the problem is regular enough (see [2]).

In our paper, following the approach of [3], we establish on such sets a result on H¨older continuity of generalized solutions of the same Dirichlet problem.

We note that for one high-order equation with degenerate nonlinear operator satisfy- ing a strengthened ellipticity condition, regularity of solutions was studied in [4,5] (non- degenerate case) and in [6,7] (degenerate case). However, it has been made for equations with right-hand sides inL^twitht >1.

2. Hypotheses

Letn_∈N,n >4, and letΩbe a bounded open set ofRⁿ. Let p,q be two real numbers such that 2< p < n/2, max(2p,^√n)< q < n.

Letν:Ω→R⁺be a measurable function such that ν∈L¹_loc(Ω), 1

ν 1/(q−1)

∈L¹_loc(Ω). (2.1)

W^1,q(ν,Ω) is the space of all functions u∈L^q(Ω) such that their derivatives, in the sense of distribution,D^αu,|α| =1, are functions for which the following properties hold:

ν^1/qD^αu∈L^q(Ω) if|α| =1;W^1,q(ν,Ω) is a Banach space with respect to the norm u1,q,ν=

Ω|u|^qdx+

|α|=1

ΩνD^αu^qdx 1/q

. (2.2)

W◦ ^1,q(ν,Ω) is the closure ofC₀^∞(Ω) inW^1,q(ν,Ω).

Letμ(x) :Ω→R⁺be a measurable function such that μ∈L¹_loc(Ω), 1

μ 1/(p−1)

∈L¹_loc(Ω). (2.3)

W_2,p^1,q(ν,μ,Ω) is the space of all functionsu∈W^1,q(ν,Ω), such that their derivatives, in the sense of distribution,D^αu,|α| =2, are functions with the following properties:

μ^{1/ p}D^αu∈L^p(Ω),|α| =2;W_2,p^1,q(ν,μ,Ω) is a Banach space with respect to the norm

u = u1,q,ν+

|α|=2

ΩμD^αu^pdx 1/ p

. (2.4)

W◦ ^1,q_2,p(ν,μ,Ω) is the closure ofC^∞₀(Ω) inW_2,p^1,q(ν,μ,Ω).

Hypothesis 2.1. Letν(x) be a measurable positive function:

1

ν^∈L^t(Ω) witht > nq q²−n, ν∈L^t(Ω) witht > nt

qt−n.

(2.5)

We putq=nqt/(n(1 +t)−qt). We can easily prove that a constantc0>0 exists such that ifu∈W^{◦ 1,q}(ν,Ω), the following inequality holds:

Ω|u|^qdx≤c0

suppu

1 ν

t

dx

q/qt

|α|=1

Ων|D^αu|^qdx q/q

. (2.6)

We setν=μ^q/(q−2p)(1/ν)^2p/(q−2p). Hypothesis 2.2. ν∈L¹(Ω).

Hypothesis 2.3. There exists a real numberr >q(q −1)/(q(q −1)(p−1)−q) such that 1

μ^∈L^r(Ω). (2.7)

For more details about weight functions, see [8,9].

LetΩ1be a nonempty open set ofRⁿsuch thatΩ1⊂Ω.

Definition 2.4. It is said thatGclosed set ofRⁿis a “regular set” ifG^◦ is nonempty and G⊂Ω1.

Denote byR^n,2the space of all setsξ= {ξα∈R:|α| =1, 2}of real numbers; if a func- tionu∈L¹_loc(Ω) has the weak derivativesD^αu,|α| =1, 2 then∇2u= {D^αu:|α| =1, 2}. Suppose thatAα:Ω×R^n,2→Rare Carath´eodory functions.

Hypothesis 2.5. There exist c1,c2>0 andg1(x), g2(x) nonnegative functions such that g1,g2∈L¹(Ω) and, for almost everyx∈Ω, for everyξ∈R^n,2, the following inequalities

hold:

|α|=1

ν(x)^−1/(q−1)Aα(x,ξ)^q/(q−1)+

|α|=2

μ(x)^−1/(p−1)Aα(x,ξ)^p/(p−1)

≤c1

|α|=1

ν(x)ξα^q+

|α|=2

μ(x)ξα^p +g1(x),

(2.8)

|α|=1,2

Aα(x,ξ)ξα≥c2

|α|=1

ν(x)ξα^q+

|α|=2

μ(x)ξα^p

−g2(x). (2.9)

Moreover, we will assume that for almost everyx∈Ωand everyξ,ξ∈R^n,2,ξ=ξ,

|α|=1,2

A_α(x,ξ)−A_α(x,ξ)ξ_α−ξ_α>0. (2.10)

LetF:Ω×R→Rbe a Carath´eodory function such that

(a) for almost everyx∈Ω, the functionF(x,·) is nonincreasing inR; (b) for everyx∈Ω, the functionF(·,s) belongs toL¹(Ω).

LetA:W^{◦ 1,q}_2,p(ν,μ,Ω)→(W^{◦ 1,q}_2,p(ν,μ,Ω))be the operator such that for everyu,v∈W^{◦ 1,q}_2,p(ν, μ,Ω),

Au,v =

Ω

|α|=1,2

Aα

x,∇2uD^αv

dx. (2.11)

We consider the following Dirichlet problem:

(P)=

⎧⎨

⎩

Au=F(x,u) inΩ

D^αu₌0, |α_{| =}0, 1, on∂Ω. (2.12) Definition 2.6. AW-solution of problem (P) is a functionu∈W^◦2,1(Ω) such that

(i)F(x,u)∈L¹(Ω);

(ii)Aα(x,∇2u)∈L¹(Ω), for everyα:|α| =1, 2;

(iii)Au,φ = F(x,u),φin distributional sense.

It is well known that Hypotheses2.1–2.3,2.5, and assumptions onF(x,s) imply the existence of aW-solution of problem (P) (see [1]). Moreover, a boundedness local result for such solution has been established in [2] under more restrictive hypotheses on data and weight functions.

More precisely, the following holds (see [2, Theorem 5.1]).

Theorem 2.7. Suppose that Hypotheses2.1–2.3and2.5 are satisfied. Let q1∈(q,q(q − 1)/q),τ >q/( q−q1). Assume that restrictions of the functionsν^q1/(q1−q),ν,g1,g2, and|F(·, 0)|^q1/(q1−1)onGbelong toL^τ(G), for every “regular set”G.

Then there existsu W-solution of problem (P) such that for everyG, ess_Gsup|u| ≤M_G<

+∞, withMGpositive constant depending only on known values.

3. Main result

In the sequel of paper,Gwill be a “regular set.” In order to obtain our regularity result on G, we need the following further hypotheses.

Hypothesis 3.1. There exists a constantc>0 such that for ally∈G^◦ and for allρ >0, with B(y,ρ)⊂G, we have^◦

ρ⁻ⁿ

B(y,ρ)

1 ν

t

dx 1/t

ρ⁻ⁿ

B(y,ρ)ν^τdx 1/τ

≤c. (3.1)

With regard to this assumption, see [3].

Hypothesis 3.2. There exist a real positive numberσ and two real functionsh(x)(≥0), f(x)(>0) defined onG, such that

F(x,s)≤h(x)|s|^σ+f(x), for almost everyx∈Gand everys∈R. (3.2) Moreover, we assume that

h(x),f(x)∈L^τ(G), (3.3)

withτdefined as above.

Using considerations stated in [1], following the approach of [3], we establish the fol- lowing result.

Theorem 3.3. Let all above-stated hypotheses hold and let conditions ofTheorem 2.7be satisfied. Then, theW-solutionuof Dirichlet problem (P), essentially bounded onG, is also locally H¨olderian onG.

More precisely, there exist positive constantCandλ(0< λ <1) such that for every open setΩ,Ω⊂G, and every^◦ x,y∈Ω

u(x)−u(y)≤CdΩ,∂G^◦−λ|x−y|^λ, (3.4) whereCandλdepend only onc1,c2,c0,c,n,q,p,t,τ,σ,MG, diamG, measG,fL^τ(G), hL^τ(G),g1L^τ(G),g2L^τ(G),νL^τ(G), and1/νL^t(Ω).

Proof. For everyl∈N, we define the functionFl:Ω×R→Rby

Fl(x,s)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−l ifF(x, 0)−F(x,s)<−l, F(x, 0)−F(x,s) ifF(x, 0)−F(x,s)≤l, l ifF(x, 0)−F(x,s)> l,

(3.5)

and the function fl:Ω→Rby f_l(x)=

⎧⎨

⎩

F(x, 0) ifF(x, 0)≤l,

0 ifF(x, 0)> l. (3.6)

By Lebesgue’s theorem and property (b) ofF(x,s), we have that fl(x) goes toF(x, 0) in L¹(Ω).

Next, inequalities (2.6), (2.8)–(2.10), property (a) ofF(x,s), and known results of the theory of monotone operators (see, e.g., [10]) imply that for any_◦ l∈N, there existsul∈ W^1,q_2,p(ν,μ,Ω) such that

Ω

|α|=1,2

A_αx,∇2u_lD^αv+F_lx,u_lv

dx=

Ωf_lv dx, (3.7) for everyv∈W^{◦ 1,q}_2,p(ν,μ,Ω).

From considerations stated in [1, Section 3], we deduce that there exists aW-solution uof problem (P) such that

ul−→u a.e. inΩ. (3.8)

Moreover, see proof ofTheorem 2.7,

essG supu_l≤M_G, for everyl∈N. (3.9) We setn=q²/(q−2p),a=(1/n)(q−n/t−n/τ).

Let us fixy∈G,^◦ ρ >0 andB(y, 2ρ)⊂G. Let us put^◦ ω_1,l= ess

B(y,2ρ)infu_l, ω_2,l= ess

B(y,2ρ)supu_l,

ωl=ω2,l−ω1,l. (3.10)

We will show that

oscul,B(y,ρ)≤cω l+ρ^a, (3.11) withc∈]0, 1[ independent ofl∈N.

To this aim, we fixl∈Nand we set Φl=

|α|=1

νD^αu_l^q+

|α|=2

μD^αu_l^p,

ψ(x)=ρ^−an1 +f(x) +h(x) +g1(x) +g2(x) +ν(x)+ρ^−qν. (3.12) Obviously, we will assume that

ω_l≥ρ^a (otherwise, it is clear that (3.11) is true). (3.13) We introduce now the following functions:

F1,l(x)=

⎧⎪

⎨

⎪⎩

2eωl

u_l(x)−ω_1,l+ρ^a ifx∈B(y, 2ρ), e ifx∈Ω\B(y, 2ρ);

(3.14)

ϕ∈C^∞₀(Ω): 0≤ϕ≤1 inΩ,ϕ=0 inΩ\B(y, 2ρ) and satisfying

D^αϕ≤cρ^−|α|, |α| =1, 2, (3.15) where the positive constantcdepends only onn.

Let us fixs > qandr≥0 and define vl=

lgF1,lr

F_1,l^q−1ϕ^s, zl= − 1

2eωl

rlgF1,lr−1

+ (q−1)lgF1,lr

F_1,l^qϕ^s. (3.16) FromHypothesis 2.2and (3.15), we have thatvl∈W^{◦ 1,q}_2,p(ν,μ,Ω) and the next inequal- ities are true:

D^αvl−zlD^αul≤csϕ^s−1lgF1,l

r

F_1,l^q−1ρ⁻¹ if|α| =1 a.e. inB(y, 2ρ), (3.17) D^αvl−zlD^αul≤5q²s(r+ 1)²lgF1,l

r

F_1,l^q−1ϕ^s

|β|=1

|D^βul|² u_l−ω_1,l+ρ^a2

+ 2nqs²c²ρ⁻²lgF1,l

r

F_1,l^q−1ϕ^s−2 if|α| =2 a.e. inB(y, 2ρ).

(3.18)

Sinceu_l(x) satisfies (3.7), forv=v_l, we obtain

Ω

|α|=1,2

A_αx,∇2u_lD^αv_l+F_lx,u_lv_l

dx=

Ωf_lv_ldx. (3.19) From this, taking into account (3.9) andHypothesis 3.2, we have

Ω

|α|=1,2

A_αx,∇2u_lD^αv_ldx≤

3 +M_G^σ

Ω

1 +f(x) +h(x)v_ldx. (3.20)

Hence

Ω

|α|=1,2

Aα x,∇2ul

D^αul

−zl dx≤

3 +M^σ_G

Ω

1 +f(x) +h(x)vldx+I1+I2, (3.21) where

I_i=

Ω

|α|=i

A_αx,∇2u_lD^αv_l−z_lD^αu_ldx, i=1, 2. (3.22)

UsingHypothesis 2.5and definition ofzl, we have (q−1)c2

2eωl

ΩΦl lgF1,lr

F_1,l^qϕ^sdx≤

3 +M_G^σ

Ω

1 +f(x) +h(x)lgF1,lr

F_1,l^q−1ϕ^sdx +

Ωg2(x)−zl

dx+I1+I2.

(3.23)

Note that

F_1,l^q−1≤(diamG)^a2eω_l^q−1ρ^−aq,

−z_l≤(q−1)(r+ 1)2eω_l^q−1ρ^−aqϕ^slgF_1,l^r a.e. inB(y, 2ρ), (3.24) consequently, from (3.23), we obtain

c2

2eω_l

B(y,2ρ)Φl

lgF_1,l^rF_1,l^qϕ^sdx

≤c3(r+ 1)2eωlq−1

B(y,2ρ)ρ^−aq1 +f(x) +h(x) +g2(x)lgF1,lr

ϕ^sdx+I1+I2, (3.25) wherec3=(q−1)(3 +M^σ_G)(diamG+ 1).

Let us fix|α| =1. Let>0, then, applying Young’s inequality and using (2.8) and (3.17), we establish

I1≤ c1 2eωl

B(y,2ρ)ΦlF_1,l^qlgF1,l

_r ϕ^sdx +c1

2eω_l^q−1

B(y,2ρ)ρ^−aqg1(x)lgF_1,l^rϕ^sdx +^1−q

2eωlq−1

n(cs)^q

B(y,2ρ)ρ^−qνlgF1,lr

ϕ^s−qdx.

(3.26)

Let us fix|α| =2 and estimateI2. To this aim, it will be useful to observe that the following equalities are true:

p−1 p +2

q+q−2p

qp ⁼1, q−1= p−1 p q+

q p⁻1

. (3.27)

Moreover,

ρ^−aq−2pμ≤ρ^−anν+ρ^−qν inΩ. (3.28) Furthermore, due to (2.8), (3.18), and Young’s inequality, we have

I2≤ c4 2eω_l

B(y,2ρ)ΦlF_1,l^qlgF_1,l^rϕ^sdx +c5

2eωlq−1

1 +1

n

sⁿ(r+ 1)ⁿ

B(y,2ρ)

ρ^−ang1(x) +ν(x)+ρ^−qνlgF1,lr

ϕ^s−qdx, (3.29) wherec4depends only onc1,n,q; andc5depends only onc1,n,q,p,c, and diamG.

From (3.25), (3.26), and (3.29), we get c2

2eω_l

B(y,2ρ)Φl

lgF1,l

r

F_1,l^q ϕ^sdx

≤ c1+c4

2eωl

B(y,2ρ)ΦlF_1,l^qlgF_1,l^rϕ^sdx +2eωl

_q−1

c6(r+ 1)ⁿsⁿ

1 ++1

n+1

B(y,2ρ)ψlgF1,l

_r ϕ^s−qdx,

(3.30)

where the constantc6depends only onc1,c,n,q,p,M_G,σ, and diamG.

Setting

= c2

2c1+c4

, (3.31)

from the last inequality, we deduce

B(y,2ρ)Φl

lgF_1,l^rF_1,l^qϕ^sdx≤c7

2eω_l^q(r+ 1)ⁿsⁿ

B(y,2ρ)ψlgF_1,l^rϕ^s−qdx, (3.32) where the constantc7depends only onc1,c2,c,n,q,p,MG,σ, and diamG.

Now, if we chooseϕsuch thatϕ=1 inB(y, (4/3)ρ), from (3.32), withr=0 ands= q+ 1, we get

B(y,(4/3)ρ)

|α|=1

νD^αul^q

F_1,l^qdx≤c7 2eωlq

(q+ 1)ⁿ

B(y,2ρ)ψdx. (3.33)

Moreover, if we take in (3.32) instead ofϕthe functionϕ1∈C₀^∞(Ω) with the properties 0≤ϕ1≤1 inΩ,ϕ1=0 inΩ\B(y, (4/3)ρ),ϕ1=1 inB(y,ρ), and|D^αϕ| ≤cρ^−|α|inΩ,

|α| =1, 2, we obtain that for everyr >0 ands > q,

B(y,2ρ)

|α|=1

νD^αu_l^q

lgF_1,l^rF_1,l^qdx≤c7

2eω_l^qsⁿ(r+ 1)ⁿ

B(y,2ρ)ψlgF_1,l^rϕ^s₁^−qdx.

(3.34) We fix arbitraryr >0 ands >q, and let

zl=

lgF1,lr/q

ϕ^s/₁^q. (3.35)

By means ofHypothesis 2.1, we establish thatzl∈W^◦1,q(ν,Ω) and for|α| =1, νD^αz_l^q≤2^q−1

r q

q

lgF_1,l^(r/q−1)qF_1,l^q 1

2eω_l^qD^αu_l^qνϕ^sq/₁ ^q + 2^q−1

s q

q

lgF1,lrq/q

ϕ^(s/1 ^q−1)qc^qρ^−qν.

(3.36)

Now, it is convenient to observe thatq/( q−q1)> nt/(qt−n), thenτ > nt/(qt−n);

moreover,ψ(x)∈L^τ(G). From (3.34) and (3.36), we deduce

ΩνD^αz_l^qdx

≤c8sⁿ(r+1)^n+q

B(y,2ρ)ψ^τdx 1/τ

B(y,2ρ)

lgF_1,l^{r(q/q)(τ/(τ −1))}ϕ^(s/₁ ^{q−1)q(τ/(τ−1))}dx (τ−1)/τ

, (3.37) where the constantc8depends only onc1,c2,c,n,q,p,M_G,σ, and diamG.

We set

θ=q(τ −1)

qτ , m= qτ

τ−1, (3.38)

and for everyr,s >0, we define I(r,s)=

B(y,2ρ)

lgF1,l

_r

ϕ^s₁dx. (3.39)

Consequently, last inequality can be rewritten in this manner:

ΩνD^αz_l^qdx≤c8sⁿ(r+ 1)^n+q

B(y,2ρ)ψ^τdx 1/τ

I r

θ,s θ⁻m

(τ−1)/τ

. (3.40) Due toHypothesis 2.1,

I(r,s)=

B(y,2ρ)z^q_ldx≤c0

B(y,2ρ)

1 ν

t

dx

q/qt

|α|=1

ΩνD^αzl^qdx q/q

. (3.41) Let us denote by_Gthe norm of (1 +f(x) +h(x) +g1(x) +g2(x) +ν(x)) inL^τ(G). By simple computation, we have

B(y,2ρ)ψ^τdx 1/τ

≤ρ^−q

B(y,2ρ)ν^τdx 1/τ

+_Gρ^−an. (3.42) Now, it is convenient to observe that (q−n/t−n/τ)(q/q) =n(θ−1).

Then, from (3.40)–(3.42), usingHypothesis 3.1, we get I(r,s)≤M(r+s)^mρ^n(1−θ)

I

r θ,s

θ⁻m θ

, for everyr >0,s >q, (3.43) wherem=2(q+n)qand the positive constantMdepends only onc1,c2,c,c0,c,n,q,p, t,1/νL^t(Ω),MG,σ, measG, diamG, and_G.

We set fori=0, 1, 2,. . .that ri= tq

t+ 1θⁱ, si= mθ θ−1

θⁱ⁺¹−1. (3.44)

Then by (3.43), it is trivial to establish the following iterative relation:

Iri,si

≤Mc9ρ^n(1−θ)θ^{i m}Iri−1,si−1

θ

for everyi∈N, (3.45) wherec9depends only onn,q,p,t, andτ.

Using this recurrent relation, we obtain that for everyi∈N, Ir_i,s_i≤

Mc9+ 1^1/(1−θ)θ^{S m}(diamG+ 1)ⁿρ⁻ⁿIr0,s0

^θi

, (3.46)

whereSis a positive constant depending only onn,q,t, andτ.

Now, we assume that meas

x∈B

y,4

3ρ

:ul(x)≥ω1,l+ω2,l

2

≥1 2measB

y,4

3ρ

. (3.47)

We observe that ifx∈B(y, (4/3)ρ) satisfiesul(x)≥(ω1,l+ω2,l)/2, thenF1,l(x)≤4e, so by [11, Lemma 4], we deduce

B(y,(4/3)ρ)

lgF1,l

_r0

dx≤cρⁿ+ cρr0

2eωl

B(y,(4/3)ρ)

|α|=1

D^αullgF1,l

_r0−1

F1,l

dx,

(3.48) wherecdepends only onn.

Then, using Young’s inequality, we get

B(y,(4/3)ρ)

lgF_1,l^r0dx_≤cr0ρⁿ+r0

cr0ρ 2eω_l

r0

B(y,(4/3)ρ)

|α|=1

D^αu_l r0

F_1,l^r0dx. (3.49) Last inequality, using H¨older’s inequality and (3.33), gives

B(y,(4/3)ρ)

lgF1,lr0

dx≤cr0ρⁿ+r0 cr0r0

2^r0−1c7(q+ 1)^nt/(t+1)ρ^r0

×

B(y,2ρ)ψdx

t/(t+1)

B(y,2ρ)

1 ν

t

dx 1/(t+1)

.

(3.50)

Observe that due to (3.42) andHypothesis 3.1,

B(y,2ρ)ψdx

t/(t+1)

B(y,2ρ)

1 ν

t

dx 1/(t+1)

≤c10(1 +M)ρ^n−r0, (3.51) wherec10depends only on measure of the unit ball inRⁿ.

Consequently, from (3.50), we obtain

B(y,(4/3)ρ)

lgF_1,l^r0dx≤ c10(1 +M)r0

cr0

r0

2^r0−1c7(q+ 1)^nt/(t+1)+cr0

!ρⁿ. (3.52)

Taking into account that Ir0,s0

≤

B(y,(4/3)ρ)

lgF_1,l^r0dx, (3.53)

from (3.46) we get

Ir_i,s_i≤ c11

θⁱ

, for everyi∈N. (3.54)

Last inequality allow us to conclude that

B(y,ρ)ess supF1,l(x)≤ 1 +c11

, (3.55)

and so

oscul,B(y,ρ)≤

1−2e^−1−c11ωl+ρ^a. (3.56) Recall that we proved (3.11) under assumption (3.47). If (3.47) is not true, we take instead ofF1,lthe functionF2,l:Ω→Rⁿsuch thatF2,l=2eωl(ω2,l−ul+ρ^a)⁻¹inB(y, 2ρ), and arguing as above, we establish (3.11) again.

It is important to observe that the positive constantc11depends only onc1,c2,c,c,c0, c,n,q,p,t,1/νL^t(Ω),MG,σ, diamG, and_G, and is independent ofl∈N.

Now from (3.11), taking into account [12, Chapter 2, Lemma 4.8], we deduce that there exist positive constantCandλ(<1) depending onc11 andabut independent of l∈Nsuch that

oscul,B(y,ρ)≤Cdy,∂G^◦−λρ^λ, for everyρ∈

0,dy,∂G^◦. (3.57) This and (3.8) imply that

oscu,B(y,ρ)≤Cdy,∂G^◦−λρ^λ, for everyρ∈

0,dy,∂G^◦. (3.58)

The proof is complete.

4. An example

LetΩ= {x∈Rⁿ:|x|<1}, 0< γ <min(q−n/q,q/2), and letν,μbe the restriction in Ω\ {0}of real functions

|x|^γ, |x|^2pγ/q. (4.1)