本文中主要考虑非齐次拟线性A-调和方程

-divA(x,u,Δu)=f(x)+

div(|Δu|^p-2Δu),x∈Ω(1)

很弱解的正则性.其中ΩRⁿ是有界区域,n≥2,1<p<∞,A(x,u,h):Ω×Rⁿ×R^nN→Rⁿ是一个Caratheodory函数,并且满足以下结构性条件:

(i)算子A(x,u,h)满足强制性条件.即存在常数a>0,使得

A(x,u,h)h≥a|h|^p,h∈R^nN,x∈Ω,

u∈Rⁿ;

(ii)算子A(x,u,h)是拟线性的.即存在常数β>0,使得

(A(x,u₁,h₁)-A(x,u₂,h₂))(h₁-h₂)≥

β(|h₁|+|h₂|)^p-2|h₁-h₂|²,

u₁,u₂∈Rⁿ,h₁,h₂∈R^nN,x∈Ω;

(iii)算子A(x,u,h)是有界的.即存在常数β≤γ<∞,使得

|A(x,u,h)|≤γ(|h|^p-1+|u|^(p-1)α+

φ(x))x∈Ω,u∈Rⁿ,h∈R^nN,

其中0<α<n/(n-(p-1)),φ(x)∈L^p/(p-1)(Ω).

定义1 称函数u∈W^1,r(Ω)(max{1,p-1}≤r≤p)为方程(1)的很弱解,若对所有的φ∈C₀^∞(Ω)都有

∫_Ω A(x,u,Δu)·Δφdx=∫_Ω f(x)·φdx-

∫_Ω |Δu|^p-2Δu·Δφdx.(2)

由于A-调和方程能够反映实际生活中和物理学中众多现象而被广大的专家学者所关注.在数学中关于A-调和方程的研究主要集中在对其解的性质,尤其是对经典弱解的存在性、唯一性、稳定性以及正则性等方面的研究^[1-2].

1994年,Iwaniec等^[3]通过观察发现,在积分意义下,A-调和方程弱解的可积指数可以降低,甚至可以降低到比弱解的自然指标小1的情况,从而提出了比经典弱解更弱的解的定义,即很弱解的定义.在同一年,Iwaniec等^[3]研究了A-调和方程-divA(x,Δu)=0的很弱解和经典弱解之间的关系.之后Greco等^[4-5]将文献[3]的结果进行了推广,发现方程div[(G(x)Δu,Δu)⁽p-2)/2G(x)Δu ]=0的很弱解实际上就是经典弱解.

Stroffolini^[6]发现文献[4-5]的结论对于方程div[(A(x)Δu,Δu)⁽p-2)/2A(x)Δu]=div((A(x))^1/2F(x))同样成立.

周树清等^[7]研究发现A-调和方程组-D_i(A^ij(x,Du))+D_i fⁱ_j(x)=0,j=1,2,…,m很弱解的性质亦成立.

佟玉霞等^[8]将同样的结果推广到一类非齐次障碍问题divA(x,Δu)=B(x,Δu).之后佟玉霞等^[9]又得到了非齐次A-调和方程divA(x,Δu)=f(x)很弱解的性质.

但是他们所考虑的经典弱解和很弱解之间的联系的这些A-调和方程中,A-调和算子都是和很弱解u无关的.受到这些问题的启发,本文中主要考虑A-调和算子和很弱解u有关的二阶非齐次A-调和方程(1)很弱解的正则性.这些结论都是在方程(1)很弱解存在的前提条件下进行的,有关A-调和方程很弱解的存在性,在文献[10]中有着详细的证明,在这里不做讨论,直接证明很弱解的正则性问题并得到以下结论.

定理1 设f∈L(nq)/(n(p-1)+q)loc(Ω),q>p存在可积指数1<r1=r1(n,p,α,β)<p<r2=r2(n,p,α,β)<∞,使得方程(1)的每一个很弱解u∈W1,r1loc(Ω)都属于W1,r2loc(Ω),从而u是经典弱解.

引理1^[11] 设1≤p<n.若u∈W^1,p_loc(Ω),则对任意球B_R Ω,有

=u-u_BR=_{L^(np)/(n-p)(BR)}≤

C(n)p/(n-p)(p/(n-p))^(n-p)/np=Δu=L^p(B_R).

引理2^[12-13] 设u(x)∈L^p(B_R),B_R Ω,f∈L^t(B_R),t>p并且满足如下不等式:

f_BR/2|u|^pdx≤K(f_BR |u|^sdx)^p/s+

θf_BR|u|^pdx+f_{BR |f|^pdx.}

其中:1≤s<p,0≤θ≤1,则存在可积指数p'=p'(K,n,p,θ),(t≥p'>p),使得u∈L^p'_loc(Ω),且

(f_BR/2|u|^p'dx)^1/(p')≤C'(f_BR|u|^pdx)^1/p+

C'(f_{BR|f|^p'dx)^1/(p'),(3)}

其中C'=C'(n,p,K,θ).

证明 设η∈C₀^∞(B_R),0≤η≤1,当x∈B_R/2 时η≡1,|Δη|≤4/R为截断函数.首先假设u∈W^1,p-ε_loc(Ω)(0<ε<1/2)为方程(1)的一个很弱解.考虑下面的Hodge分解^[3]

|Δ(ηu)|^-εΔ(ηu)=Δ+H,(4)

这里∈W^{1,(p-ε)/(1-ε)}_loc(Ω),H∈L^{(p-ε)/(1-ε)}(Ω)是一个散度为零的向量场,且满足:

=Δ=_{(p-ε)/(1-ε)}≤C(n,p)=Δ(ηu)=^1-ε_p-ε,(5)

=H=_{(p-ε)/(1-ε)}≤C(n,p)ε=Δ(ηu)=₍p-ε)/(¹-ε).(6)

令

E(η,u)=|Δ(ηu)|^-εΔ(ηu)-|ηΔu|^-εηΔu,(7)

则由一个基本的关系式

||X|^-εX-|Y|^-εY|≤2^ε(1+ε)/(1-ε)|X-Y|^1-ε,

0<ε<1,X,Y∈Rⁿ(8)

得到

|E(η,u)|≤2^ε(1+ε)/(1-ε)|uΔη|^1-ε.(9)

取Hodge分解中的为弱解定义中的检验函数,于是

∫_{BR A(x,u,Δu)(E(η,u)+|ηΔu|^-εηΔu-}

H)dx=∫_BRfdx-∫_BR|Δu|^p-2Δu(E(η,u)+

|ηΔu|^-εηΔu-H)dx.(10)

即

∫_BRA(x,u,Δu)|ηΔu|^-εηΔudx+

∫_BR |Δu|^p|ηΔu|^-εηdx=-∫_{BR A(x,u,Δu)}

E(η,u)dx+∫_{BR A(x,u,Δu)Hdx+}

∫_BRfdx-∫_{BR |Δu|^p-2ΔuE(η,u)dx+}

∫_BR |Δu|^p-2ΔuHdx≤I₁+I₂+I₃+

I₄+I₅.(11)

其中

I₁=∫_BR |A(x,u,Δu)E(η,u)|dx,

I₂=∫_BR |A(x,u,Δu)H|dx,

I₃=∫_BR |f|dx,

I₄=∫_BR ||Δu|^p-2ΔuE(η,u)| dx,

I₅=∫_BR ||Δu|^p-2ΔuH| dx.

先估计式(11)左边,由假设条件(i)得到

∫_BR A(x,u,Δu)|ηΔu|^-εηΔudx=

∫_BR η^1-ε|Δu|^-εA(x,u,Δu)Δudx≥

a∫_BR η^1-ε|Δu|^-ε|Δu|^pdx,

由η的定义可得

∫_BR A(x,u,Δu)|ηΔu|^-εηΔudx+

∫_BR |Δu|^p|ηΔu|^-εηdx≥

(a+1)∫_BR η^1-ε |Δu|^p-εdx≥

(a+1)∫_BR/2|Δu|^p-ε dx.(12)

下面估计I₁,由条件(iii)和式(9)可得

I₁=∫_BR |A(x,u,Δu)E(η,u)|dx≤

∫_BR |γ(|Δu|^p-1+|u|^(p-1)α+φ(x)|

|E(η,u)|dx≤∫_BR 〖JB<2|〗γ(|Δu|^p-1+|u|^(p-1)α+

φ(x)〖JB>2|〗 2^ε(1+ε)/(1-ε)|uΔη|^1-εdx≤

γ2^ε(1+ε)/(1-ε)∫_BR |Δu|^p-1|uΔη|^1-εdx+

γ2^ε(1+ε)/(1-ε)∫_BR |u|^(p-1)α|uΔη|^1-εdx+

γ2^ε(1+ε)/(1-ε)∫_BR |φ(x)| |uΔη|^1-εdx=

γ2^ε(1+ε)/(1-ε)(J₁+J₂+J₃),(13)

其中

J₁=∫_BR |Δu|^p-1|uΔη|^1-εdx,

J₂=∫_BR |u|^(p-1)α|uΔη|^1-εdx,

J₃=∫_BR |φ(x)| |uΔη|^1-εdx.

由η的定义可得

J₁=∫_BR |Δu|^p-1|uΔη|^1-εdx≤

(4^1-ε)/(R^1-ε)∫_BR |Δu|^p-1|u|^1-εdx.

令

p'=(n(p-ε))/((n+1-ε)(p-1)),

q'=(n(p-ε))/((n-p+1)(1-ε)),

则1<p',q'<∞,1/(p')+1/(q')=1.

由Hölder不等式可得

J₁≤(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^{((n+1-ε)(p-1))/(n(p-ε))}

(∫_BR |u|^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε))/(n(p-ε))}≤

(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^{((n+1-ε)(p-1))/(n(p-ε))}

(∫_BR |u-u_BR|^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε))/(n(p-ε))}+

(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^{((n+1-ε)(p-1))/(n(p-ε))}

(∫_BR u_BR^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε))/(n(p-ε))}.(14)

令p″=(n(p-ε))/(n+1-ε),则(np″)/(n-p″)=(n(p-ε))/(n-p+1),当p″<n时,应用引理2可得

( ∫_BR |u-u_BR|^{(n(p-ε))/(n-p+1)}dx)^{(n-p+1)/(n(p-ε))}=

( ∫_BR |u-u_BR|^{(np″)/(n-p″)}dx)^{(n-p″)/(np″)}≤

C(n)(p″)/(n-p″)((p″)/(p″-1))^{(n-p″)/(np″)}( ∫_BR |Δu|^p″dx)¹/(p″).

由Young不等式可得

J₁≤C(n,p)(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^{((n-p+1)(1-ε))/(n(p-ε))}

(∫_BR u_BR^{(n(p-ε))/(n-p+1)}dx)((n-p+1)(1-ε))/(n(p-ε))≤

C(n,p)(4^1-ε)/(R^1-ε)(∫_BR|Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(4^1-ε)/(R^1-ε)(∫_BRu_BR^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε)(n+1-ε))/(n[(n+1-ε)(p-ε)-(n-p+1)(1-ε)])}.

注意到η的定义,则由Hölder不等式可得

J₂=∫_BR |u|^(p-1)α|uΔη|^1-εdx≤

(4^1-ε)/(R^1-ε)∫_BR |u|^(p-1)α|u|^1-εdx=

(4^1-ε)/(R^1-ε)∫_BR |u|^(p-1)α+1-εdx≤

(4^1-ε)/(R^1-ε)∫_BR|u-u_BR|^(p-1)α+1-εdx+

(4^1-ε)/(R^1-ε)∫_BR u_BR^(p-1)α+1-εdx≤

(4^1-ε)/(R^1-ε)(∫_BR |u-u_BR|^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)[(p-1)α+1-ε])/(n(p-ε))}

(∫_BR dx)^{((p-1)[n(1-α)+(p-1)α+1-ε])/(n(p-ε))}+(4^1-ε)/(R^1-ε)

∫_BR u_BR^(p-1)α+1-εdx≤C(n,p)(4^1-ε)/(R^1-ε)

(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)⁽[(p-1)α+1-ε](n+1-ε))/(n(p-ε))

(∫_BR dx)^{((p-1)[n(1-α)+(p-1)α+1-ε])/(n(p-ε))}+

(4^1-ε)/(R^1-ε)∫_BR u_BR^(p-1)α+1-εdx.(15)

下面对α分情况讨论.

情形1 当 0<α<1 时,由Young不等式可得

J₂≤C(n,p)(4^1-ε)/(R^1-ε)[(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

C|B_R|]+(4^1-ε)/(R^1-ε)∫_BR u_BR^(p-1)α+1-εdx=

C(n,p)(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(4^1-ε)/(R^1-ε)C(n,p,|B_R|)+(4^1-ε)/(R^1-ε)∫_BRu_BR^(p-1)α+1-εdx.

情形2 当 1≤α<n/(n-(p-1))时,由Sobolev空间的定义,可得

J₂≤C(n,p,|B_R|,=u=_W¹,p″(BR))(4^1-ε)/(R^1-ε)

(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(4^1-ε)/(R^1-ε)∫_BR u_BR^(p-1)α+1-εdx.

综上,

J₂≤C(n,p,|B_R|,=u=_W¹,p″(BR))(4^1-ε)/(R^1-ε)

(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(4^1-ε)/(R^1-ε)C(n,p,|B_R|)+(4^1-ε)/(R^1-ε)∫_BR u_BR^(p-1)α+1-εdx.

由Hölder不等式、Poincáre不等式和Young不等式,得

J₃=∫_BR |φ(x)| |uΔη|^1-εdx≤

(4^1-ε)/(R^1-ε)∫_BR |φ(x)| |u|^1-εdx≤(4^1-ε)/(R^1-ε)

(∫_BR |φ(x)|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)(∫_BR |u|^p-ε dx)^{(1-ε)/(p-ε)}≤

(4^1-ε)/(R^1-ε)(∫_BR |φ(x)|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)

(∫_BR |u-u_BR|^p-ε dx)^{(1-ε)/(p-ε)}+

(4^1-ε)/(R^1-ε)(∫_BR |φ(x)|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)

(∫_BR u_BR^p-εdx)^{(1-ε)/(p-ε)}≤(4^1-ε)/(R^1-ε)

(∫_BR |φ(x)|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)(∫_BR |Δu|^p-εdx)^{(1-ε)/(p-ε)}+

(4^1-ε)/(R^1-ε)(∫_BR |φ(x)|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)(∫_BR u_BR^p-εdx)^{(1-ε)/(p-ε)}≤

(4^1-ε)/(R^1-ε)ε₁ ∫_BR |Δu|^p-εdx+(4^1-ε)/(R^1-ε)C(ε₁)

∫_BR |φ(x)|^(p-ε)/(p-1)dx+(4^1-ε)/(R^1-ε)∫_BR |φ(x)|^(p-ε)/(p-1)dx+

(4^1-ε)/(R^1-ε)∫_BR u_BR^p-εdx≤(4^1-ε)/(R^1-ε)ε₁ ∫_BR |Δu|^p-εdx+

(4^1-ε)/(R^1-ε)C(ε₁)∫_BR |φ(x)|^(p-ε)/(p-1)dx+

(4^1-ε)/(R^1-ε)∫_BR u_BR^p-εdx.(16)

因此

I₁≤γ2^ε(1+ε)/(1-ε)(4^1-ε)/(R^1-ε)

{C(n,p)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

(∫_BRu_BR^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε)(n+1-ε))/(n[(n+1-ε)(p-ε)-(n-p+1)(1-ε)])}+

C(n,p,|B_R|,=u=_W¹,p″(BR))

(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+C(n,p,|B_R|)+

ε₁ ∫_BR |Δu|^p-εdx+C(ε₁)∫_BR |φ(x)|^(p-ε)/(p-1)dx+

∫_BR u_BR^p-εdx}≤C(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

Cε₁ ∫_BR|Δu|^p-εdx+C+C(ε₁)

∫_BR |φ(x)|^(p-ε)/(p-1)dx+C∫_BRu_BR^p-εdx,(17)

其中C=C(n,p,|B_R|,=u=_W¹,p″(BR)).

下面估计I₂,由假设条件(iii)、Hölder不等式和式(6),可得

I₂=∫_BR |A(x,u,Δu)H|dx≤

∫_BR 〖JB<2|〗γ(|Δu|^p-1+|u|^(p-1)α+φ(x)〖JB>2|〗

|H|dx≤∫_BR γ|Δu|^p-1|H|dx+

γ∫_BR |u-u_BR|^(p-1)α|H|dx+

∫_BR |γ[φ(x)+u_BR^(p-1)α]| |H|dx=

K₁+K₂+K₃.(18)

由Hölder不等式以及估计式(6)可得

K₁=∫_BR γ|Δu|^p-1|H|dx≤

γ(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε)(∫_BR |H|^{(p-ε)/(1-ε)}dx)^{(1-ε)/(p-ε)}≤

γC(n,p)ε(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε)

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}.(19)

由Hölder不等式、Sobolev嵌入定理和估计式(6)可得

K₂=γ∫_BR |u-u_BR|^(p-1)α|H|dx≤

γ(∫_BR |u-u_BR|^(p-ε)αdx)^(p-1)/(p-ε)

(∫_BR |H|^{(p-ε)/(1-ε)}dx)^{(1-ε)/(p-ε)}≤

γ(∫_BR |Δu|^p-εdx)^{((p-1)α)/(p-ε)}(∫_BR |H|^{(p-ε)/(1-ε)}dx)^{(1-ε)/(p-ε)}≤

C(n,p)γε(∫_BR |Δu|^p-εdx)^{((p-1)α)/(p-ε)}

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}.(20)

由于φ(x)∈L^p/(p-1)(Ω),则由Hölder不等式和估计式(6)可得

K₃=∫_BR |γ[φ(x)+u_BR^(p-1)α]| |H|dx≤

γ(∫_BR |φ(x)+u_BR^(p-1)α|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)

(∫_BR |H|^{(p-ε)/(1-ε)}dx)^{(1-ε)/(p-ε)}≤C(n,p)

γε(∫_BR |φ(x)+u_BR^(p-1)α|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}.(21)

结合估计式(19),(20)和(21)可得

I₂≤C(n,p,γ)ε(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε)

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}+

C(n,p,γ)ε(∫_BR |Δu|^p-εdx)^{((p-1)α)/(p-ε)}

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}+C(n,p,γ)ε

(∫_BR |φ(x)+u_BR^(p-1)α|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}.(22)

由Poincáre不等式以及η的定义可得

∫_BR |Δ(ηu)|^p-εdx=

∫_BR |uΔη+ηΔu|^p-εdx≤

∫_BR |u-u_BR|^p-ε|Δη|^p-εdx+

∫_BR |ηΔu|^p-εdx+∫_BR u_BR^p-εdx≤

C_p ∫_BR |Δu|^p-εdx+∫_BR |Δu|^p-εdx+

∫_BR u_BR^p-εdx≤C∫_BR |Δu|^p-εdx+

∫_BR u_BR^p-εdx.(23)

下面对α的取值范围分情况讨论.

情形1 当0<α≤1时,由Young不等式可得

(∫_BR |Δu|^p-εdx)^{((p-1)α)/(p-ε)}≤

(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε)+1.

情形2 当1<α<n/(n-(p-1))时,由u∈W^1,p-ε(B_R)可得

(∫_BR |Δu|^p-εdx)^{((p-1)α)/(p-ε)}=

(∫_BR |Δu|^p-εdx)^{((p-1)(1+δ))/(p-ε)}≤

C(=u=_W^1,p-ε)(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε).

将上面这些估计式代入I₂ 中,并由Young不等式可得

I₂≤Cε∫_BR|Δu|^p-εdx+

Cε{∫_BRu_BR^p-εdx+1+

∫_BR|φ(x)+u_BR^(p-1)α|^(p-ε)/(p-1)dx)^(p-1)/(p-ε)},(24)

其中C=C(n,p,γ,C_p,=u=_W^1,p-ε).

下面估计I₃.由Hölder不等式、引理2、式(5)和(15)可得

I₃=∫_BR |f|dx≤

(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-ε))}

(∫_BR ||^{(n(p-ε))/(n(1-ε)-p+ε)}dx)^{(n(1-ε)-p+ε)/(n(p-ε))}≤

C(n,p)(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-ε))}

(∫_BR |Δ|^{(p-ε)/(1-ε)}dx)^{(1-ε)/(p-ε)}≤

C(n,p)(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-ε))}

(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}≤

C(n,p)[C(ε₁)(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-1))}+

ε₁ ∫_BR |Δ(ηu)|^p-εdx]≤C(n,p,ε₁)

(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-1))}+C(n,p)ε₁

[C∫_BR |Δu|^p-εdx+∫_B u_BR^p-εdx]≤

C(n,p,ε₁)(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-1))}+

C(n,p)ε₁ ∫_BR |Δu|^p-εdx+C(n,p)ε₁

∫_BR u_BR^p-εdx.

下面估计I₄.由式(9)可得

I₄=∫_BR ||Δu|^p-2ΔuE(η,u)|dx=

∫_BR |Δu|^p-1|E(η,u)|dx≤

2^ε(1+ε)/(1-ε)∫_BR |Δu|^p-1|uΔη|^1-εdx≤

(4^1-ε)/(R^1-ε)2^ε(1+ε)/(1-ε)∫_BR |Δu|^p-1|u|^1-εdx.

类似J₁ 的估计可知

I₄≤C(n,p)2^ε(1+ε)/(1-ε)(4^1-ε)/(R^1-ε)

(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+2^ε(1+ε)/(1-ε)(4^1-ε)/(R^1-ε)

(∫_BR u_BR^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε)(n+1-ε))/(n[(n+1-ε)(p-ε)-(n-p+1)(1-ε)])}.(25)

下面估计I₅.类似I₂ 的估计可知

I₅=∫_BR||Δu|^p-2ΔuH|dx=

∫_BR|Δu|^p-1|H|dx≤C(n,p)ε

(∫_BR |Δu|^p-εdx)^(p-1)/(p-ε)(∫_BR |Δ(ηu)|^p-εdx)^{(1-ε)/(p-ε)}≤

Cε∫_BR |Δu|^p-εdx+Cε∫_BRu_BR^p-εdx.

综合估计式I₁-I₅ 和式(12)可得

(a+1)∫_BR/2|Δu|^p-εdx≤

C(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+Cε₁ ∫_BR |Δu|^p-ε

dx+C+C(ε₁)∫_BR |φ(x)|^(p-ε)/(p-1)dx+

C∫_BR u_BR^p-ε dx+Cε∫_BR |Δu|^p-εdx+

Cε{∫_BR u_BR^p-εdx+1+∫_BR 〖JB<2|〗φ(x)+

u_BR^(p-1)α〖JB>2|〗^(p-ε)/(p-1)dx)^(p-1)/(p-ε)}+C(n,p,ε₁)

(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-1))}+C(n,p)ε₁

∫_BR |Δu|^p-εdx+C(n,p)ε₁ ∫_BR u_BR^p-εdx+

C(n,p)2^ε(1+ε)/(1-ε)(4^1-ε)/(R^1-ε)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

2^ε(1+ε)/(1-ε)(4^1-ε)/(R^1-ε)(∫_BR u_BR^{(n(p-ε))/(n-p+1)}dx)^{((n-p+1)(1-ε)(n+1-ε))/(n[(n+1-ε)(p-ε)-(n-p+1)(1-ε)])}+

Cε∫_BR |Δu|^p-εdx+Cε∫_BR u_BR^p-εdx≤

2C(ε₁+ε)∫_BR |Δu|^p-εdx+

((24)/RC(n,p)+C)(∫_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+

C(n,p,ε₁)(∫_BR |f|^{(n(p-ε))/(n(p-1)+p-ε)}dx)^{(n(p-1)+p-ε)/(n(p-1))}+

(C(ε₁)+Cε)∫_BR |φ(x)|^(p-ε)/(p-1)dx+

C(n,p,|B_R|,u_BR,=u=_W^1,p″(BR)),(26)

整理可得

f_{BR/2|Δu|^1-εdx≤θfBR |Δu|^p-εdx+}

K(f_BR |Δu|^{(n(p-ε))/(n+1-ε)}dx)^(n+1-ε)/n+f_BR |F|^pdx.(27)

选取ε,ε₁ 充分小,使得0<θ=(2ⁿ⁺¹(ε₁+ε))/(a+1)<1,其中K=[(24)/RC(n,p)+C]2ⁿ,注意到

1=(∫_BR1dx)/(|B_R|)=f_BR1dx,则

|F|^p=[C(n,p,ε₁,=f=_{(n(p-ε))/(n(p-1)+p-ε)})|f|^{(n(p-ε))/(n(p-1)+p-ε)}+

(C(ε₁)+Cε)|φ(x)|^(p-ε)/(p-1)+

C(n,p,|B_R|,u_BR,=u=_W^1,p″(BR))]·2ⁿ,

其中1<τ=(n(p-ε))/(n+1-ε)<p-ε.

利用引理2,可得存在r'>r₁,使u∈W^1,r'_loc(Ω,Rⁿ).注意到f∈L^{1,(nq)/(n(p-1)+q)}_loc(Ω),φ∈L^p/(p-1)(Ω)_,其中q>p,可以发现函数F∈L^∞(Ω),且根据引理2,有

(f_BR/2|u|^r'dx)^1/r'≤C'(f_BR|u|^p-εdx)^1/p-ε+

C'(f_BR|F|^r'dx)^1/r',(28)

即存在r'∈(p-ε,p)(p-ε,+∞),使式(28)成立.下面只须证区间(p-ε,p)是右闭的.实际上,当r'=p时,由引理2以及式(28)可以发现,结论显然成立.故反复应用引理2可得定理的结论.