# Strong Unique Continuation for Solutions of  ap(x)-Laplacian Problem

Authors: Johnny Cuadro; Gabriel López
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We study the strong unique continuation property for solutions to the quasilinear elliptic equation-div(|∇u|p(x)-2∇u)+V(x)|u|p(x)-2u=0inΩ
where V(x)∈LN/p(x)(Ω), Ω is a smooth bounded domain in ℝN, and 1<p(x)<N for x in Ω.


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## Body

## 1. Introduction and Preliminary Results

LetΩ be an open, connected subset in ℝN. Consider the Schrödinger Operator H=-Δ+V.  If Hu=0, and if u vanishes of infinite order at one point x0∈Ω (see definitions in Section 3) imply that u≡0 in Ω, then H has the Strong Unique Continuation Property (S.U.C.P). If, on the other hand, Hu=0 in Ω, and u=0 in Ω′, an open subset of Ω, imply that u≡0 in Ω, we say that H has the Weak Unique Continuation Property (W.U.C.P). In 1939 Carleman [1] showed that H=-Δ+V has the S.U.C.P whenever V∈Lloc∞(ℝ2). In order to prove this result he introduced a method, the so-called Carleman estimates, which has permeated almost all the subsequent works in the subject. For instance,  Jerison and Kenig [2] showed that if n>2,p⩾N/2 and V∈Llocp, then H has the S.U.C.P.;  Fabes et al. in [3] gave a positive answer for a radial potential V to Simon's conjecture, which stated that for a potential V in the Stummel-Kato class and u∈H1(Ω) then H has the S.U.C.P. Other results were obtained by de Figueiredo and Gossez, but for Linear Elliptic Operators in the case V∈LN/2(Ω),N>2, [4]. Also, Loulit extended this property to N=2 [5]. More recently, Hadi and Tsouli [6] proved Strong Unique Continuation Property for the p-Laplacian in the case V∈LN/p(Ω),p<N and p constant.Equations involving variable exponent growth conditions have been intensively discussed in the last decade. A strong motivation in the study of such kind of problems is due to the fact that they can model with high accuracy various phenomena which arise from the study of elastic mechanics, electrorheological fluids, or image restoration; for information on modeling physical phenomena by equations involvingp(x)-growth condition we refer to [7–12]. The understanding of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces, Lp(x) and W1,p(x), where p(x) is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in literature as early as 1931 in an article by Orlicz [13]. The spaces Lp(x) are special cases of Orlicz spaces Lφ originated by Nakano [14] and developed by Musielak and Orlicz [15, 16], where f∈Lφ if and only if ∫‍φ(x,|f(x)|)dx<∞ for a suitable φ. For some interesting results on elliptic equation involving variable exponent growth conditions see [17–19]. We point out the presence of the p(x)-Laplace operator. This is a natural extension of the p-Laplace operator, with p positive constant. However, such generalizations are not trivial since the p(x)-Laplace operator possesses a more complicated structure than p-Laplace operator; for example, it is inhomogeneous.In this paper we prove Strong Unique Continuation Property of the solutions of the quasilinear elliptic equation:(1.1)-div(|∇u|p(x)-2∇u)+V(x)|u|p(x)-2u=0inΩ,
where 1<p(x)<N,V∈LN/p(x)(Ω) and Ω⊂ℝN is a bounded domain with smooth boundary.Finally, we recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spacesLp(·)(Ω) and W01,p(·)(Ω), where Ω is a bounded domain in ℝN.SetC+(Ω¯)={h∈C(Ω¯):minx∈Ω¯h(x)>1}. For any h∈C+(Ω¯) we define
(1.2)h+=supx∈Ωh(x),h-=infx∈Ωh(x).
For p∈C+(Ω¯), we introduce the variable exponent Lebesgue space:(1.3)Lp(·)(Ω)={u:uisameasurablereal-valuedfunctionsuchthat∫Ω‍|u(x)|p(x)dx<∞},
endowed with the so-called Luxemburg norm:(1.4)|u|p(·)=inf{μ>0;∫Ω‍|u(x)μ|p(x)dx≤1},
which is a separable and reflexive Banach space. For basic properties of the variable exponent Lebesgue spaces we refer to [20]. If 0<|Ω|<∞ and p1, p2 are variable exponents in C+(Ω¯) such that p1≤p2 in Ω, then the embedding Lp2(·)(Ω)↪Lp1(·)(Ω) is continuous [20, Theorem 2.8].LetLp'(·)(Ω) be the conjugate space of Lp(·)(Ω), obtained by conjugating the exponent pointwise, that is, 1/p(x)+1/p'(x)=1, [20, Corollary 2.7]. For any u∈Lp(·)(Ω) and v∈Lp'(·)(Ω) the following Hölder type inequality
(1.5)|∫Ω‍uvdx|≤(1p-+1p′-)|u|p(·)|v|p′(·)
is valid.An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by thep(·)-modular of the Lp(·)(Ω) space, which is the mapping ρp(·):Lp(·)(Ω)→ℝ defined by
(1.6)ρp(·)(u)=∫Ω‍|u|p(x)dx.
If (un),u∈Lp(·)(Ω) then the following relations hold:
(1.7)|u|p(·)<1(=1;>1)⟺ρp(·)(u)<1(=1;>1),(1.8)|u|p(·)>1⇒|u|p(·)p-≤ρp(·)(u)≤|u|p(·)p+,(1.9)|u|p(·)<1⇒|u|p(·)p+≤ρp(·)(u)≤|u|p(·)p-,(1.10)|un-u|p(·)→0⟺ρp(·)(un-u)→0,
since p+<∞. For a proof of these facts see [20]. Spaces with p+=∞ have been studied by Edmunds et al. [21].Next, we defineW01,p(x)(Ω) as the closure of C0∞(Ω) under the norm
(1.11)∥u∥p(x)=|∇u|p(x).
The space (W01,p(x)(Ω),∥·∥p(x)) is a separable and reflexive Banach space. We note that if q∈C+(Ω¯) and q(x)<p*(x) for all x∈Ω¯ then the embedding W01,p(x)(Ω)↪Lq(x)(Ω) is continuous, where p*(x)=Np(x)/(N-p(x)) if p(x)<N or p*(x)=+∞ if p(x)≥N [20, Theorems 3.9 and 3.3] (see also [22, Theorems 1.3 and 1.1]).The bounded variable exponentp is said to be Log-Hölder continuous if there is a constant C>0 such that
(1.12)|p(x)-p(y)|≤C-log(|x-y|)
for all x,y∈ℝN, such that |x-y|≤1/2.A bounded exponentp is Log-Hölder continuous in Ω if and only if there exists a constant C>0 such that
(1.13)|B|pB--pB+≤C
for every ball B⊂Ω [23, Lemma 4.1.6, page 101].As a result of the Log-Hölder continuous condition we have(1.14)r-(pB+-pB-)≤C,C-1r-p(y)≤rp(x)≤Cr-p(y),
for all x,y∈B:=B(x0,r)⊂Ω and the constant C depends only on the constant Log-Hölder continuous. It's well known that Smooth Functions are dense in Variable Exponent Sobolev Spaces if the exponent p satisfies the Log-Hölder condition [23, Proposition 11.2.3, page 346].


## 2. On Fefferman's Type Inequality

For everyu∈W01,p(·)(Ω) the norm Poincaré inequality
(2.1)|u|Lp(·)(Ω)≤cdiam(Ω)|∇u|Lp(·),c=C(N,Ω,clog(p)) holds (we refer to [24] for notations and proofs). Nevertheless, the modular inequality
(2.2)∫Ω‍|u|p(x)dx≤C∫Ω‍|∇u|p(x)dx,∀u∈W01,p(·)(Ω)
not always holds (see [18, Theorem 3.1]). It is known that (2.2) holds if, for instance (i) N>1, and the function f(t):=p(xo+tw) is monotone [18, Theorem 3.4] with xo+tw with an appropriate setting in Ω; (ii) if there exists a function ξ≥0 such that ∇p·∇ξ≥0,∥∇ξ∥≠0 [25, Theorem 1]; (iii) If there exists a:Ω→ℝN bounded such that diva(x)≥a0>0 for all x∈Ω- and a(x)·∇p(x)=0 for all x∈Ω, [26, Theorem 1]. To the best of our knowledge necessary and sufficient conditions in order to ensure that
(2.3)infu∈W1,p(·)(Ω)/{0}∫Ω‍|∇u|p(x)∫Ω‍|u|p(x)>0
have not been obtained yet, except in the case N=1, [18, Theorem 3.2]. The following definition is in order.Definition 2.1.
We say thatp(·) belongs to the Modular Poincaré Inequality Class, MPIC(Ω), if there exist necessary conditions to ensure that
(2.4)∫Ω‍|u|p(x)≤C∫Ω‍|∇u|p(x),∀u∈W01,p(·)(Ω),C=C(N,Ω,clog(p))>0 holds.In [27] Fefferman proved the following inequality:
(2.5)∫ℝN‍|u(x)|p|f(x)|dx≤C∫ℝN‍|∇u(x)|pdx∀u∈C0∞(ℝN).
in the case p=2, assuming f in the Morrey's space Lr,N-2r(ℝN), with 1<r≤N/2. Later in [28] Schechter showed the same result taking f in the Stummel-Kato class S(ℝN). Chiarenza and Frasca [29] generalized Fefferman’s result proving (2.5) under the assumption f∈Lr,N-pr(ℝN), with 1<r<N/p and 1<p<N. Zamboni in [30] generalized Schecter's result proving (2.5) under the assumption f∈M~p(ℝN), with 1<p<N. We stress out that it is not possible to compare the assumptions f∈Lr,N-pr(ℝN), the Morrey class, and f∈S(ℝN), the Stumel-Kato class. All the mentioned results were obtained for fixed p. The theory for a variable exponent spaces is a growing area but Modular Fefferman-type inequalities are more scarce than Poincaré inequalities in variable exponent setting. In [31]  Cuadro and López proved inequality (2.6) for variable exponent spaces. We use such inequality in order to prove S.U.C.P. We include the proof for the convenience of the reader.Theorem 2.2.
Letp be a Log-Hölder continuous exponent with 1<p(x)<N, and p∈MPIC(Ω). Let V∈Lloc1(Ω) with 0<ε<V(x) almost everywhere. Then there exists a positive constant C=C(N,Ω,clog(p)) such that
(2.6)∫Ω‍V(x)|u(x)|p(x)dx≤C∫Ω‍|∇u(x)|p(x)dx
for any u∈W01,p(x)(Ω).Proof.
Letu∈W01,p(x)(Ω) supported in B(x0,r). Given that V∈Lloc1(Ω) the function
(2.7)w(x):=(∫x10x1‍V(ξ1,x2,…,xn)dξ1,…,∫xN0xN‍V(x1,…,xN-1,ξN)dξN),
where x0=(x10,…,xN0) and x=(x1,…,xN)∈B(x0,r), is well defined. Notice that ∫xi0xi‍V(x1,…,ξi,…,xn)dξi∈C[xi0,xi] for i=1,…,N (Lemme VIII.2 [32]) so that divw(x)=NV(x). Moreover,
(2.8)|V(x)|L1(B(x0,r))≥∫x10x1‍⋯∫xN0xN‍V(ξ)dξn⋯dξ1,
where ξ=(ξ1,…,ξN). Therefore, |w(x)|≤N|V(x)|L1(B(x0,r)).
A direct calculation leads to(2.9)div(|u|p(x)w(x))=|u(x)|p(x)divw(x)+p(x)|u|p(x)-2u∇u·w(x)+|u|p(x)logu∇p(x)·w(x).
Now the Divergence Theorem implies ∫B(x0,r)‍div(|u|p(x)w(x))=0, and so
(2.10)∫B(x0,r)‍|u(x)|p(x)divw(x)dx≤p+∫B(x0,r)‍|u(x)|p(x)-1|∇u(x)||w(x)|dx+∫B(x0,r)‍|u(x)|p(x)log|u(x)||∇p(x)||w(x)|dx.
Set
(2.11)I1:=p+∫B(x0,r)‍|u(x)|p(x)-1|∇u(x)||w(x)|dx,I2:=∫B(x0,r)‍|u(x)|p(x)log|u(x)||∇p(x)||w(x)|dx.
Now we estimateI2 by distinguishing the case when |u(x)|≤1 and |u(x)|>1. Notice that the relations
(2.12)sup0≤t≤1tη|logt|<∞(2.13)supt>1t-ηlogt<∞
hold for η>0.
LetΩ1=:{x∈Br:|u(x)|≤1} and Ω2=:{x∈Br:|u(x)|>1}, then for (2.12) and (2.13) we have
(2.14)I2≤C1∫Ω1‍|w(x)||u(x)|p(x)-η1dx+C2∫Ω2‍|w(x)||u(x)|p(x)+η2dx.
We can choose k∈ℕ such that p(x)-1/k≥p-. Since u∈Lp-(B(x0,r)) and in Ω1,|u(x)|≤1 we have
(2.15)|u(x)|p(x)-1/n≤|u(x)|p-,
for n>k. The Lebesgue Dominated Convergence Theorem implies
(2.16)limn→∞∫Ω1‍|u(x)|p(x)-1/ndx=∫Ω1‍|u(x)|p(x)dx.
For Ω2 we can choose k' such that p(x)+1/k'≤(p(x))*=Np(x)/(N-p(x)). So
(2.17)|u(x)|p(x)+1/n≤|u(x)|(p(x))*,n>k', and x∈Ω2. Since u∈L(p(x))*(B(x0,r)) [23, Theorem 8.3.1] we may use the Lebesgue Theorem again to obtain
(2.18)limn→∞∫Ω2‍|u(x)|p(x)+1/ndx=∫Ω2‍|u(x)|p(x)dx.
Given that p∈MPIC(Ω) we have
(2.19)I2≤C∫B(x0,r)‍|u|p(x)dx≤C∫B(x0,r)‍|∇u|p(x)dx.
Now we estimate I1 by using the modular Young's inequality [24, equation (3.2.21)]:
(2.20)I1≤p+C1∫B(x0,r)‍|w(x)|p(x)/(p(x)-1)|u(x)|p(x)+p+C2∫B(x0,r)‍|∇u(x)|p(x).
Again, since p∈MPIC(Ω) we obtain
(2.21)I1≤C∫B(x0,r)‍|∇u|p(x)dx.
Finally, recalling that divw(x)=NV(x) we get
(2.22)N∫B(xo,r)‍V(x)|u(x)|p(x)≤C∫B(x0,r)‍|∇u(x)|p(x)dx,
which leads to the claim of the theorem.


## 3. Strong Unique Continuation

Consider the equation(3.1)Hu:=div(|∇u|p(x)-2∇u)+V(x)|u|p(x)-2u=0,x∈Ω,u∈Wloc1·p(x)(Ω),1<p(x)<N,V∈LN/p(x)(Ω).A weak solution of (3.1) is the function u∈Wloc1·p(x)(Ω) such that
(3.2)∫Ω‍|∇u|p(x)-2∇u·∇φdx+∫Ω‍V(x)|u|p(x)-2u·φdx=0,
for all φ∈W01,p(x)(Ω).The main interest of this section is to prove a unique continuation result for solutions of (3.1) according to the following definition.Definition 3.1.
A functionu∈Llocp(x)(Ω) has a zero of infinite order in the p(x)-mean at a point x0∈Ω if, for each k∈ℕ,
(3.3)∫|x-x0|⩽R‍|u|p(x)dx=O(Rk).Recall thatΩ⊂ℝN is a bounded open set. We want to prove estimates’ independency of p+ for bounded solutions. For this purpose we assume throughout this section that 1<p-≤p+<∞ and p is Lipschitz continuous. In particular, p is Log-Hölder continuous. The new feature in the estimate is the choice of a test function which includes the variable exponent. This has both advantages and disadvantages: we need to assume that p is differentiable almost everywhere, but, on the other hand, we avoid terms involving p+, which would be impossible to control later, see [24].Before proving Theorem3.5 which is the main result of this paper we require the following Lemmas.Lemma 3.2.
Letp be a Log-Hölder continuous exponent with 1<p(x)<N, and p∈MPIC(Ω). Let V∈LN/p(x),0<ϵ<V(x), almost everywhere and u∈W01,p(x)(Ω). Then, for each ϵo>0, there exists K such that
(3.4)∫Ω‍V|u|p(x)dx≤ϵo∫Ω‍|∇u|p(x)+K∫Ω‍|u|p(x)dx.Proof.
Letϵo>0 be given. We have
(3.5)∫Ω‍V(x)|u|p(x)=∫{x:V(x)>t}‍V(x)|u|p(x)+∫{x:V(x)≤t}‍V(x)|u|p(x)≤∫{x:V(x)>t}‍V(x)|u|p(x)+t∫Ω‍|u|p(x)≤C∫{x:V(x)>t}‍|∇u|p(x)+t∫Ω‍|u|p(x),
where the last inequality follows from Theorem 2.2. Now, notice that the measure λ(E)=∫E‍|∇u|p(x) is absolutely continuous with respect to the Lebesgue measure μ. It follows that for ϵ1:=(ϵo/C)∫Ω‍|∇u|p(x)>0 there exists δ>0 such that ∫E‍|∇u|p(x)dμ<ϵ1 whenever μ(E)<δ. Moreover, by Chebyshev's type inequality,
(3.6)μ({x:V(x)>t})≤t-N/p+∫Ω‍V(x)N/p(x).
So taking t sufficiently big, we get the desired inequality.Lemma 3.3.
Letp:Ω→(1,N) be an exponent with 1<p-≤p+<∞ and such that p∈MPIC(Ω) is Lipschitz continuous. Let u be solution of (3.1) in Ω,  and Br and B2r two concentric balls contained in Ω. Then
(3.7)∫Br‍|∇u|p(x)≤Crp(x0)∫B2r‍|u|p(x),
where the constant C does not depend on r,x0∈B2r and V∈LN/p(x).Proof.
Takeη∈C0∞(Ω), with suppη⊂B2r,0≤η≤1 such that η(x)=1 for any x∈Br and |∇η|≤C/r. We want to use as test function ψ=ηp(x)u. To this end we show first that ψ∈W01,p(x)(Ω); it is clear that ψ∈Lp(x)(Ω) since u is solution of (3.1). Furthermore, since 0⩽η⩽1 then |ηlogη|⩽a for some constant a, so
(3.8)|∇ψ|≤|∇uηp(x)|+|up(x)ηp(x)-1∇η|+|uηp(x)logη∇p(x)|≤|∇uηp(x)|+|up(x)ηp(x)-1∇η|+|uηp(x)-1∇p(x)ηlogη|≤|∇uηp(x)|+|up(x)ηp(x)-1∇η|+|uηp(x)-1∇p(x)|a≤|∇uηp(x)|+|uηp(x)-1|(|∇η|p(x)+a|∇p(x)|)≤|∇u|+|u|(Cp++aL)≤|∇u|+Cp|u|.
Hence,|∇ψ|p(x)⩽2p+-1|∇u|p(x)+Cpp+2p+-1|u|p(x). Therefore, |∇ψ|∈Lp(x)(Ω).
Now we can useψ=ηp(x)u as a test function to obtain
(3.9)0=∫B2r‍|∇u|p(x)-2∇u·∇ψdx+∫B2r‍V|u|p(x)-2uψdx=∫B2r‍|∇u|p(x)ηp(x)dx+∫B2r‍|∇u|p(x)-2u∇u·(p(x)ηp(x)-1∇η+ηp(x)logη∇p(x))dx+∫B2r‍Vηp(x)|u|p(x)dx.
Let
(3.10)I1:=∫B2r‍|∇u|p(x)-2u∇u·(p(x)ηp(x)-1∇η+ηp(x)logη∇p(x))dx,I2:=∫B2r‍Vηp(x)|u|p(x)dx.
We can estimate I1 by
(3.11)I1≤∫B2r‍|∇u|p(x)-2|u||∇u|(|p(x)ηp(x)-1∇η|+|ηp(x)logη∇p(x)|)dx≤∫B2r‍(|∇u||η|)p(x)-1(|u||∇η|)(|p(x)|+a|∇p(x)|)dx≤∫B2r‍(|∇u||η|)p(x)-1(|u||∇η|)(p++aL)dx≤Cp∫B2r‍(|∇u||η|)p(x)-1(|u||∇η|)dx≤Cp∫B2r‍ϵ(|∇u||η|)p(x)+(1ϵ)p(x)-1(|u||∇η|)p(x)dx,
where the Young-type inequality
(3.12)fg≤ϵfp(x)/(p(x)-1)+(1ϵ)p(x)-1gp(x)
was used in the last inequality. Moreover,
(3.13)I1≤Cpϵ∫B2r‍|∇u|p(x)ηp(x)dx+Cp∫B2r‍(1ϵ)p(x)-1|u|p(x)|∇η|p(x)dx≤Cpϵ∫B2r‍|∇u|p(x)ηp(x)dx+Cp(1ϵ)p+-1∫B2r‍|u|p(x)|∇η|p(x)dx.
We estimate I2:
(3.14)I2=∫B2r‍Vηp(x)|u|p(x)dx=∫B2r‍V|ηu|p(x)dx≤ϵ∫B2r‍|∇(uη)|p(x)dx+Cϵ∫B2r‍|ηu|p(x)dx,
where Lemma (3.2) was used in the last inequality.
Now using the estimates forI1 and I2, we have
(3.15)∫B2r‍|∇u|p(x)ηp(x)dx≤Cpϵ∫B2r‍|∇u|p(x)ηp(x)dx+Cp(1ϵ)p+-1∫B2r‍|u|p(x)|∇η|p(x)dxdx+ϵ∫B2r‍|∇(uη)|p(x)+Cϵ∫B2r‍|ηu|p(x)dx≤ϵ(C1+C2)∫B2r‍|∇u|p(x)ηp(x)dx+ϵϵp+(C1+C2ϵp+)∫B2r‍|u|p(x)|∇η|p(x)dx+Cϵ∫B2r‍ηp(x)|u|p(x)dx
for 0<ϵ≤1. By choosing ϵ≤min{1,1/2(C1+C2)}, we have
(3.16)∫B2r‍|∇u|p(x)ηp(x)dx≤1ϵp+∫B2r‍|u|p(x)|∇η|p(x)dx+2Cϵ∫B2r‍|u|p(x)ηp(x)dx≤1ϵp+∫B2r‍|u|p(x)(Cr)p(x)dx+2Cϵ∫B2r‍|u|p(x)(Cr)p(x)dx.
Since p(x) is Log-Hölder r-p(x)≤Cr-p(x0) for all x0∈B2r, then
(3.17)∫B2r‍|∇u|p(x)ηp(x)dx≤1ϵp+∫B2r‍|u|p(x)(Cr)p(x0)dx+2Cϵ∫B2r‍|u|p(x)(Cr)p(x0)dx≤(1ϵp++Cϵ)Cp(x0)rp(x0)∫B2r‍|u|p(x)dx≤Crp(x0)∫B2r‍|u|p(x)dx.
Therefore,
(3.18)∫Br‍|∇u|p(x)dx≤Crp(x0)∫B2r‍|u|p(x)dx.Lemma 3.4.
Letu∈W1,1(B(x0,r)), where B(x0,r) is the ball of radius r>0 in ℝN and E={x∈B(x0,r):u(x)=0}. Then there exists a constant β>0 depending only on N, such that
(3.19)∫D‍|u(x)|dx≤βrN|E||D|1/N∫B(x0,r)‍|∇u(x)|dx
for all B(x0,r), u as above, and all mensurable sets D⊂B(x0,r).Proof.
See [33, Lemma 3.4, page 54].Now we are ready to prove the main result in this paper.Theorem 3.5.
LetΩ be a bounded domain in ℝN, p:Ω→(1,N) an exponent with 1<p-≤p+<∞ and such that p∈MPIC(Ω) is Lipschitz continuous, and u∈Wloc1,p(x)(Ω) a solution of (3.1). If u vanishes on set E⊂Ω of positive measure, then u has a zero of infinite order in the p(x)-mean.Proof.
We know that almost every point ofE is a point of density, let x0 be such a point, that is,
(3.20)|EC∩B(x0,r)||B(x0,r)|→0,|E∩B(x0,r)||B(x0,r)|→1
as r→0.
LetBr:=B(x0,r). So for a given ϵ>0, there exists r0=r0(ϵ) such that for r≤r0,
(3.21)|EC∩Br||Br|<ϵ,|E∩Br||Br|>1-ϵ,
where EC denote the complement of E in Ω. Taking r0 smaller if necessary, we may assume that Br0⊂Ω. Since u=0 on E, and using Lemma 3.4 we have
(3.22)∫Br‍|u(x)|p(x)dx=∫Br∩EC‍|u(x)|p(x)dx+∫Br∩E‍|u(x)|p(x)dx=∫Br∩EC‍|u(x)|p(x)dx≤βrN|Br∩E||Br∩EC|1/N∫Br‍|∇(|u(x)|p(x))|dx≤CβrNrN-1ϵ1/N1-ϵ∫Br‍|∇(|u(x)|p(x))|dx=Crϵ1/N1-ϵ∫Br‍|∇(|u(x)|p(x))|dx.
But|∇(|u(x)|p(x))|≤|p(x)|||u(x)|p(x)-1||∇u(x)|+|∇p(x)|||u(x)|p(x)log|u(x)||. Hence,
(3.23)∫Br‍|u(x)|p(x)dx≤Crϵ1/N1-ϵ[∫Br‍p+|u(x)|p(x)-1|∇u(x)|dx+∫Br‍L|u(x)|p(x)|log|u(x)||dx].
Let
(3.24)I1:=∫Br‍|u(x)|p(x)-1|∇u(x)|dx,I2:=∫Br‍||u(x)|p(x)log|u(x)||dx.I1 can be estimated using the Young type inequality with ϵ=1/r:
(3.25)∫Br‍|u(x)|p(x)-1|∇u(x)|dx⩽∫Br‍1r|u(x)|p(x)dx+∫Br‍rp(x)-1|∇u(x)|p(x)dx.
Now we estimate I2 by distinguishing the case when |u(x)|≤1 and |u(x)|>1, using the relations (2.2) and (2.13).
LetΩ1=:{x∈Br:|u(x)|≤1} and Ω2=:{x∈Br:|u(x)|>1}, then
(3.26)I2≤C1∫Ω1‍|u(x)|p(x)-η1dx+C2∫Ω2‍|u(x)|p(x)+η2dx.
We can choose k∈ℕ such that p(x)-1/k≥p-. Since u∈Lp-(B(x0,r)) and in Ω1,|u(x)|≤1 we have
(3.27)|u(x)|p(x)-1/n≤|u(x)|p-,
for n>k. The Lebesgue Dominated Convergence Theorem implies
(3.28)limn→∞∫Ω1‍|u(x)|p(x)-1/ndx=∫Ω1‍|u(x)|p(x)dx.
For Ω2 we can choose k' such that p(x)+1/k'≤(p(x))*=Np(x)/(N-p(x)). So
(3.29)|u(x)|p(x)+1/n≤|u(x)|(p(x))*,
for n>k', and x∈Ω2. Since u∈L(p(x))*(B(x0,r)) [23, Theorem 8.3.1] we may use the Lebesgue Theorem again to obtain
(3.30)limn→∞∫Ω2‍|u(x)|p(x)+1/ndx=∫Ω2‍|u(x)|p(x)dx.
Therefore,
(3.31)I2≤C∫B(x0,r)‍|u|p(x)dx.
Now, using estimates for I1 and I2, and noticing that for 0<r<1 we have rp(x)<rp-, we get
(3.32)∫Br‍|u(x)|p(x)dx≤ϵ1/N1-ϵ{Cp+∫Br‍|u(x)|p(x)dxε1/N1-ε21+Cp+rp-∫Br‍|∇u(x)|p(x)dx+Cp∫Br‍|u(x)|p(x)dx}
and by Lemma (3.3) we have
(3.33)∫Br‍|u(x)|p(x)dx≤ϵ1/N1-ϵ{Cp+∫B2r‍|u(x)|p(x)dx+Cp+rp-r-p(x0)∫B2r‍|u(x)|p(x)dxϵ1/N1-ϵ21+Cp∫B2r‍|u(x)|p(x)dx}≤Cϵ1/N1-ϵ∫B2r‍|u(x)|p(x)dx,
where C is independent of ε and of r as r→0. Note that rp--p(x0)<C where C is the Log-Hölder constant. From this point the argument in the proof is standard, see, for instance, in [4] the proof of Lemma 1, page 344-345 from equation (10) to the end of the proof, or the proof of Theorem 2.1 [6], from inequality (2.18) to (2.23), page 216; we include this last part of the proof for the sake of completeness. Set f(r):=∫Br‍|u(x)|p(x)dx. Let us fix n∈ℕ and choose ϵ>0 such that (Cϵ1/N)/(1-ϵ)≤2-n. Now, observe that r0 depends on n, hence by the last inequality we deduce
(3.34)f(r)≤2-nf(2r),forr≤r0.
Iterating (3.34), we get
(3.35)f(ρ)≤2-knf(2kρ),if2k-1ρ≤r0.
Thus, given that 0<r<r0(n) and choosing k∈ℕ such that
(3.36)2-kr0≤r≤2-(k-1)r0.
From (3.35), we conclude that
(3.37)f(r)≤2-knf(2kr)≤2-knf(2r0),
and since 2-k≤r/r0, we get
(3.38)f(r)≤(rr0)nf(2r0),
which shows that x0 is a zero infinite order in p(x)-mean.



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*Source: 108671-2012-11-05.xml*