Strong Unique Continuation for Solutions of a 𝑝(𝑥)-Laplacian Problem

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)−2}𝑢=0inΩ where 𝑉(𝑥)∈𝐿^{𝑁/𝑝(𝑥)}(Ω), Ω is a smooth bounded domain in ℝ^𝑁, and 1<𝑝(𝑥)<𝑁 for 𝑥 in Ω.

1. Introduction and Preliminary Results

Let Ω be an open, connected subset in ℝ^𝑁. 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∈𝐿^∞_loc(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∈𝐿^𝑝_loc, 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∈𝐿^𝑁/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∈𝐿^𝑁/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 involving 𝑝(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, 𝐿^𝑝(𝑥) and 𝑊^1,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 𝐿^𝑝(𝑥) are special cases of Orlicz spaces 𝐿^𝜑 originated by Nakano [14] and developed by Musielak and Orlicz [15, 16], where 𝑓∈𝐿^𝜑 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: −div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢+𝑉(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢=0inΩ, (1.1) where 1<𝑝(𝑥)<𝑁,𝑉∈𝐿^{𝑁/𝑝(𝑥)}(Ω) and Ω⊂ℝ^𝑁 is a bounded domain with smooth boundary.

Finally, we recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spaces 𝐿^𝑝(⋅)(Ω) and 𝑊₀^1,𝑝(⋅)(Ω), where Ω is a bounded domain in ℝ^𝑁.

Set 𝐶₊(Ω)={ℎ∈𝐶(Ω)∶min_𝑥∈Ωℎ(𝑥)>1}. For any ℎ∈𝐶₊(Ω) we define ℎ⁺=sup_𝑥∈Ωℎ(𝑥),ℎ⁻=inf_𝑥∈Ωℎ(𝑥). (1.2) For 𝑝∈𝐶₊(Ω), we introduce the variable exponent Lebesgue space:𝐿^𝑝(⋅)(Ω)=𝑢∶𝑢isameasurablereal-valuedfunctionsuchthat_Ω|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥<∞, (1.3) endowed with the so-called Luxemburg norm:|𝑢|_𝑝(⋅)=inf𝜇>0;_Ω𝑢(𝑥)𝜇^𝑝(𝑥)𝑑𝑥≤1, (1.4) 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 𝐿^𝑝2(⋅)(Ω)↪𝐿^𝑝1(⋅)(Ω) is continuous [20, Theorem 2.8].

Let 𝐿^𝑝′(⋅)(Ω) be the conjugate space of 𝐿^𝑝(⋅)(Ω), obtained by conjugating the exponent pointwise, that is, 1/𝑝(𝑥)+1/𝑝^′(𝑥)=1, [20, Corollary 2.7]. For any 𝑢∈𝐿^𝑝(⋅)(Ω) and 𝑣∈𝐿^𝑝′(⋅)(Ω) the following Hölder type inequality _Ω≤1𝑢𝑣𝑑𝑥𝑝⁻+1𝑝^′−|𝑢|_𝑝(⋅)|𝑣|_𝑝′(⋅) (1.5) is valid.

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the 𝑝(⋅)-modular of the 𝐿^𝑝(⋅)(Ω) space, which is the mapping 𝜌_𝑝(⋅)∶𝐿^𝑝(⋅)(Ω)→ℝ defined by 𝜌_𝑝(⋅)(𝑢)=_Ω|𝑢|^𝑝(𝑥)𝑑𝑥. (1.6) If (𝑢_𝑛),𝑢∈𝐿^𝑝(⋅)(Ω) then the following relations hold: |𝑢|_𝑝(⋅)<1(=1;>1)⟺𝜌_𝑝(⋅)(𝑢)<1(=1;>1),|𝑢|_𝑝(⋅)>1⇒|𝑢|^𝑝−_𝑝(⋅)≤𝜌_𝑝(⋅)(𝑢)≤|𝑢|^𝑝+_𝑝(⋅),|𝑢|_𝑝(⋅)<1⇒|𝑢|^𝑝+_𝑝(⋅)≤𝜌_𝑝(⋅)(𝑢)≤|𝑢|^𝑝−_𝑝(⋅),𝑢_𝑛−𝑢_𝑝(⋅)→0⟺𝜌_𝑝(⋅)𝑢_𝑛−𝑢→0, (1.7)(1.8)(1.9)(1.10) since 𝑝⁺<∞. For a proof of these facts see [20]. Spaces with 𝑝⁺=∞ have been studied by Edmunds et al. [21].

Next, we define 𝑊₀^1,𝑝(𝑥)(Ω) as the closure of 𝐶^∞₀(Ω) under the norm ‖𝑢‖_𝑝(𝑥)=|∇𝑢|_𝑝(𝑥). (1.11) The space (𝑊₀^1,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 𝑊₀^1,𝑝(𝑥)(Ω)↪𝐿^𝑞(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 exponent 𝑝 is said to be Log-Hölder continuous if there is a constant 𝐶>0 such that 𝑝(𝑥)𝑦≤𝐶−𝑝−log𝑥−𝑦 (1.12) for all 𝑥,𝑦∈ℝ^𝑁, such that |𝑥−𝑦|≤1/2.

A bounded exponent 𝑝 is Log-Hölder continuous in Ω if and only if there exists a constant 𝐶>0 such that |𝐵|^{𝑝−𝐵−𝑝+𝐵}≤𝐶 (1.13) for every ball 𝐵⊂Ω [23, Lemma 4.1.6, page 101].

As a result of the Log-Hölder continuous condition we have 𝑟^{−(𝑝+𝐵−𝑝−𝐵)}𝐶≤𝐶,₋₁𝑟^{−𝑝(𝑦)}≤𝑟^𝑝(𝑥)≤𝐶𝑟^{−𝑝(𝑦)}, (1.14) 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 every 𝑢∈𝑊₀^1,𝑝(⋅)(Ω) the norm Poincaré inequality |𝑢|_𝐿^𝑝(⋅)_(Ω)≤𝑐diam(Ω)|∇𝑢|_𝐿^𝑝(⋅), (2.1)𝑐=𝐶(𝑁,Ω,𝑐log(𝑝)) holds (we refer to [24] for notations and proofs). Nevertheless, the modular inequality _Ω|𝑢|^𝑝(𝑥)𝑑𝑥≤𝐶_Ω|∇𝑢|^𝑝(𝑥)𝑑𝑥,∀𝑢∈𝑊₀^1,𝑝(⋅)(Ω) (2.2) 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(𝑥_𝑜+𝑡𝑤) is monotone [18, Theorem 3.4] with 𝑥_𝑜+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→ℝ^𝑁 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 inf_{𝑢∈𝑊1,𝑝(⋅)(Ω)/{0}}∫_Ω|∇𝑢|^𝑝(𝑥)∫_Ω|𝑢|^𝑝(𝑥)>0 (2.3) 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 that 𝑝(⋅) belongs to the Modular Poincaré Inequality Class, MPIC(Ω), if there exist necessary conditions to ensure that _Ω|𝑢|^𝑝(𝑥)≤𝐶_Ω|∇𝑢|^𝑝(𝑥),∀𝑢∈𝑊₀^1,𝑝(⋅)(Ω), (2.4)𝐶=𝐶(𝑁,Ω,𝑐_log(𝑝))>0 holds.

In [27] Fefferman proved the following inequality: _ℝ𝑁|𝑢(𝑥)|^𝑝𝑓(𝑥)𝑑𝑥≤𝐶_ℝ𝑁|∇𝑢(𝑥)|^𝑝𝑑𝑥∀𝑢∈𝐶^∞₀ℝ^𝑁. (2.5) in the case 𝑝=2, assuming 𝑓 in the Morrey's space 𝐿^{𝑟,𝑁−2𝑟}(ℝ^𝑁), with 1<𝑟≤𝑁/2. Later in [28] Schechter showed the same result taking 𝑓 in the Stummel-Kato class 𝑆(ℝ^𝑁). Chiarenza and Frasca [29] generalized Fefferman’s result proving (2.5) under the assumption 𝑓∈𝐿^{𝑟,𝑁−𝑝𝑟}(ℝ^𝑁), 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(ℝ^𝑁), with 1<𝑝<𝑁. We stress out that it is not possible to compare the assumptions 𝑓∈𝐿^{𝑟,𝑁−𝑝𝑟}(ℝ^𝑁), the Morrey class, and 𝑓∈𝑆(ℝ^𝑁), 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. Let 𝑝 be a Log-Hölder continuous exponent with 1<𝑝(𝑥)<𝑁, and 𝑝∈MPIC(Ω). Let 𝑉∈𝐿¹_loc(Ω) with 0<𝜀<𝑉(𝑥) almost everywhere. Then there exists a positive constant 𝐶=𝐶(𝑁,Ω,𝑐_log(𝑝)) such that _Ω𝑉(𝑥)|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥≤𝐶_Ω|∇𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥 (2.6) for any 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω).

Proof. Let 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω) supported in 𝐵(𝑥₀,𝑟). Given that 𝑉∈𝐿¹_loc(Ω) the function 𝑤(𝑥)∶=^𝑥1_𝑥01𝑉𝜉₁,𝑥₂,…,𝑥_𝑛𝑑𝜉₁,…,^𝑥𝑁_𝑥0𝑁𝑉𝑥₁,…,𝑥_𝑁−1,𝜉_𝑁𝑑𝜉_𝑁, (2.7) where 𝑥₀=(𝑥⁰₁,…,𝑥⁰_𝑁) and 𝑥=(𝑥₁,…,𝑥_𝑁)∈𝐵(𝑥₀,𝑟), is well defined. Notice that ∫^𝑥𝑖_𝑥0𝑖𝑉(𝑥₁,…,𝜉_𝑖,…,𝑥_𝑛)𝑑𝜉_𝑖∈C[𝑥⁰_𝑖,𝑥_𝑖] for 𝑖=1,…,𝑁 (Lemme VIII.2 [32]) so that div𝑤(𝑥)=𝑁𝑉(𝑥). Moreover, |𝑉(𝑥)|_𝐿¹_{(𝐵(𝑥0,𝑟))}≥^𝑥1_𝑥01⋯^𝑥𝑁_𝑥0𝑁𝑉(𝜉)𝑑𝜉_𝑛⋯𝑑𝜉₁, (2.8) where 𝜉=(𝜉₁,…,𝜉_𝑁). Therefore, |𝑤(𝑥)|≤√𝑁|𝑉(𝑥)|_𝐿¹_{(𝐵(𝑥0,𝑟))}.
A direct calculation leads to div|𝑢|^𝑝(𝑥)𝑤(𝑥)=|𝑢(𝑥)|^𝑝(𝑥)div𝑤(𝑥)+𝑝(𝑥)|𝑢|^{𝑝(𝑥)−2}(𝑥)+|𝑢|𝑢∇𝑢⋅𝑤_𝑝(𝑥)log𝑢∇𝑝(𝑥)⋅𝑤(𝑥). (2.9) Now the Divergence Theorem implies ∫_{𝐵(𝑥0,𝑟)}div(|𝑢|^𝑝(𝑥)𝑤(𝑥))=0, and so _{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^𝑝(𝑥)div𝑤(𝑥)𝑑𝑥≤𝑝⁺_{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^{𝑝(𝑥)−1}|(𝑥𝑤(𝑥+∇𝑢)||)|𝑑𝑥_{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^𝑝(𝑥)log|𝑢(𝑥)|∇𝑝(𝑥)|𝑤(𝑥)|𝑑𝑥. (2.10) Set 𝐼₁∶=𝑝⁺_{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^{𝑝(𝑥)−1}|(𝑥𝑤(𝑥𝐼∇𝑢)||)|𝑑𝑥,₂∶=_{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^𝑝(𝑥)log|𝑢(𝑥)|∇𝑝(𝑥)|𝑤(𝑥)|𝑑𝑥. (2.11)
Now we estimate 𝐼₂ by distinguishing the case when |𝑢(𝑥)|≤1 and |𝑢(𝑥)|>1. Notice that the relations sup_0≤𝑡≤1𝑡^𝜂log𝑡<∞sup_𝑡>1𝑡^−𝜂log𝑡<∞ (2.12)(2.13) hold for 𝜂>0.
Let Ω₁=∶{𝑥∈𝐵_𝑟∶|𝑢(𝑥)|≤1} and Ω₂=∶{𝑥∈𝐵_𝑟∶|𝑢(𝑥)|>1}, then for (2.12) and (2.13) we have 𝐼₂≤𝐶_1Ω1|𝑤(𝑥)||𝑢(𝑥)|^{𝑝(𝑥)−𝜂1}𝑑𝑥+𝐶_2Ω2|𝑤(𝑥)||𝑢(𝑥)|^{𝑝(𝑥)+𝜂2}𝑑𝑥. (2.14) We can choose 𝑘∈ℕ such that 𝑝(𝑥)−1/𝑘≥𝑝⁻. Since 𝑢∈𝐿^𝑝−(𝐵(𝑥₀,𝑟)) and in Ω₁,|𝑢(𝑥)|≤1 we have |𝑢(𝑥)|^{𝑝(𝑥)−1/𝑛}≤|𝑢(𝑥)|^𝑝−, (2.15) for 𝑛>𝑘. The Lebesgue Dominated Convergence Theorem implies lim_{𝑛→∞Ω1}|𝑢(𝑥)|^{𝑝(𝑥)−1/𝑛}𝑑𝑥=_Ω1|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (2.16) For Ω₂ we can choose 𝑘^′ such that 𝑝(𝑥)+1/𝑘^′≤(𝑝(𝑥))^∗=𝑁𝑝(𝑥)/(𝑁−𝑝(𝑥)). So |𝑢(𝑥)|^{𝑝(𝑥)+1/𝑛}≤|𝑢(𝑥)|^{(𝑝(𝑥))∗}, (2.17)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 lim_{𝑛→∞Ω2}|𝑢(𝑥)|^{𝑝(𝑥)+1/𝑛}𝑑𝑥=_Ω2|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (2.18) Given that 𝑝∈MPIC(Ω) we have 𝐼₂≤𝐶_{𝐵(𝑥0,𝑟)}|𝑢|^𝑝(𝑥)𝑑𝑥≤𝐶_{𝐵(𝑥0,𝑟)}|∇𝑢|^𝑝(𝑥)𝑑𝑥. (2.19) Now we estimate 𝐼₁ by using the modular Young's inequality [24, equation (3.2.21)]: 𝐼₁≤𝑝⁺𝐶_{1𝐵(𝑥0,𝑟)}|𝑤(𝑥)|^{𝑝(𝑥)/(𝑝(𝑥)−1)}|𝑢(𝑥)|^𝑝(𝑥)+𝑝⁺𝐶_{2𝐵(𝑥0,𝑟)}|∇𝑢(𝑥)|^𝑝(𝑥). (2.20) Again, since 𝑝∈MPIC(Ω) we obtain 𝐼₁≤𝐶_{𝐵(𝑥0,𝑟)}|∇𝑢|^𝑝(𝑥)𝑑𝑥. (2.21) Finally, recalling that div𝑤(𝑥)=𝑁𝑉(𝑥) we get 𝑁_{𝐵(𝑥𝑜,𝑟)}𝑉(𝑥)|𝑢(𝑥)|^𝑝(𝑥)≤𝐶_{𝐵(𝑥0,𝑟)}|∇𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥, (2.22) which leads to the claim of the theorem.

3. Strong Unique Continuation

Consider the equation 𝐻𝑢∶=div|∇𝑢|^{𝑝(𝑥)−2}∇𝑢+𝑉(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢=0,𝑥∈Ω, (3.1)𝑢∈𝑊^{1⋅𝑝(𝑥)}_loc(Ω),1<𝑝(𝑥)<𝑁,𝑉∈𝐿^{𝑁/𝑝(𝑥)}(Ω).

A weak solution of (3.1) is the function 𝑢∈𝑊^{1⋅𝑝(𝑥)}_loc(Ω) such that _Ω|∇𝑢|^{𝑝(𝑥)−2}∇𝑢⋅∇𝜑𝑑𝑥+_Ω𝑉(𝑥)|𝑢|^{𝑝(𝑥)−2}𝑢⋅𝜑𝑑𝑥=0, (3.2) for all 𝜑∈𝑊₀^1,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 function 𝑢∈𝐿^𝑝(𝑥)_loc(Ω) has a zero of infinite order in the 𝑝(𝑥)-mean at a point 𝑥₀∈Ω if, for each 𝑘∈ℕ, _{|𝑥−𝑥0|⩽𝑅}|𝑢|^𝑝(𝑥)𝑑𝑥=𝑂𝑅^𝑘. (3.3)

Recall that Ω⊂ℝ^N 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 Theorem 3.5 which is the main result of this paper we require the following Lemmas.

Lemma 3.2. Let 𝑝 be a Log-Hölder continuous exponent with 1<𝑝(𝑥)<𝑁, and 𝑝∈MPIC(Ω). Let 𝑉∈𝐿^{𝑁/𝑝(𝑥)},0<𝜖<𝑉(𝑥), almost everywhere and 𝑢∈𝑊₀^1,𝑝(𝑥)(Ω). Then, for each 𝜖_𝑜>0, there exists 𝐾 such that _Ω𝑉|𝑢|^𝑝(𝑥)𝑑𝑥≤𝜖_𝑜Ω|∇𝑢|^𝑝(𝑥)+𝐾_Ω|𝑢|^𝑝(𝑥)𝑑𝑥. (3.4)

Proof. Let 𝜖_𝑜>0 be given. We have _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), (3.5) 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∶=(𝜖_𝑜/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, 𝜇({𝑥∶𝑉(𝑥)>𝑡})≤𝑡^{−𝑁/𝑝+}_Ω𝑉(𝑥)^{𝑁/𝑝(𝑥)}. (3.6) So taking 𝑡 sufficiently big, we get the desired inequality.

Lemma 3.3. Let 𝑝∶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 𝐵_𝑟 and 𝐵_2𝑟 two concentric balls contained in Ω. Then _𝐵𝑟|∇𝑢|^𝑝(𝑥)≤𝐶𝑟^𝑝(𝑥0)_𝐵2𝑟|𝑢|^𝑝(𝑥), (3.7) where the constant 𝐶 does not depend on 𝑟,𝑥₀∈𝐵_2𝑟 and 𝑉∈𝐿^{𝑁/𝑝(𝑥)}.

Proof. Take 𝜂∈𝐶^∞₀(Ω), with sup𝑝𝜂⊂𝐵_2𝑟,0≤𝜂≤1 such that 𝜂(𝑥)=1 for any 𝑥∈𝐵_𝑟 and |∇𝜂|≤𝐶/𝑟. We want to use as test function 𝜓=𝜂^𝑝(𝑥)𝑢. To this end we show first that 𝜓∈𝑊₀^1,𝑝(𝑥)(Ω); it is clear that 𝜓∈𝐿^𝑝(𝑥)(Ω) since 𝑢 is solution of (3.1). Furthermore, since 0⩽𝜂⩽1 then |𝜂log𝜂|⩽𝑎 for some constant 𝑎, so ∇𝜓≤∇𝑢𝜂^𝑝(𝑥)+𝑢𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝑢𝜂^𝑝(𝑥)(𝑥≤log𝜂∇𝑝)∇𝑢𝜂^𝑝(𝑥)+𝑢𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝑢𝜂^{𝑝(𝑥)−1}(𝑥)≤∇𝑝𝜂log𝜂∇𝑢𝜂^𝑝(𝑥)+𝑢𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝑢𝜂^{𝑝(𝑥)−1}(𝑥𝑎≤∇𝑝)∇𝑢𝜂^𝑝(𝑥)+𝑢𝜂^{𝑝(𝑥)−1}𝑝(𝑥)(𝑥≤||+|𝑢|∇𝜂+𝑎∇𝑝)∇𝑢𝐶𝑝⁺≤||+𝑎𝐿∇𝑢+𝐶_𝑝|𝑢|. (3.8)
Hence, |∇𝜓|^𝑝(𝑥)⩽2^𝑝+−1|∇𝑢|^𝑝(𝑥)+𝐶^𝑝+_𝑝2^𝑝+−1|𝑢|^𝑝(𝑥). Therefore, |∇𝜓|∈𝐿^𝑝(𝑥)(Ω).
Now we can use 𝜓=𝜂^𝑝(𝑥)𝑢 as a test function to obtain 0=_𝐵2𝑟|∇𝑢|^{𝑝(𝑥)−2}∇𝑢⋅∇𝜓𝑑𝑥+_𝐵2𝑟𝑉|𝑢|^{𝑝(𝑥)−2}=𝑢𝜓𝑑𝑥_𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)𝑑𝑥+_𝐵2𝑟|∇𝑢|^{𝑝(𝑥)−2}𝑢∇𝑢⋅𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝜂^𝑝(𝑥)(𝑥)+log𝜂∇𝑝𝑑𝑥_𝐵2𝑟𝑉𝜂^𝑝(𝑥)|𝑢|^𝑝(𝑥)𝑑𝑥. (3.9) Let 𝐼₁∶=_𝐵2𝑟|∇𝑢|^{𝑝(𝑥)−2}𝑢∇𝑢⋅𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝜂^𝑝(𝑥)(𝑥)𝐼log𝜂∇𝑝𝑑𝑥,₂∶=_𝐵2𝑟𝑉𝜂^𝑝(𝑥)|𝑢|^𝑝(𝑥)𝑑𝑥. (3.10) We can estimate 𝐼₁ by 𝐼₁≤_𝐵2𝑟|∇𝑢|^{𝑝(𝑥)−2}|𝑢||∇𝑢|𝑝(𝑥)𝜂^{𝑝(𝑥)−1}∇𝜂+𝜂^𝑝(𝑥)(𝑥≤log𝜂∇𝑝)𝑑𝑥_𝐵2𝑟|∇𝑢|𝜂^{𝑝(𝑥)−1}|𝑢𝑝(𝑥(𝑥≤|∇𝜂)+𝑎∇𝑝)𝑑𝑥_𝐵2𝑟|∇𝑢|𝜂^{𝑝(𝑥)−1}|𝑢|∇𝜂𝑝⁺+𝑎𝐿𝑑𝑥≤𝐶_{𝑝𝐵2𝑟}|∇𝑢|𝜂^{𝑝(𝑥)−1}|𝑢|∇𝜂𝑑𝑥≤𝐶_{𝑝𝐵2𝑟}𝜖|∇𝑢|𝜂^𝑝(𝑥)+1𝜖^{𝑝(𝑥)−1}|𝑢|∇𝜂^𝑝(𝑥)𝑑𝑥, (3.11) where the Young-type inequality 𝑓𝑔≤𝜖𝑓^{𝑝(𝑥)/(𝑝(𝑥)−1)}+1𝜖^{𝑝(𝑥)−1}𝑔^𝑝(𝑥) (3.12) was used in the last inequality. Moreover, 𝐼₁≤𝐶_𝑝𝜖_𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)𝑑𝑥+𝐶_{𝑝𝐵2𝑟}1𝜖^{𝑝(𝑥)−1}|𝑢|^𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑑𝑥≤𝐶_𝑝𝜖_𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)𝑑𝑥+𝐶_𝑝1𝜖^𝑝+−1_𝐵2𝑟|𝑢|^𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑑𝑥. (3.13) We estimate 𝐼₂: 𝐼₂=_𝐵2𝑟𝑉𝜂^𝑝(𝑥)|𝑢|^𝑝(𝑥)=𝑑𝑥_𝐵2𝑟𝑉𝜂𝑢^𝑝(𝑥)𝑑𝑥≤𝜖_𝐵2𝑟∇𝑢𝜂^𝑝(𝑥)𝑑𝑥+𝐶_{𝜖𝐵2𝑟}𝜂𝑢^𝑝(𝑥)𝑑𝑥, (3.14) where Lemma (3.2) was used in the last inequality.
Now using the estimates for 𝐼₁ and 𝐼₂, we have _𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)𝑑𝑥≤𝐶_𝑝𝜖_𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)𝑑𝑥+𝐶_𝑝1𝜖^𝑝+−1_𝐵2𝑟|𝑢|^𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑑𝑥𝑑𝑥+𝜖_𝐵2𝑟∇𝑢𝜂^𝑝(𝑥)+𝐶_{𝜖𝐵2𝑟}𝜂𝑢^𝑝(𝑥)𝐶𝑑𝑥≤𝜖₁+𝐶_2𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)+𝜖𝑑𝑥𝜖^𝑝+𝐶₁+𝐶₂𝜖^𝑝+_𝐵2𝑟|𝑢|^𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑑𝑥+𝐶_{𝜖𝐵2𝑟}𝜂^𝑝(𝑥)|𝑢|^𝑝(𝑥)𝑑𝑥 (3.15) for 0<𝜖≤1. By choosing 𝜖≤min{1,1/2(𝐶₁+𝐶₂)}, we have _𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)1𝑑𝑥≤𝜖^𝑝+_𝐵2𝑟|𝑢|^𝑝(𝑥)∇𝜂^𝑝(𝑥)𝑑𝑥+2𝐶_{𝜖𝐵2𝑟}|𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)≤1𝑑𝑥𝜖^𝑝+_𝐵2𝑟|𝑢|^𝑝(𝑥)𝐶𝑟^𝑝(𝑥)𝑑𝑥+2𝐶_{𝜖𝐵2𝑟}|𝑢|^𝑝(𝑥)𝐶𝑟^𝑝(𝑥)𝑑𝑥. (3.16) Since 𝑝(𝑥) is Log-Hölder 𝑟^{−𝑝(𝑥)}≤𝐶𝑟^{−𝑝(𝑥0)} for all 𝑥₀∈𝐵_2𝑟, then _𝐵2𝑟|∇𝑢|^𝑝(𝑥)𝜂^𝑝(𝑥)1𝑑𝑥≤𝜖^𝑝+_𝐵2𝑟|𝑢|^𝑝(𝑥)𝐶𝑟^𝑝(𝑥0)𝑑𝑥+2𝐶_{𝜖𝐵2𝑟}|𝑢|^𝑝(𝑥)𝐶𝑟^𝑝(𝑥0)≤1𝑑𝑥𝜖^𝑝++𝐶_𝜖𝐶^𝑝(𝑥0)𝑟^𝑝(𝑥0)_𝐵2𝑟|𝑢|^𝑝(𝑥)≤𝐶𝑑𝑥𝑟^𝑝(𝑥0)_𝐵2𝑟|𝑢|^𝑝(𝑥)𝑑𝑥. (3.17) Therefore, _𝐵𝑟|∇𝑢|^𝑝(𝑥)𝐶𝑑𝑥≤𝑟^𝑝(𝑥0)_𝐵2𝑟|𝑢|^𝑝(𝑥)𝑑𝑥. (3.18)

Lemma 3.4. Let 𝑢∈𝑊^1,1(𝐵(𝑥₀,𝑟)), where 𝐵(𝑥₀,𝑟) is the ball of radius 𝑟>0 in ℝ^𝑁 and 𝐸={𝑥∈𝐵(𝑥₀,𝑟)∶𝑢(𝑥)=0}. Then there exists a constant 𝛽>0 depending only on 𝑁, such that _𝐷|𝑢(𝑥𝑟)|𝑑𝑥≤𝛽^𝑁|𝐸||𝐷|^1/𝑁_{𝐵(𝑥0,𝑟)}|∇𝑢(𝑥)|𝑑𝑥 (3.19) 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. Let Ω be a bounded domain in ℝ^𝑁, 𝑝∶Ω→(1,𝑁) an exponent with 1<𝑝⁻≤𝑝⁺<∞ and such that 𝑝∈MPIC(Ω) is Lipschitz continuous, and 𝑢∈𝑊^1,𝑝(𝑥)_loc(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 of 𝐸 is a point of density, let 𝑥₀ be such a point, that is, 𝐸^𝐶∩𝐵𝑥₀,𝑟𝐵𝑥₀𝑥,𝑟→0,𝐸∩𝐵₀,𝑟𝐵𝑥₀,𝑟→1 (3.20) as 𝑟→0.
Let 𝐵_𝑟∶=𝐵(𝑥₀,𝑟). So for a given 𝜖>0, there exists 𝑟₀=𝑟₀(𝜖) such that for 𝑟≤𝑟₀, 𝐸^𝐶∩𝐵_𝑟𝐵_𝑟<𝜖,𝐸∩𝐵_𝑟𝐵_𝑟>1−𝜖, (3.21) where 𝐸^𝐶 denote the complement of 𝐸 in Ω. Taking 𝑟₀ smaller if necessary, we may assume that 𝐵_𝑟0⊂Ω. Since 𝑢=0 on 𝐸, and using Lemma 3.4 we have _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)𝑟𝑑𝑥≤𝛽^𝑁𝐵_𝑟𝐵∩𝐸^𝑟∩𝐸^𝐶1/𝑁_𝐵𝑟∇|𝑢(𝑥)|^𝑝(𝑥)𝑟𝑑𝑥≤𝐶𝛽^𝑁𝑟^𝑁−1𝜖^1/𝑁1−𝜖_𝐵𝑟∇|𝑢(𝑥)|^𝑝(𝑥)𝜖𝑑𝑥=𝐶𝑟^1/𝑁1−𝜖_𝐵𝑟∇|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (3.22)
But |∇(|𝑢(𝑥)|^𝑝(𝑥))|≤|𝑝(𝑥)|||𝑢(𝑥)|^{𝑝(𝑥)−1}||∇𝑢(𝑥)|+|∇𝑝(𝑥)|||𝑢(𝑥)|^𝑝(𝑥)log|𝑢(𝑥)||. Hence, _𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)𝜖𝑑𝑥≤𝐶𝑟^1/𝑁1−𝜖_𝐵𝑟𝑝⁺|𝑢(𝑥)|^{𝑝(𝑥)−1}|∇𝑢(𝑥)|𝑑𝑥+_𝐵𝑟𝐿|𝑢(𝑥)|^𝑝(𝑥)log|𝑢(𝑥)|𝑑𝑥. (3.23) Let 𝐼₁∶=_𝐵𝑟|𝑢(𝑥)|^{𝑝(𝑥)−1}|(𝑥𝐼∇𝑢)|𝑑𝑥,₂∶=_𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)log|𝑢(𝑥)|𝑑𝑥. (3.24)𝐼₁ can be estimated using the Young type inequality with 𝜖=1/𝑟: _𝐵𝑟|𝑢(𝑥)|^{𝑝(𝑥)−1}|∇𝑢(𝑥)|𝑑𝑥⩽_𝐵𝑟1𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥+_𝐵𝑟𝑟^{𝑝(𝑥)−1}|∇𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (3.25) Now we estimate 𝐼₂ by distinguishing the case when |𝑢(𝑥)|≤1 and |𝑢(𝑥)|>1, using the relations (2.2) and (2.13).
Let Ω₁=∶{𝑥∈𝐵_𝑟∶|𝑢(𝑥)|≤1} and Ω₂=∶{𝑥∈𝐵_𝑟∶|𝑢(𝑥)|>1}, then 𝐼₂≤𝐶_1Ω1|𝑢(𝑥)|^{𝑝(𝑥)−𝜂1}𝑑𝑥+𝐶_2Ω2|𝑢(𝑥)|^{𝑝(𝑥)+𝜂2}𝑑𝑥. (3.26) We can choose 𝑘∈ℕ such that 𝑝(𝑥)−1/𝑘≥𝑝⁻. Since 𝑢∈𝐿^𝑝−(𝐵(𝑥₀,𝑟)) and in Ω₁,|𝑢(𝑥)|≤1 we have |𝑢(𝑥)|^{𝑝(𝑥)−1/𝑛}≤|𝑢(𝑥)|^𝑝−, (3.27) for 𝑛>𝑘. The Lebesgue Dominated Convergence Theorem implies lim_{𝑛→∞Ω1}|𝑢(𝑥)|^{𝑝(𝑥)−1/𝑛}𝑑𝑥=_Ω1|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (3.28) For Ω₂ we can choose 𝑘^′ such that 𝑝(𝑥)+1/𝑘^′≤(𝑝(𝑥))^∗=𝑁𝑝(𝑥)/(𝑁−𝑝(𝑥)). So |𝑢(𝑥)|^{𝑝(𝑥)+1/𝑛}≤|𝑢(𝑥)|^{(𝑝(𝑥))∗}, (3.29) 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 lim_{𝑛→∞Ω2}|𝑢(𝑥)|^{𝑝(𝑥)+1/𝑛}𝑑𝑥=_Ω2|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥. (3.30) Therefore, 𝐼₂≤𝐶_{𝐵(𝑥0,𝑟)}|𝑢|^𝑝(𝑥)𝑑𝑥. (3.31) Now, using estimates for 𝐼₁ and 𝐼₂, and noticing that for 0<𝑟<1 we have 𝑟^𝑝(𝑥)<𝑟^𝑝−, we get _𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)𝜖𝑑𝑥≤^1/𝑁1−𝜖𝐶𝑝⁺_𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥+𝐶𝑝⁺𝑟^𝑝−_𝐵𝑟|∇𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥+𝐶_𝑝𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥 (3.32) and by Lemma (3.3) we have _𝐵𝑟|𝑢(𝑥)|^𝑝(𝑥)𝜖𝑑𝑥≤^1/𝑁1−𝜖𝐶𝑝⁺_𝐵2𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥+𝐶𝑝⁺𝑟^𝑝−𝑟^{−𝑝(𝑥0)}_𝐵2𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥+𝐶_{𝑝𝐵2𝑟}|𝑢(𝑥)|^𝑝(𝑥)𝜖𝑑𝑥≤𝐶^1/𝑁1−𝜖_𝐵2𝑟|𝑢(𝑥)|^𝑝(𝑥)𝑑𝑥, (3.33) where 𝐶 is independent of 𝜀 and of 𝑟 as 𝑟→0. Note that 𝑟^{𝑝−−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 (𝐶𝜖^1/𝑁)/(1−𝜖)≤2^−𝑛. Now, observe that 𝑟₀ depends on 𝑛, hence by the last inequality we deduce 𝑓(𝑟)≤2^−𝑛𝑓(2𝑟),for𝑟≤𝑟₀. (3.34) Iterating (3.34), we get 𝑓𝜌≤2^−𝑘𝑛𝑓2^𝑘𝜌,if2^𝑘−1𝜌≤𝑟₀. (3.35) Thus, given that 0<𝑟<𝑟₀(𝑛) and choosing 𝑘∈ℕ such that 2^−𝑘𝑟₀≤𝑟≤2^{−(𝑘−1)}𝑟₀. (3.36) From (3.35), we conclude that 𝑓(𝑟)≤2^−𝑘𝑛𝑓2^𝑘𝑟≤2^−𝑘𝑛𝑓2𝑟₀, (3.37) and since 2^−𝑘≤𝑟/𝑟₀, we get 𝑓(𝑟)≤𝑟𝑟₀^𝑛𝑓2𝑟₀, (3.38) which shows that 𝑥₀ is a zero infinite order in 𝑝(𝑥)-mean.

Acknowledgments

The authors want thank to Peter Hästö for his careful reading and corrections to a draft of this paper. J. Cuadro was supported by CONACYT México's Ph.D. Schoolarship.