Nonlinear Kato Class and Unique Continuation of Eigenfunctions for 𝑝-Laplacian Operator

Abstract

We study some basic properties of nonlinear Kato class 𝑀_𝑝(ℝ^𝑛) and 𝑀_𝑝(ℝ^𝑛), respectively, for 1<𝑝<𝑛. Also, we study the problem −div(|∇𝑢|^𝑝−2∇𝑢)+𝑉|𝑢|^𝑝−2𝑢=0 in Ω, where Ω is a bounded domain in ℝ^𝑛 and the weight function 𝑉 is assumed to be not equivalent to zero and lies in 𝑀_𝑝(Ω), in the case where 𝑝<𝑛. Finally, we establish the strong unique continuation property of the eigenfunction for the 𝑝-Laplacian operator in the case where 𝑉∈𝑀_𝑝(Ω).

1. Introduction

The Kato class 𝐾_𝑛 was introduced and studied by Aizenman and Simon (see [1]). For 𝑛≥3, it consists of locally integrable functions 𝑓 on ℝ^𝑛, such that lim_𝑟→0sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}𝑓𝑦𝑥−𝑦^𝑛−2𝑑𝑦=0. (1) For 1<𝑝<𝑛, the following classes were defined by Zamboni (see [2]): the class 𝑀_𝑝 of functions 𝑓, such that sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^{𝑛−1𝐵(𝑥,𝑟)}𝑓(𝑧)𝑦−𝑧^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1<∞, (2) and the class 𝑀_𝑝 of functions 𝑓 such that 𝑓∈𝑀_𝑝 and lim_𝑟→0sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^𝑛−1×_{𝐵(𝑥,𝑟)}𝑓(𝑧)𝑦−𝑧^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1=0. (3)

Section 2 of the present paper is devoted to the study of some basic properties of the nonlinear Kato class 𝑀_𝑝(ℝ^𝑛) and 𝑀_𝑝(ℝ^𝑛), respectively, for 1<𝑝<𝑛.

Among other things, we show that the Lorentz space 𝐿(𝑛/2,1) is embedded into 𝑀_𝑝(ℝ^𝑛) (see Lemma 10) as well as that 𝑀_𝑝(ℝ^𝑛) is a complete topological vector space (see Remark 11 and Lemma 13).

The 𝑝-Laplacian operator is a generalization of the Laplace operator, where 𝑝 is allowed to range over 1<𝑝<∞; in our case where 1<𝑝<𝑛, it is written as div|∇𝑢|^𝑝−2||∇𝑢=∇⋅∇𝑢^𝑝−2=||∇𝑢∇𝑢^𝑝−4|∇𝑢|²Δ𝑢+𝑝−2^𝑛_𝑖,𝑗=1𝜕𝑢𝜕𝑥_𝑖𝜕𝑢𝜕𝑥_𝑗𝜕²𝑢𝜕𝑥_𝑖𝜕𝑥_𝑗, (4) where ∇𝑢=𝜕𝑢𝜕𝑥₁,…,𝜕𝑢𝜕𝑥_𝑛,|∇𝑢|=^𝑛_𝑖=1𝜕𝑢𝜕𝑥_𝑖^21/2, (5) and 𝑢∈𝐶^∞₀(Ω), with Ω bounded domain in ℝ^𝑛.

We are concerned with the following problem: −div|∇𝑢|^𝑝−2∇𝑢+𝑉|𝑢|^𝑝−2𝑢=0,inΩ, (6) and the weight function 𝑉 is assumed to be not equivalent to zero and lies in 𝑀_𝑝(ℝ^𝑛) in the case 𝑝<𝑛.

Specifically, we are interested in studying a family of functions which enjoys the strong unique continuation property, that is, functions besides the possible zero functions which have zero of infinite order.

Definition 1. We say that a function 𝑢∈𝐿^𝑝_loc(Ω) vanishes of infinite order at point 𝑥₀ if for any natural number 𝑁 there exists a constant 𝐶_𝑁, such that _{𝐵(𝑥0,𝑟)}|𝑢(𝑥)|^𝑝𝑑𝑥≤𝐶_𝑁𝑟^𝑁, (7) for all 𝑁∈ℕ and for small positive number 𝑟. Here, 𝐵𝑥₀,𝑟=𝑦∈ℝ^𝑛∶𝑦−𝑥₀<𝑟. (8)

Definition 2. We say that (6) has a strong unique continuation property if and only if any solution 𝑢 of (6) in Ω is identically zero in Ω provided that 𝑢 vanishes of infinite order at a point in Ω.

There is an extensive literature on unique continuation. We refer to the work of Zamboni on unique continuation for nonnegative solutions of quasilinear elliptic equation [3], also the work of Jerison-Kenig on the unique continuation for Schrödinger operators [4]. The same work is done by Chiarenza and Frasca, but for linear elliptic operator in the case where 𝑉∈𝐿^𝑛/2 when 𝑛>2 [5].

Let us recall some known results concerning Fefferman's inequality as follows: _ℝ𝑛|𝑢(𝑥)|^𝑝|𝑉(𝑥)|𝑑𝑥≤𝐶_ℝ𝑛|∇𝑢(𝑥)|^𝑝𝑑𝑥∀𝑢∈𝐶^∞₀ℝ^𝑛. (9)

In [6], de Figueiredo and Gossez prove (9) in the case where 𝑝=2, assuming that 𝑉∈𝐿^{𝑟,𝑛−2𝑟}(ℝ^𝑛) with 1<𝑟≤𝑛/2. Later in [7], Jerison and Kening showed the same result taking 𝑉 in the Stummel-Kato class 𝑆(ℝ^𝑛). We point out that it is not possible to compare the assumptions 𝑉∈𝐿^{𝑟,𝑛−2𝑟}(ℝ^𝑛) and 𝑓∈𝑆(ℝ^𝑛). Chiarenza and Frasca [5] generalized Fefferman's result proving (9) under the assumption that 𝑉∈𝐿^{𝑟,𝑛−2𝑟}(ℝ^𝑛) with 𝑟∈(1,𝑛/𝑝) and 𝑝∈(1,𝑛). In [3], Schechter gave a new proof of (9) assuming that 𝑉∈𝑀_𝑝(ℝ^𝑛).

2. Definitions and Notation

In this section, we gather definitions and notations that will be used throughout the paper. We also include several simple lemmas. By 𝐿¹_loc(ℝ^𝑛), we will denote the space of functions which are locally integrable on ℝ^𝑛, and by 𝐿¹_loc,𝑢 the space of functions 𝑓, such that sup_{𝑥∈ℝ𝑛𝐵(𝑥,1)}𝑓𝑦𝑑𝑦<∞. (10)

Definition 3. Let 𝑓∈𝐿¹_loc(ℝ^𝑛). For any 1<𝑝<𝑛 and 𝑟>0, we set Φ(𝑟)=sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^{𝑛−1𝐵(𝑥,𝑟)}𝑓(𝑧)𝑑𝑧𝑧−𝑦^{𝑛−11/(𝑝−1)}𝑑𝑦^𝑝−1, (11) where 𝐵(𝑥,𝑟)={𝑦∶|𝑥−𝑦|<𝑟}. We say that 𝑓 belongs to the space 𝑀_𝑝(ℝ^𝑛) if Φ(𝑟)<∞ for all 𝑟>0.

Definition 4. We say that a function 𝑓∈𝑀_𝑝(ℝ^𝑛) if lim_𝑟→0Φ(𝑟)=0. (12)

We are now ready to formulate some simple properties of the classes𝑀_𝑝and𝑀_𝑝.

Lemma 5 (see [3], page 152). For 1<𝑝<𝑛, one has (i)𝑀_𝑝(ℝ^𝑛)⊂𝑀_𝑝(ℝ^𝑛),(ii)𝑀₂(ℝ^𝑛)=𝐾_𝑛.

From Lemma 5 we conclude that both 𝑀_𝑝(ℝ^𝑛) and 𝑀_𝑝(ℝ^𝑛) are generalizations of 𝐾_𝑛.

Remark 6. The following example shows that𝐾_𝑛is properly contained in 𝑀_𝑝(ℝ^𝑛) for 𝑝>2. It is known that the function 𝑓(𝑥)=|𝑥|⁻² is not in the Kato class 𝐾_𝑛. However, 𝑓∈𝑀_𝑝. Indeed, lim_𝑟→0sup_{𝑥𝐵(𝑥,𝑟)}1𝑥−𝑦^𝑛−2×∫_{𝐵(𝑥,𝑟)}𝑑𝑧|𝑧|²𝑧−𝑦^{𝑛−11/(𝑝−1)}𝑑𝑦^𝑝−1=0. (13) This can be shown by splitting the domain of integration in the interior integral into the following three parts: 𝐵(𝑥,𝑟)⋂{|𝑧|<(1/2)|𝑦|}, 𝐵(𝑥,𝑟)⋂{(1/2)|𝑦|≤|𝑧|≤(3/2)|𝑦|}, and 𝐵(𝑥,𝑟)⋂{|𝑧|>(3/2)|𝑦|}.
After routine calculations, we can see that _{𝐵(𝑥,𝑟)}𝑑𝑧|𝑧|²𝑧−𝑦^𝑛−1 (14) is majorized by 𝐶|𝑦|⁻¹. Finally, we have 𝐶sup_{𝑥𝐵(𝑥,𝑟)}𝑑𝑦𝑦^1/(𝑝−1)𝑥−𝑦^{𝑛−1𝑝−1}⟶0as𝑟⟶0. (15) This shows that (13) holds. Thus,𝑓∈⋂_𝑝>2𝑀_𝑝.

Definition 7. The distribution function 𝐷_𝑓 of a measurable function 𝑓 is given by 𝐷_𝑓(𝜆)=𝑚𝑥∈ℝ^𝑛∶𝑓(𝑥)>𝜆, (16) where 𝑚 denotes the Lebesgue measure on ℝ^𝑛. The distribution function 𝐷_𝑓 provides information about the size of 𝑓 but not about the behavior of 𝑓 itself near any given point. For instance, a function on ℝ^𝑛 and each of its translates have the same distribution function. It follows from Definition 7 that 𝐷_𝑓 is a decreasing function of 𝜆 (not strictly necessary).

Definition 8. Let 𝑓 be a measurable function in ℝ^𝑛. The decreasing rearrangement of 𝑓 is the function 𝑓 defined on [0,∞) by 𝑓^∗(𝑡)=inf𝜆∶𝐷_𝑓(𝜆)≤𝑡(𝑡≥0). (17) We use here the convention that inf∅=∞.

Definition 9 (Lorentz space). Let 𝑓 be a measurable function; we say that 𝑓 belongs to 𝐿_(𝑛/2,1) if 𝑓_(𝑛/2,1)=^∞₀𝑡^2/𝑛−1𝑓^∗(𝑡)𝑑𝑡<∞. (18) And it belongs to 𝐿(𝑛/(𝑛−2),∞) if 𝑓_{(𝑛/(𝑛−2),∞)}=sup_𝑡>1𝑡^1−2/𝑛𝑓^∗(𝑡)<∞. (19)

Lemma 10. Consider 𝐿_(𝑛/2,1)⊂𝑀_𝑝(ℝ^𝑛).

Proof. Let 𝑓∈𝐿_(𝑛/2,1); then ^∞₀𝑡^2/𝑛−1𝑓^∗(𝑡)𝑑𝑡<∞. (20) Since |𝑓|𝜒_{𝐵(𝑥,𝑟)}≤|𝑓|, we have 𝑓𝜒_{𝐵(𝑥,𝑟)}^∗(𝑡)≤𝑓^∗(𝑡); (21) then, ^∞₀𝑡^2/𝑛−1𝑓𝜒_{𝐵(𝑥,𝑟)}^∗(𝑡)𝑑𝑡≤^∞₀𝑡^2/𝑛−1𝑓^∗(𝑡)𝑑𝑡<∞. (22) Thus, |𝑓|𝜒_{𝐵(𝑥,𝑟)}∈𝐿_(𝑛/2,1).
On the other hand, let 𝑔(𝑥)=|𝑥|^{−(1−𝑛)}; then 𝑚𝑥∶𝑔(𝑥)>𝜆=𝑚𝑥∶|𝑥|^{−(1−𝑛)},𝑚|𝑥|<1>𝜆𝑥∶𝜆^1/(𝑛−1)=𝐶_𝑛1𝜆^{𝑛/(𝑛−1)}, (23) where 𝐶_𝑛=𝑚(𝐵(0,1)).
Next, we set 𝑡=𝐶_𝑛(1/𝜆)^{𝑛/(𝑛−1)}, and then 𝜆=𝐶_𝑛𝑡^1/𝑛−1. Thus, 𝑔^∗(𝑡)=𝐶_𝑛𝑡^1/𝑛−1. From this, we obtain 𝑔_{(𝑛/(𝑛−2),∞)}=1|⋅|^𝑛−1_{(𝑛/(𝑛−2),∞)}=sup_𝑡>1𝐶_𝑛𝑡^1−2/𝑛𝑡^1/𝑛−1=sup_𝑡>1𝐶_𝑛𝑡^−(1/𝑛)≤𝐶_𝑛<∞, (24) which means that 𝑔∈𝐿_{(𝑛/(𝑛−2),∞)}. Finally, by Fubini's theorem and Hölder's inequality, we have 𝜙(𝑟)=sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^𝑛−1×_{𝐵(𝑥,𝑟)}|𝑓(𝑧)|𝑦−𝑧^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1≤sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^𝑛−1×_{𝐵(𝑥,2𝑟)}𝑓(𝑧)𝑦−𝑧^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1=sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}1𝑥−𝑦^𝑛−1×_ℝ𝑛|𝑓(𝑧)|𝜒_{𝐵(0,2𝑟)}(𝑦−𝑧)𝑑𝑧𝑦−𝑧^{𝑛−11/(𝑝−1)}𝑑𝑦^𝑝−1≤sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟)}𝑑𝑦𝑥−𝑦^{𝑛−1𝑝−1}×𝑓𝜒_{𝐵(0,2𝑟)(𝑛/2,1)}1|⋅|^𝑛−1_{(𝑛/(𝑛−2),∞)}=𝐶_𝑛𝑟^𝑝−1𝑓𝜒_{𝐵(0,2𝑟)(𝑛/2,1)}⟶0as𝑟⟶0, (25) which means that 𝑓∈𝑀_𝑝(ℝ^𝑛) and the proof is complete.

Remark 11. (i) For 0<𝑟<1, it is not hard to check that for 1<𝑝≤2, the expression 𝑓𝑀_𝑝(ℝ^𝑛₎=sup_{𝑥∈ℝ𝑛𝐵(𝑥,1)}1𝑥−𝑦^𝑛−1×_𝐵(𝑥,1)𝑓(𝑧)𝑧−𝑦^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1 (26) defines a norm on 𝑀_𝑝(ℝ^𝑛).
(ii) For 𝑝>2, the expression (26) satisfies the following inequality: 𝑀𝑓+𝑔_{𝑝(ℝ𝑛)}≤2^𝑝−2𝑓𝑀_𝑝(ℝ^𝑛₎+𝑔𝑀_𝑝(ℝ^𝑛₎ (27) for all 𝑓 and 𝑔 in 𝑀_𝑝(ℝ^𝑛).
If 𝑈 is a neighborhood of 0 from (27), we have 2^𝑝−1𝑈+2^𝑝−1𝑈⊂𝑈; (28) then, 𝑀_𝑝(ℝ^𝑛) is a topological vector space.

Lemma 12. Consider 𝑀_𝑝(ℝ^𝑛)⊂𝐿¹_loc,𝑢(ℝ^𝑛) for 1<𝑝<𝑛.

Proof. Let 𝑓∈𝑀_𝑝(ℝ^𝑛), and fix 𝑟₀>0. Then, there exists a positive constant 𝐶 such that Φ(𝑟₀)≤𝐶. It follows that sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟0)}1𝑥−𝑦^{𝑛−1𝐵(𝑥,𝑟}₀⁾𝑓(𝑧)𝑧−𝑦^𝑛−1𝑑𝑧^1/(𝑝−1)𝑑𝑦^𝑝−1≥sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟0)}𝑑𝑦𝑟₀^𝑛−1_{𝐵(𝑥,𝑟0)}𝑓(𝑧)(2𝑟₀)^𝑛−1𝑑𝑧^{1/(𝑝−1)𝑝−1}≥sup_{𝑥∈ℝ𝑛}12𝑟₀^𝑛−1𝑚𝐵𝑥,𝑟₀𝑟₀^{𝑛−1𝑝−1}_{𝐵(𝑥,𝑟0)}𝑓(𝑧)𝑑𝑧. (29) Therefore, sup_{𝑥∈ℝ𝑛𝐵(𝑥,𝑟0)}𝑓(𝑧)𝑑𝑧<𝐵𝐶, (30) where 𝐵=2𝑟₀^𝑛−1𝑟₀𝑚(𝐵(0,1))^𝑝−1. (31) Finally, let 𝐵(𝑥,1)⊆⋃^𝑛_𝑘=1𝐵(𝑥_𝑘,𝑟₀); then sup_{𝑥∈ℝ𝑛𝐵(𝑥,1)}𝑓(𝑧)𝑑𝑧≤^𝑛_𝑘=1sup_{𝑥∈ℝ𝑛𝐵(𝑥𝑘,𝑟0)}𝑓(𝑧)𝑑𝑧, (32) so sup_{𝑥∈ℝ𝑛𝐵(𝑥,1)}𝑓(𝑧)𝑑𝑧<∞. (33) Therefore, 𝑀_𝑝ℝ^𝑛⊂𝐿¹_loc,𝑢ℝ^𝑛. (34)

Lemma 13. For 1<𝑝<𝑛, 𝑀_𝑝(ℝ^𝑛) is a complete space.

Proof. Let {𝑓_𝑛}_𝑛∈𝒩 be a Cauchy sequence in 𝐵(0,𝑟)=𝑓∈𝑀_𝑝ℝ^𝑛∶𝑓𝑀_𝑝(ℝ^𝑛₎≤𝑟. (35) By Lemma 10, {𝑓_𝑛}_𝑛∈𝒩 is a Cauchy sequence in 𝐿¹_loc,𝑢(ℝ^𝑛). Since this space is complete, there exists a function 𝑓∈𝐿¹_loc,𝑢(ℝ^𝑛) such that 𝑓_𝑛→𝑓 in 𝐿¹_loc,𝑢(ℝ^𝑛). By Fatous's lemma, we have 𝑓𝑀_𝑝(ℝ^𝑛₎≤liminf𝑓_𝑛𝑀_𝑝(ℝ^𝑛₎≤𝑟. (36) Thus, 𝑓∈𝐵(0,𝑟), which means that 𝐵(0,𝑟) is complete with respect to the topology generated by 𝐿¹_loc,𝑢(ℝ^𝑛)-norm. By Corollary 2 of Proposition 9 in [8, Chapter III, Section 3, no. 5] we obtain the assertion.

Lemma 14. If 1<𝑝<𝑛, then 𝑀_𝑝(ℝ^𝑛) is closed in 𝑀_𝑝(ℝ^𝑛).

Proof. Let us define the map 𝜑∶𝑀_𝑝(ℝ^𝑛)→[0,∞) by 𝜑(𝑓)=lim_𝑟→0𝜙_𝑓(𝑟) (see Definition 3).
It is not hard to prove that the family {𝜑_𝑟}_𝑟>0 where 𝜑_𝑟(𝑓)=𝜙_𝑓(𝑟) is equicontinuous and 𝜑_𝑟→𝜑 pointwise as 𝑟→0. Since 𝑀_𝑝(ℝ^𝑛)=𝜑⁻¹(0), we obtain the result.

For more details on nonlinear Kato class, we refer the readers to [9].

3. Some Useful Inequalities

For the sake of completeness and convenience of the reader, we include the proof of the next result which is due to Schechter [3].

Theorem 15. Assume that𝑉∈𝑀_𝑝(ℝ^𝑛). Then, for any𝑟>0 there exists a positive constant𝐶(𝑛,𝑝), such that _ℝ𝑛|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑑𝑥≤𝐶𝑛,𝑝Φ(2𝑟)_ℝ𝑛|∇𝑢(𝑥)|^𝑝𝑑𝑥, (37) for any𝑢∈𝐶^∞₀(ℝ^𝑛)supported in𝐵(𝑥₀,𝑟).

Proof. For any 𝑢∈𝐶^∞₀(ℝ^𝑛) supported in 𝐵(𝑥₀,𝑟), using the well-known inequality |𝑢(𝑥)|≤𝐶𝑛,𝑝_{𝐵(𝑥0,𝑟)}∇𝑢𝑦𝑥−𝑦^𝑛−1𝑑𝑦, (38) Fubini's theorem, and Hölder's inequality, we obtain _ℝ𝑛|𝑉(𝑥)||𝑢(𝑥)|^𝑝=𝑑𝑥_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑑𝑥≤𝐶𝑛,𝑝_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝−1×_{𝐵(𝑥0,𝑟)}∇𝑢𝑦𝑥−𝑦^𝑛−1𝑑𝑦𝑑𝑥≤𝐶𝑛,𝑝_{𝐵𝑥0,𝑟}𝑦×∇𝑢_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝−11𝑥−𝑦^𝑛−1𝑑𝑥𝑑𝑦≤𝐶𝑛,𝑝_{𝐵(𝑥0,𝑟)}∇𝑢𝑦^𝑝𝑑𝑦^1/𝑝×_{𝐵(𝑥0,𝑟)𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝−1×1𝑥−𝑦^𝑛−1𝑑𝑥^{𝑝/(𝑝−1)}𝑑𝑦. (39)
On the other hand, using Hölder's inequality one more time, we have _{𝐵(𝑥0,𝑟)𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝−11|𝑥−𝑦|^𝑛−1𝑑𝑥^{𝑝/(𝑝−1)}≤𝑑𝑦_{𝐵(𝑥0,𝑟)𝐵(𝑥0,𝑟)}|𝑉(𝑧)|𝑧−𝑦^𝑛−1𝑑𝑧^1/(𝑝−1)×_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑥−𝑦^𝑛−1=𝑑𝑥𝑑𝑦_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝×_{𝐵(𝑥0,𝑟)}1𝑥−𝑦^{𝑛−1𝐵(𝑥}₀^,𝑟)|𝑉(𝑧)|𝑧−𝑦^𝑛−1𝑑𝑧^1/(𝑝−1)≤[Φ(𝑑𝑦𝑑𝑥2𝑟)]^1/(𝑝−1)_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑑𝑥. (40)
By (39) and (40), we obtain _{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝[Φ(𝑑𝑥≤𝐶𝑛,𝑝2𝑟)]^1/(𝑝−1)_{𝐵(𝑥0,𝑟)}∇𝑢𝑦^𝑝𝑑𝑦^1/𝑝×_{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑑𝑥^1−1/𝑝. (41)
Hence _{𝐵(𝑥0,𝑟)}|𝑉(𝑥)||𝑢(𝑥)|^𝑝[Φ(𝑑𝑥≤𝐶𝑛,𝑝2𝑟)]^𝑞/𝑝_{𝐵(𝑥0,𝑟)}|∇𝑢(𝑥)|^𝑝𝑑𝑥. (42)

The next corollary is an easy consequence of the previous theorem. It can be obtained via a standard partition of unity.

Corollary 16. Let 𝑉∈𝑀_𝑝(ℝ^𝑛) and let Ω be a bounded subset of ℝ^𝑛, supp𝑉⊆Ω. Then, for any 𝜎>0 there exists a positive constant 𝐾 depending on 𝜎, such that _Ω|𝑉(𝑥)||𝑢(𝑥)|^𝑝𝑑𝑥≤𝜎_Ω|∇𝑢(𝑥)|^𝑝𝑑𝑥+𝐾(𝜎)_Ω|𝑢(𝑥)|^𝑝𝑑𝑥, (43) for all 𝑢∈𝐶^∞₀(Ω).

Proof. Let 𝜎>0. Let 𝑟 be a positive number that will be chosen later. Let {𝛼^𝑝_𝑘}, 𝑘=1,2,…,𝑁(𝑟), be a finite partition of the unity of Ω, such that supp𝑉⊆𝐵(𝑥_𝑘,𝑟) with 𝑥_𝑘∈Ω. We apply Theorem 15 to the functions 𝛼_𝑘 and we get _Ω|𝑉(𝑥)||𝑢(𝑥)|^𝑝=𝑑𝑥_Ω|𝑉(𝑥)||𝑢(𝑥)|^{𝑝𝑁(𝑟)}_𝑘=1𝛼^𝑝_𝑘(𝑥)=𝑑𝑥_{𝑁(𝑟)𝑘=1Ω}|𝑉(𝑥)|𝑢(𝑥)𝛼^𝑝_𝑘(𝑥)^𝑝≤𝑑𝑥_{𝑁(𝑟)𝑘=1}𝐶Φ_𝑉(2𝑟)_Ω|∇𝑢(𝑥)|^𝑝𝛼^𝑝_𝑛(𝑥)+𝑑𝑥_Ω|∇𝑢(𝑥)|^𝑝|𝑢(𝑥)|^𝑝𝑑𝑥≤𝐶Φ_𝑉(2𝑟)_Ω|∇𝑢(𝑥)|^𝑝𝑁(𝑟)𝑑𝑥+𝑟^𝑝_Ω|𝑢(𝑥)|^𝑝𝑑𝑥. (44) Finally, to obtain the result, it is sufficient to choose 𝑟 such that 𝐶Φ_𝑉(2𝑟)=𝜎. After that, we note that 𝑁(𝑟)≈𝑟^−𝑛 and the corollary follows.

Lemma 17. Let 𝐵_𝑟 and 𝐵_2𝑟 be two concentric balls contained in Ω. Then, _𝐵𝑟|∇𝑢(𝑥)|^𝑝𝐶𝑑𝑥≤𝑟^𝑝_𝐵2𝑟|𝑢|^𝑝, (45) where the constant 𝐶 does not depend on 𝑟 and 𝑢∈𝐶^∞₀(𝐵_𝑟).

Proof. Take 𝜑∈𝐶^∞₀(Ω), with supp𝜑⊂𝐵_2𝑟, 𝜑(𝑥)=1 for 𝑥∈𝐵_𝑟 and |∇𝜑|≤𝐶/𝑟 using 𝜑^𝑝 as a test function in (6); we get −div|_𝐵2𝑟∇𝑢|^𝑝−2∇𝑢𝜑^𝑝𝑢+_𝐵2𝑟𝑉|𝑢|^𝑝−2𝑢𝜑^𝑝𝑢=0. (46) Thus, _𝐵2𝑟|∇𝑢|^𝑝𝜑^𝑝=−𝑝_𝐵2𝑟|∇𝑢|^𝑝−2𝜑^𝑝−2∇𝑢⋅∇𝜑𝜑𝑢−_𝐵2𝑟𝑉𝜑𝑢^𝑝. (47)
Using Young's inequalities for (𝑝−1)/𝑝+1/𝑝=1, we can estimate the first integral in the right-hand side of (47) by 𝑝−1𝜀^{𝑝/(𝑝−1)}_𝐵2𝑟|∇𝑢|^𝑝𝜑^𝑝+𝜀^−𝑝_𝐵2𝑟∇𝜑^𝑝|𝑢|^𝑝. (48)
Also by result of Corollary 16, we can estimate the second integral in the right-hand side of (47) by 𝜀_𝐵2𝑟∇𝜑𝑢^𝑝+𝐶_{𝜀𝐵2𝑟}𝜑𝑢^𝑝. (49)
Using these estimates in (47), we have _𝐵2𝑟|∇𝑢|^𝑝𝜑^𝑝≤𝑝−1𝜀^{𝑝/(𝑝−1)}+𝜀_𝐵2𝑟|∇𝑢|^𝑝𝜑^𝑝+𝜀^𝑝+𝜀_𝐵2𝑟|𝑢|^𝑝|∇𝑢|^𝑝+𝐶_{𝜀𝐵2𝑟}|∇𝑢|^𝑝𝜑^𝑝. (50)
Using the fact that |∇𝜑|≤𝐶/𝑟, |𝜑|≤𝐶/𝑟, and 𝜑=1 in 𝐵_𝑟, we immediately have inequality (45).

Lemma 18. Let 𝑢∈𝐶^∞₀(𝐵_𝑟) where 𝐵_𝑟 is the ball of radius 𝑟 in ℝ^𝑛 and let 𝐸={𝑥∈𝐵_𝑟∶𝑢(𝑥)=0}. Then, there exists a constant 𝛽 depending only on 𝑛, such that _𝐴|𝑢|𝑟≤𝛽^𝑛𝑚(𝐸)[𝑚(𝐴)]^1/𝑛_𝐵2𝑟|∇𝑢|, (51) for all ball 𝐵_𝑟, u as above, and all measurable sets 𝐴⊂𝐵_𝑟.

To prove this lemma see [5]. Note that 𝑚(𝐴) and 𝑚(𝐸) denote the Lebesgue measure of the sets 𝐴 and 𝐸.

4. Strong Unique Continuation

In this section, we proceed to establish the strong unique continuation property of the eigenfunction for the 𝑝-Laplacian operator in the case 𝑉∈𝑀_𝑝(Ω).

Theorem 19. Let 𝑢∈𝐶^∞₀(Ω) be a solution of (6). If 𝑢=0 on a set 𝐸 of positive measures, then 𝑢 has zero of infinite order in 𝑝-mean.

Proof. We know that almost every point of 𝐸 is a point of density of 𝐸. Let 𝑥₀∈𝐸 be such point. This means that lim_𝑟→0𝑚𝐸∩𝐵_𝑟𝑚𝐵_𝑟=1, (52) where 𝐵_𝑟 denotes the ball of radius 𝑟 centered at 𝑥₀, and thus, given that 𝜀>0, there is an 𝑟₀=𝑟₀(𝜀), such that 𝑚𝐸^𝑐∩𝐵_𝑟𝑚𝐵_𝑟𝑚<𝜀,𝐸∩𝐵_𝑟𝑚𝐵_𝑟>1−𝜀for𝑟≤𝑟₀, (53) where 𝐸^𝑐 denotes the complement of the set 𝐸. Taking 𝑟₀, smaller if necessary, we can assume that 𝐵_𝑟0⊂Ω. Since𝑢=0 on𝐸, by Lemmas 18 and 11, we have _𝐵𝑟|𝑢|^𝑝=_{𝐵𝑟∩𝐸𝑐}|𝑢|^𝑝𝑟≤𝛽^𝑛𝑚𝐸∩𝐵_𝑟𝑚𝐸^𝑐∩𝐵_𝑟^1/𝑛×_𝐵𝑟|∇(𝑢)|^𝑝,𝑟𝑝𝛽^𝑛𝜀^1/𝑛𝑚𝐵_𝑟^1−1/𝑛(1−𝜀)^𝐵_𝑟|𝑢|^𝑝−1|∇(𝑢)|. (54) By Hölder inequality _𝐵𝑟|𝑢|^𝑝𝜀≤𝐶^1/𝑛1−𝜀_𝐵𝑟|∇(𝑢)|^𝑝1/𝑝_𝐵𝑟|𝑢|^{𝑝(𝑝−1)/𝑝}, (55) and by using the Young inequality, we get _𝐵𝑟|𝑢|^𝑝𝜀≤𝐶^1/𝑛𝑟𝑟1−𝜀^𝑝−1_𝐵𝑟|∇(𝑢)|^𝑝+𝑝−1𝑟_𝐵𝑟𝑢^𝑝. (56) Finally, by Lemma 17, we have _𝐵𝑟|𝑢|^𝑝𝜀≤𝐶^1/𝑛1−𝜀_𝐵2𝑟𝑢^𝑝, (57) where 𝐶 is independent of 𝜀 and of 𝑟, as 𝑟→0. Now, let us introduce the following function: 𝑓(𝑟)=_𝐵2𝑟|𝑢|^𝑝. (58)
And let us fix 𝑛∈ℕand choose 𝜀>0 such that (𝐶𝜀^1/𝑛)(1−𝜀)≤2^−𝑛. Observe that consequently 𝑟₀ depends on 𝑛. Then, (57) can be written as 𝑓(𝑟)≤2^−𝑛𝑓(2𝑟),for𝑟≤𝑟₀. (59) Iterating (59), we get 𝑓𝜌≤2^−𝑘𝑛𝑓2^𝑘𝜌,if2^𝑘−1𝜌≤𝑟₀. (60)
Now, given that0<𝑟<𝑟₀(𝑛), choose 𝑘∈ℕ, such that 2^−𝑘𝑟₀≤𝑟≤2^−𝑘+1𝑟₀. (61)
From (60), we obtain 𝑓(𝑟)≤2^−𝑘𝑛𝑓2^𝑘𝑟≤2^−𝑘𝑛𝑓(2𝑟). (62) Since2^−𝑘≤𝑟/𝑟₀, we finally obtain 𝑓(𝑟)≤𝑟𝑟₀^𝑛𝑓2𝑟₀. (63) And thus, we have _{𝐵𝑟(𝑥0)}𝑢𝑦^𝑝𝑟𝑑𝑦≤𝑟₀^𝑛𝑓2𝑟₀, (64) and this shows that (7) holds, which means that 𝑢 has a zero of infinite order in 𝑝-mean at 𝑥₀.

Corollary 20. Equation (6) has a strong unique continuation property.

Acknowledgments

The authors would like to thank the referee for the useful comments and suggestions which improved the presentation of this paper. The authors were supported by the Banco Central de Venezuela.