Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

Abstract

We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of ℝ^𝑛, 𝑛≥3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.

1. Introduction

Let Ω be an open subset of 𝑅^𝑛, 𝑛≥3. Assign in Ω the uniformly elliptic second-order linear differential operator𝐿=−^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑛_𝑖=1𝑎_𝑖𝜕𝜕𝑥_𝑖+𝑎, (1.1)with coefficients 𝑎_𝑖𝑗=𝑎_𝑗𝑖∈𝐿^∞(Ω), 𝑖,𝑗=1,…,𝑛, and consider the associate Dirichlet problem:𝑢∈𝑊^2,𝑝(Ω)∩𝑊^∘1,𝑝(Ω),𝐿𝑢=𝑓,𝑓∈𝐿^𝑝(Ω), (1.2)where 𝑝∈]1,+∞[.

It is well known that if Ω is a bounded and sufficiently regular set, the above problem has been widely investigated by several authors under various hypotheses of discontinuity on the leading coefficients, in the case 𝑝=2 or 𝑝 sufficiently close to 2. In particular, some 𝑊^2,𝑝-bounds for the solutions of the problem (1.2) and related existence and uniqueness results have been obtained. Among the other results on this subject, we quote here those proved in [1], where the author assumed that 𝑎_𝑖𝑗's belong to 𝑊^1,𝑛(Ω) (and considered the case 𝑝=2) and in [2–4] (where the coefficients belong to some classes wider than 𝑊^1,𝑛(Ω)). More recently, a relevant contribution has been given in [5–8], where the coefficients 𝑎_𝑖𝑗 are assumed to be in the class VMO and 𝑝∈]1,+∞[; observe here that VMO contains the space 𝑊^1,𝑛(Ω).

If the set Ω is unbounded and regular enough, under assumptions similar to those required in [1], problem (1.2) has for instance been studied in [9–11] with 𝑝=2, and in [12] with 𝑝∈]1,+∞[. Instead, in [13, 14] the leading coefficients satisfy restrictions similar to those in [5, 6].

In this paper, we extend some results of [13, 14] to a weighted case. More precisely, we denote by 𝜌 a weight function belonging to a suitable class such thatinf_Ω𝜌>0,lim_{|𝑥|→+∞}𝜌(𝑥)=+∞, (1.3)and consider the Dirichlet problem:𝑢∈𝑊_𝑠^2,𝑝(Ω)∩𝑊^∘_𝑠^1,𝑝(Ω),𝐿𝑢=𝑓,𝑓∈𝐿^𝑝_𝑠(Ω), (1.4)where 𝑠∈𝑅, 𝑊_𝑠^2,𝑝(Ω), 𝑊^∘_𝑠^1,𝑝(Ω), and 𝐿^𝑝_𝑠(Ω) are some weighted Sobolev spaces and the weight functions are a suitable power of 𝜌. We obtain an a priori bound for the solutions of (1.4). Moreover, we state a regularity result that allows us to deduce a uniqueness theorem for the problem (1.4). A similar weighted case was studied in [15] with the leading coefficients satisfying hypotheses of Miranda's type and when 𝑝=2.

2. Weight Functions and Weighted Spaces

Let 𝐺 be any Lebesgue measurable subset of 𝑅^𝑛 and let Σ(𝐺) be the collection of all Lebesgue measurable subsets of 𝐺. If 𝐹∈Σ(𝐺), denote by |𝐹| the Lebesgue measure of 𝐹, by 𝜒_𝐹 the characteristic function of 𝐹, by 𝐹(𝑥,𝑟) the intersection 𝐹∩𝐵(𝑥,𝑟) (𝑥∈𝑅^𝑛, 𝑟∈𝑅₊)—where 𝐵(𝑥,𝑟) is the open ball of radius 𝑟 centered at 𝑥—and by 𝒟(𝐹) the class of restrictions to 𝐹 of functions 𝜁∈𝐶^∞_∘(𝑅^𝑛) with 𝐹∩supp𝜁⊆𝐹. Moreover, if 𝑋(𝐹) is a space of functions defined on 𝐹, we denote by 𝑋_loc(𝐹) the class of all functions 𝑔∶𝐹→𝑅, such that 𝜁𝑔∈𝑋(𝐹) for any 𝜁∈𝒟(𝐹).

We introduce a class of weight functions defined on an open subset Ω of 𝑅^𝑛. Denote by 𝒜(Ω) the set of all measurable functions 𝜌∶Ω→𝑅₊, such that𝛾⁻¹𝜌(𝑦)≤𝜌(𝑥)≤𝛾𝜌(𝑦)∀𝑦∈Ω,∀𝑥∈Ω(𝑦,𝜌(𝑦)), (2.1)where 𝛾∈𝑅₊ is independent of 𝑥 and 𝑦. Examples of functions in 𝒜(Ω) are the function𝑥∈𝑅^𝑛→1+𝑎|𝑥|,𝑎∈]0,1[, (2.2)and, if Ω≠𝑅^𝑛 and 𝑆 is a nonempty subset of 𝜕Ω, the function𝑥∈Ω→𝑎dist(𝑥,𝑆),𝑎∈]0,1[. (2.3)For 𝜌∈𝒜(Ω), we put𝑆_𝜌={𝑧∈𝜕Ω∶lim_𝑥→𝑧𝜌(𝑥)=0}. (2.4)It is known that𝜌∈𝐿^∞_loc(Ω),𝜌⁻¹∈𝐿^∞_loc(Ω⧵𝑆_𝜌) (2.5)(see [16, 17]).

We assign an unbounded open subset Ω of 𝑅^𝑛.

Let 𝜌₁ be a function, such that 𝜌₁∈𝒜(𝑅^𝑛) andinf_Ω𝜌₁>0,lim_{|𝑥|→+∞}𝜌₁(𝑥)=+∞. (2.6)We put𝜌=𝜌_1∣Ω. (2.7)For any 𝑎∈]0,1] and 𝑥∈𝑅^𝑛, we set𝐼_𝑎(𝑥)=Ω(𝑥,𝑎𝜌₁(𝑥)). (2.8)

If 𝑘∈𝑁₀, 1≤𝑝<+∞, 𝑠∈𝑅, and 𝜌∈𝒜(Ω), consider the space 𝑊_𝑠^𝑘,𝑝(Ω) of distributions 𝑢 on Ω, such that 𝜌^𝑠𝜕^𝛼𝑢∈𝐿^𝑝(Ω) for |𝛼|≤𝑘, equipped with the norm‖𝑢‖_𝑊𝑠^𝑘,𝑝_(Ω)=_{|𝛼|≤𝑘}‖𝜌^𝑠𝜕^𝛼𝑢‖_𝐿^𝑝_(Ω). (2.9)Moreover, denote by 𝑊^∘_𝑠^𝑘,𝑝(Ω) the closure of 𝐶^∞_∘(Ω) in 𝑊_𝑠^𝑘,𝑝(Ω) and put 𝑊_𝑠^0,𝑝(Ω)=𝐿^𝑝_𝑠(Ω). A more detailed account of properties of the above defined spaces can be found, for instance, in [18].

From [15, Lemmas 1.1 and 2.1], we deduce the following two lemmas, respectively.

Lemma 2.1. For any 𝑝∈[1,+∞[, 𝑠∈𝑅, and 𝑎∈]0,1], 𝑔∈𝐿^𝑝_𝑠(Ω) if and only if 𝑔∈𝐿^𝑝_loc(Ω) and the function 𝑥∈𝑅^𝑛→𝜌₁^{𝑠−𝑛/𝑝}(𝑥)||𝑔||_𝐿^𝑝_{(𝐼𝑎(𝑥))} belongs to 𝐿^𝑝(𝑅^𝑛). Moreover, there exist 𝑐₁,𝑐₂∈𝑅₊, such that𝑐₁‖𝑔‖^𝑝_{𝐿𝑝𝑠(Ω)}≤_𝑅𝑛𝜌₁^{𝑠𝑝−𝑛}(𝑥)‖𝑔‖^𝑝_{𝐿𝑝(𝐼𝑎(𝑥))}𝑑𝑥≤𝑐₂‖𝑔‖^𝑝_{𝐿𝑝𝑠(Ω)}∀𝑔∈𝐿^𝑝_𝑠(Ω), (2.10) where 𝑐₁ and 𝑐₂ depend on 𝑛, 𝑝, 𝑠, 𝑎, and 𝜌.

Lemma 2.2. If Ω has the segment property, then for any 𝑘∈𝑁₀, 𝑝∈[1,+∞[, and 𝑠∈𝑅 one has𝑊_𝑠^𝑘,𝑝(Ω)∩𝑊^∘𝑘,𝑝_loc(Ω)=𝑊^∘_𝑠^𝑘,𝑝(Ω). (2.11)

3. Some Embedding Lemmas

We now recall the definitions of the function spaces in which the coefficients of the operator will be chosen. If Ω has the property|Ω(𝑥,𝑟)|≥𝐴𝑟^𝑛∀𝑥∈Ω,∀𝑟∈]0,1], (3.1)where 𝐴 is a positive constant independent of 𝑥 and 𝑟, it is possible to consider the space BMO(Ω,𝜏) (𝜏∈𝑅₊) of functions 𝑔∈𝐿¹_loc(Ω) such that[𝑔]_BMO(Ω,𝜏)=sup_{𝑥∈Ω𝑟∈]0,𝜏]Ω(𝑥,𝑟)}|𝑔−_{Ω(𝑥,𝑟)}𝑔|<+∞, (3.2)where_{Ω(𝑥,𝑟)}𝑔=|Ω(𝑥,𝑟)|⁻¹_{Ω(𝑥,𝑟)}𝑔. (3.3)If 𝑔∈BMO(Ω)=BMO(Ω,𝜏_𝐴), where𝜏_𝐴=sup{𝜏∈𝑅₊∶sup_{𝑥∈Ω𝑟∈]0,𝜏]}𝑟^𝑛≤1|Ω(𝑥,𝑟)|𝐴}, (3.4)we will say that 𝑔∈VMO(Ω) if [𝑔]_BMO(Ω,𝜏)→0 for 𝜏→0⁺. A function 𝜂[𝑔]∶]0,1]→𝑅₊ is called a modulus of continuity of 𝑔 in VMO(Ω) if[𝑔]_BMO(Ω,𝜏)≤𝜂[𝑔](𝜏)∀𝜏∈]0,1],lim_𝜏→0+𝜂[𝑔](𝜏)=0. (3.5)For 𝑡∈[1,+∞[ and 𝜆∈[0,𝑛[, we denote by 𝑀^𝑡,𝜆(Ω) the set of all functions 𝑔 in 𝐿^𝑡_loc(Ω) such that‖𝑔‖_𝑀^𝑡,𝜆_(Ω)=sup_{𝑟∈]0,1]𝑥∈Ω}𝑟^−𝜆/𝑡‖𝑔‖_𝐿^𝑡_{(Ω(𝑥,𝑟))}<+∞, (3.6)endowed with the norm defined by (3.6). Then, we define 𝑀_∘^𝑡,𝜆(Ω) as the closure of 𝐶^∞_∘(Ω) in 𝑀^𝑡,𝜆(Ω). In particular, we put 𝑀^𝑡(Ω)=𝑀^𝑡,0(Ω), and 𝑀^𝑡_∘(Ω)=𝑀_∘^𝑡,0(Ω). In order to define the modulus of continuity of a function 𝑔 in 𝑀_∘^𝑡,𝜆(Ω), recall first that for a function 𝑔∈𝑀^𝑡,𝜆(Ω) the following characterization holds:𝑔∈𝑀_∘^𝑡,𝜆(Ω)⟺lim_𝜏→0+(𝑝_𝑔(𝜏)+‖(1−𝜁_1/𝜏)𝑔‖_𝑀^𝑡,𝜆_(Ω))=0, (3.7)where𝑝_𝑔(𝜏)=sup_{𝐸∈Σ(Ω)sup𝑥∈Ω|𝐸(𝑥,1)|≤𝜏}‖𝜒_𝐸𝑔‖_𝑀^𝑡,𝜆_(Ω),𝜏∈𝑅₊, (3.8)and 𝜁_𝑟, 𝑟∈𝑅₊, is a function in 𝐶^∞_∘(𝑅^𝑛) such that0≤𝜁_𝑟≤1,𝜁_{𝑟∣𝐵𝑟}=1,supp𝜁_𝑟⊂𝐵_2𝑟, (3.9)with the position 𝐵_𝑟=𝐵(0,𝑟). Thus, the modulus of continuity of 𝑔∈𝑀_∘^𝑡,𝜆(Ω) is a function𝜎_∘[𝑔]∶]0,1]→𝑅₊, (3.10)such that𝑝_𝑔(𝜏)+‖(1−𝜁_1/𝜏)𝑔‖_𝑀^𝑡,𝜆_(Ω)≤𝜎_∘[𝑔](𝜏)∀𝜏∈]0,1],lim_𝜏→0+𝜎_∘[𝑔](𝜏)=0. (3.11)A more detailed account of properties of the above defined function spaces can be found in [9, 19, 20].

We consider the following condition:(ℎ₀)Ω has the cone property, 𝑝∈]1,+∞[, 𝑠∈𝑅, 𝑘, ℎ, 𝑡 are numbers such that𝑛𝑘∈𝑁,ℎ∈{0,1,…,𝑘−1},𝑡≥𝑝,𝑡>𝑝if𝑝=𝑘−ℎ,𝑔∈𝑀^𝑡(Ω). (3.12)

From [21, Theorem 3.1] we have the following.

Lemma 3.1. If the assumption (ℎ₀) holds, then for any 𝑢∈𝑊_𝑠^𝑘,𝑝(Ω) one has 𝑔𝜕^ℎ𝑢∈𝐿^𝑝_𝑠(Ω) and ‖𝑔𝜕^ℎ𝑢‖_𝐿^𝑝_𝑠(Ω)≤𝑐‖𝑔‖_𝑀^𝑡_(Ω)‖𝑢‖_𝑊𝑠^𝑘,𝑝_(Ω), (3.13)with 𝑐 dependent only on Ω, 𝑛, 𝑘, ℎ, 𝑝, and 𝑡.

From [21, Theorem 3.2] it follows Lemma 3.2.

Lemma 3.2. If the assumption (ℎ₀) is satisfied and in addition 𝑔∈𝑀^𝑡_∘(Ω), then for any 𝜀∈𝑅₊ there exist a constant 𝑐(𝜀)∈𝑅₊ and a bounded open set Ω_𝜀⊂⊂Ω, with the cone property, such that ‖𝑔𝜕^ℎ𝑢‖_𝐿^𝑝_𝑠(Ω)≤𝜀‖𝑢‖_𝑊𝑠^𝑘,𝑝_(Ω)+𝑐(𝜀)‖𝑢‖_𝐿^𝑝_(Ω𝜀)∀𝑢∈𝑊_𝑠^𝑘,𝑝(Ω), (3.14)where 𝑐(𝜀), Ω_𝜀 depend on 𝜀, Ω, 𝑛, 𝑘, ℎ, 𝑝, 𝑡, 𝜌, 𝑠, and 𝜎_∘[𝑔].

4. An a Priori Bound

Assume that Ω is an unbounded open subset of 𝑅^𝑛, 𝑛≥3, with the uniform 𝐶^1,1-regularity property, and let 𝜌 be the function defined by (2.7). Moreover, let 𝑝∈]1,+∞[ and 𝑠∈𝑅. Consider in Ω the differential operator:𝐿=−^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑛_𝑖=1𝑎_𝑖𝜕𝜕𝑥_𝑖+𝑎, (4.1)with the following conditions on the coefficients:(ℎ₁)𝑎_𝑖𝑗=𝑎_𝑗𝑖∈𝐿^∞(Ω)∩VMO_loc(Ω),𝑖,𝑗=1,…,𝑛,∃𝜈>0∶∑^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜉_𝑖𝜉_𝑗≥𝜈|𝜉|²a.e.inΩ,∀𝜉∈𝑅^𝑛, (4.2) there exist functions 𝑒_𝑖𝑗, 𝑖,𝑗=1,…,𝑛, 𝑔 and 𝜇∈𝑅₊ such that(ℎ₂)𝑒_𝑖𝑗=𝑒_𝑗𝑖∈𝐿^∞(Ω),(𝑒_𝑖𝑗)_𝑥ℎ∈𝑀₀^{𝑡,𝑛−𝑡}(Ω),with𝑡∈]2,𝑛],𝑖,𝑗,ℎ=1,…,𝑛,_𝑛𝑖,∑_𝑗=1𝑒_𝑖𝑗𝜉_𝑖𝜉_𝑗≥𝜇|𝜉|²a.e.inΩ,∀𝜉∈𝑅^𝑛,𝑔∈𝐿^∞(Ω),𝑔₀=essinf_Ω𝑔>0,lim^𝑛_𝑟→+∞∑_𝑖,𝑗=1‖𝑒_𝑖𝑗−𝑔𝑎_𝑖𝑗‖_𝐿^∞_{(Ω⧵𝐵𝑟)}=0, (4.3)(ℎ₃)𝑎_𝑖∈𝑀^𝑡1_∘(Ω),𝑖=1,…,𝑛,𝑎=𝑎^′+𝑎^′′,𝑎^′∈𝑀^𝑡2_∘(Ω),𝑎^′′∈𝐿^∞(Ω),essinf_Ω𝑎^′′=𝑎₀^′′>0, (4.4)where𝑡₁≥𝑛if𝑝<𝑛,𝑡₁>𝑛if𝑝=𝑛,𝑡₁𝑡=𝑝if𝑝>𝑛,₂≥𝑛/2if𝑝<𝑛/2,𝑡₂>𝑛/2if𝑝=𝑛/2,𝑡₂=𝑝if𝑝>𝑛/2. (4.5)

Observe that under the assumptions (ℎ₁)-(ℎ₃), it follows that the operator 𝐿∶𝑊_𝑠^2,𝑝(Ω)→𝐿^𝑝_𝑠(Ω) is bounded from Lemma 3.1.

Theorem 4.1. If the hypotheses (ℎ₁), (ℎ₂), and (ℎ₃) are verified, then there exist a constant 𝑐∈𝑅₊ and a bounded open subset Ω₀⊂⊂Ω, with the cone property, such that‖𝑢‖_𝑊𝑠^2,𝑝_(Ω)≤𝑐(‖𝐿𝑢‖_𝐿^𝑝_𝑠(Ω)+‖𝑢‖_𝐿^𝑝_(Ω0)),∀𝑢∈𝑊_𝑠^2,𝑝(Ω)∩𝑊^∘_𝑠^1,𝑝(Ω), (4.6) with 𝑐 and Ω₀ depending on 𝑛, 𝑝, 𝜌, 𝑠, Ω, 𝜈, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, 𝑡₁, 𝑡₂, ‖𝑎_𝑖𝑗‖_𝐿∞(Ω), ‖𝑒_𝑖𝑗‖_𝐿∞(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜂[𝜁_2𝑟0𝑎_𝑖𝑗], 𝜎_∘[(𝑒_𝑖𝑗)_𝑥], 𝜎_∘[𝑎_𝑖], 𝜎_∘[𝑎^′], where 𝑟₀∈𝑅₊ depends on 𝑛, 𝑝, Ω, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜎_∘[(𝑒_𝑖𝑗)_𝑥]. 

Proof. We consider a function 𝜙∈𝐶^∞_∘(𝑅^𝑛), such that𝜙_|𝐵1/2=1,supp𝜙⊂𝐵₁,sup_𝑅𝑛|𝜕^𝛼𝜙|≤𝑐_𝛼∀𝛼∈𝑁^𝑛₀, (4.7) where 𝑐_𝛼∈𝑅₊ depends only on 𝛼, fix 𝑦∈𝑅^𝑛 and put𝜓=𝜓_𝑦∶𝑥∈𝑅^𝑛→𝜙(𝑥−𝑦𝜌₁(𝑦)). (4.8)Clearly we have𝜓_{|𝐵(𝑦,(1/2)𝜌1(𝑦))}=1,supp𝜓⊂𝐵(𝑦,𝜌₁(𝑦)),sup_𝑅𝑛|𝜕^𝛼𝜓|≤𝑐_𝛼𝜌₁^−|𝛼|(𝑦)∀𝛼∈𝑁^𝑛₀. (4.9) Now, we put𝐿₀=−^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗 (4.10)and fix 𝑢∈𝑊_𝑠^2,𝑝(Ω)∩𝑊^∘_𝑠^1,𝑝(Ω). Since 𝜓𝑢∈𝑊^2,𝑝(Ω)∩𝑊^∘1,𝑝(Ω), from [14, Theorem 3.3], it follows that there exist 𝑐₁∈𝑅₊ and a bounded open subset Ω₁⊂⊂Ω, with the cone property, such that‖𝜓𝑢‖_{𝑊2,𝑝(Ω)}≤𝑐₁(‖(𝐿₀+𝑎^′′)(𝜓𝑢)‖_𝐿𝑝(Ω)+‖𝜓𝑢‖_{𝐿𝑝(Ω1)}), (4.11)with 𝑐₁ and Ω₁ depending on 𝑛, 𝑝, Ω, 𝜈, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, ‖𝑎_𝑖𝑗‖_𝐿∞(Ω), ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜂[𝜁_2𝑟0𝑎_𝑖𝑗], 𝜎_∘[(𝑒_𝑖𝑗)_𝑥], where 𝑟₀∈𝑅₊ depends on 𝑛, 𝑝, Ω, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜎_∘[(𝑒_𝑖𝑗)_𝑥]. Since𝐿₀(𝜓𝑢)=𝜓𝐿₀𝑢−2^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜓_𝑥𝑖𝑢_𝑥𝑗−^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜓_{𝑥𝑖𝑥𝑗}𝑢, (4.12)from (4.11) and (4.12), we have‖𝜓𝑢‖_{𝑊2,𝑝(Ω)}≤𝑐₂(‖𝜓(𝐿₀+𝑎^′′)𝑢‖_𝐿𝑝(Ω)+∑^𝑛_𝑖,𝑗=1‖𝜓_𝑥𝑖𝑢_𝑥𝑗‖_𝐿𝑝(Ω)+∑^𝑛_𝑖,𝑗=1‖𝜓_{𝑥𝑖𝑥𝑗}𝑢‖_𝐿𝑝(Ω)+‖𝜓𝑢‖_{𝐿𝑝(Ω1)}), (4.13) with 𝑐₂ dependent on the same parameters of 𝑐₁.
On the other hand, since 𝜌∈𝐿^∞_loc(Ω), we have that‖𝜓𝑢‖_{𝐿𝑝(Ω1)}≤𝑐₃𝜌₁⁻²(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}, (4.14)where 𝑐₃∈𝑅₊ depends only on 𝜌.
Therefore, by (4.13) and (4.14), we deduce the bound:‖𝑢‖_𝑊^2,𝑝_{(𝐼1/2(𝑦))}≤‖𝜓𝑢‖_{𝑊2,𝑝(Ω)}≤𝑐₄(‖𝐿₀𝑢+𝑎^′′𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}+𝜌₁⁻¹(𝑦)‖𝑢_𝑥‖_{𝐿𝑝(𝐼1(𝑦))}+𝜌₁⁻²(𝑦)‖𝑢‖_𝐿^𝑝_(𝐼1(𝑦)), (4.15) where 𝑐₄∈𝑅₊ depends on the same parameters of 𝑐₂ and on 𝜌.
From (4.15) it follows_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛}(𝑦)‖𝑢‖^𝑝_{𝑊2,𝑝(𝐼1/2(𝑦))}𝑑𝑦≤𝑐₅(_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛}(𝑦)‖𝐿₀𝑢+𝑎^′′𝑢‖^𝑝_{𝐿𝑝(𝐼1(𝑦))}+𝑑𝑦_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛−𝑝}(𝑦)‖𝑢_𝑥‖^𝑝_{𝐿𝑝(𝐼1(𝑦))}𝑑𝑦+_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛−2𝑝}(𝑦)‖𝑢‖^𝑝_{𝐿𝑝(𝐼1(𝑦))}𝑑𝑦), (4.16) where 𝑐₅∈𝑅₊ depends on the same parameters of 𝑐₄.
Since𝐿^𝑝_𝑠(Ω)↪𝐿^𝑝_𝑠−1(Ω),𝐿^𝑝_𝑠(Ω)↪𝐿^𝑝_𝑠−2(Ω), (4.17)from (4.16) and from Lemma 2.1 we have that‖𝑢‖_𝑊𝑠^2,𝑝_(Ω)≤𝑐₆(‖𝐿₀𝑢+𝑎^′′𝑢‖_𝐿^𝑝_𝑠(Ω)+‖𝑢_𝑥‖_𝐿^𝑝_𝑠−1(Ω)+‖𝑢‖_𝐿^𝑝_𝑠−2(Ω)), (4.18)with 𝑐₆∈𝑅₊ dependent on the same parameters of 𝑐₅ and also on 𝑠.
Moreover, from Lemma 3.2 it follows that for any 𝜀∈𝑅₊, there exist 𝑐^′(𝜀), 𝑐^′′(𝜀)∈𝑅₊, and two bounded open sets Ω^′_𝜀, Ω_𝜀^′′⊂⊂Ω, both with the cone property, such that‖𝑢_𝑥‖_𝐿^𝑝_𝑠−1(Ω)+‖𝑢‖_𝐿^𝑝_𝑠−2(Ω)≤𝜀‖𝑢‖_𝑊𝑠^2,𝑝_(Ω)+𝑐^′(𝜀)‖𝑢‖_{𝐿𝑝(Ω′𝜀)},‖^𝑛_𝑖=1𝑎_𝑖𝑢_𝑥𝑖+𝑎^′𝑢‖_{𝐿𝑝𝑠(Ω)}≤𝜀‖𝑢‖_𝑊𝑠^2,𝑝_(Ω)+𝑐^′′(𝜀)‖𝑢‖_{𝐿𝑝(Ω𝜀′′)}, (4.19) where 𝑐^′(𝜀), Ω^′_𝜀 depend on 𝜀, Ω, 𝑛, 𝑝, 𝜌, 𝑠, and 𝑐^′′(𝜀), Ω_𝜀^′′ depend on 𝜀, Ω, 𝑛, 𝑝, 𝑡₁, 𝑡₂, 𝜌, 𝑠, 𝜎_∘[𝑎_𝑖], and 𝜎_∘[𝑎^′].
From (4.18) and (4.19) it follows (4.6) and then we have the result.

5. A Uniqueness Result

In this section, we will prove our uniqueness theorem. We begin to prove a regularity result.

Lemma 5.1. Suppose that the assumptions (ℎ₁), (ℎ₂), and (ℎ₃) (with 𝑡₁>𝑛 and 𝑡₂>𝑛/2) hold, and let 𝑢 be a solution of the problem 𝑢∈𝑊^2,𝑞_loc(Ω)∩𝑊^∘1,𝑞_loc(Ω)∩𝐿^𝑝_𝑚(Ω),𝐿𝑢∈𝐿^𝑝_𝑠(Ω), (5.1)where 𝑞∈]1,𝑝] and 𝑚∈𝑅. Then, 𝑢 belongs to 𝑊_𝑠^2,𝑝(Ω).

Proof. By [13, Lemma 4.1] we have𝑢∈𝑊^2,𝑝_loc(Ω)∩𝑊^∘1,𝑝_loc(Ω). (5.2)We choose 𝑟,𝑟^′∈𝑅₊, with 𝑟<𝑟^′<1, and a function 𝜙∈𝐶^∞_∘(𝑅^𝑛), such that𝜙_|𝐵𝑟=1,supp𝜙⊂𝐵_𝑟′,sup_𝑅𝑛|𝜕^𝛼𝜙|≤𝑐_𝛼(𝑟^′−𝑟)^−|𝛼|∀𝛼∈𝑁^𝑛₀, (5.3) where 𝑐_𝛼∈𝑅₊ depends only on 𝛼.
We fix 𝑦∈𝑅^𝑛 and put𝜓=𝜓_𝑦∶𝑥∈𝑅^𝑛→𝜙(𝑥−𝑦𝜌₁(𝑦)). (5.4)Clearly we have𝜓_{|𝐵(𝑦,𝑟𝜌1(𝑦))}=1,supp𝜓⊂𝐵(𝑦,𝑟^′𝜌₁(𝑦)),sup_𝑅𝑛|𝜕^𝛼𝜓|≤𝑐_𝛼𝜌₁^−|𝛼|(𝑦)(𝑟^′−𝑟)^−|𝛼|∀𝛼∈𝑁^𝑛₀. (5.5)
Since 𝜓𝑢∈𝑊^2,𝑝(Ω)∩𝑊^∘1,𝑝(Ω), from [14, Theorem 3.3] it follows that there exist 𝑐₁∈𝑅₊ and a bounded open subset Ω₁⊂⊂Ω, with the cone property, such that‖𝜓𝑢‖_𝑊^2,𝑝_(Ω)≤𝑐₁(‖𝐿(𝜓𝑢)‖_𝐿𝑝(Ω)+‖𝜓𝑢‖_{𝐿𝑝(Ω1)}), (5.6)with 𝑐₁ and Ω₁ depending on 𝑛, 𝑝, Ω, 𝜈, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, 𝑡₁, 𝑡₂, ‖𝑎_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜂[𝜁_2𝑟0𝑎_𝑖𝑗], 𝜎_∘[(𝑒_𝑖𝑗)_𝑥], 𝜎_∘[𝑎_𝑖], 𝜎_∘[𝑎^′], where 𝑟₀∈𝑅₊ depends on 𝑛, 𝑝, Ω, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜎_∘[(𝑒_𝑖𝑗)_𝑥].
Since𝐿(𝜓𝑢)=−^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝜓𝑢)_{𝑥𝑖𝑥𝑗}+^𝑛_𝑖=1𝑎_𝑖(𝜓𝑢)_𝑥𝑖+𝑎𝜓𝑢=𝜓𝐿𝑢−2^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝜓_𝑥𝑖𝑢)_𝑥𝑗+^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜓_{𝑥𝑖𝑥𝑗}𝑢+^𝑛_𝑖=1𝑎_𝑖𝜓_𝑥𝑖𝑢, (5.7) from (5.6) and (5.7), we have‖𝜓𝑢‖_{𝑊2,𝑝(Ω)}≤𝑐₂(‖𝜓𝐿𝑢‖_𝐿𝑝(Ω)+∑^𝑛_𝑖,𝑗=1‖(𝜓_𝑥𝑖𝑢)_𝑥𝑗‖_𝐿𝑝(Ω)+^𝑛_𝑖,𝑗=1‖𝜓_{𝑥𝑖𝑥𝑗}𝑢‖_𝐿𝑝(Ω)+^𝑛_𝑖=1‖𝑎_𝑖𝜓_𝑥𝑖𝑢‖_𝐿𝑝(Ω)+‖𝜓𝑢‖_𝐿^𝑝_(Ω1)), (5.8) with 𝑐₂ dependent on the same parameters of 𝑐₁.
From Lemma 3.1 with 𝑠=0, we have that‖𝑎_𝑖𝜓_𝑥𝑖𝑢‖_𝐿𝑝(Ω)≤𝑐₃‖𝑎_𝑖‖_𝑀^𝑡_1(Ω)(‖𝜓_𝑥𝑖𝑢‖_𝐿𝑝(Ω)+‖(𝜓_𝑥𝑖𝑢)_𝑥‖_𝐿^𝑝_(Ω)), (5.9)with 𝑐₃ dependent on Ω, 𝑛, 𝑝, and 𝑡₁.
Using [22, Corollary 4.5], we can obtain the following interpolation estimate:‖𝜓_𝑥𝑖𝑢‖_𝐿𝑝(Ω)+‖(𝜓_𝑥𝑖𝑢)_𝑥𝑗‖_𝐿𝑝(Ω)≤𝑐₄(‖(𝜓_𝑥𝑖𝑢)_𝑥𝑥‖_𝐿^1/2_𝑝(Ω)‖𝜓_𝑥𝑖𝑢‖_𝐿^1/2_𝑝(Ω)+‖𝜓_𝑥𝑖𝑢‖_𝐿𝑝(Ω)), (5.10) where the constant 𝑐₄ depends on Ω, 𝑛, 𝑝.
Thus, by (5.8)–(5.10), with easy computations, we deduce the bound:‖𝑢‖_𝑊^2,𝑝_{(𝐼𝑟(𝑦))}≤‖𝜓𝑢‖_{𝑊2,𝑝(Ω)}≤𝑐₅(𝑟^′−𝑟)⁻²×(‖𝐿𝑢‖_𝐿^𝑝_{(𝐼𝑟′(𝑦))}+‖𝑢‖_𝑊^1/2_{2,𝑝(𝐼𝑟′(𝑦))}(𝜌₁⁻¹(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼𝑟′(𝑦))})^1/2+𝜌₁⁻¹(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼𝑟′(𝑦))}), (5.11) where 𝑐₅∈𝑅₊ depends on 𝑛, 𝑝, 𝜌, Ω, 𝜈, 𝜇, 𝑔₀, 𝑎₀^′′, 𝑡, 𝑡₁, 𝑡₂, ‖𝑎_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑒_𝑖𝑗‖_𝐿^∞_(Ω), ‖𝑔‖_𝐿^∞_(Ω), ‖𝑎^′′‖_𝐿^∞_(Ω), 𝜂[𝜁_2𝑟0𝑎_𝑖𝑗], 𝜎_∘[(𝑒_𝑖𝑗)_𝑥], ‖𝑎_𝑖‖_𝑀^𝑡_1(Ω), 𝜎_∘[𝑎_𝑖], 𝜎_∘[𝑎^′].
By a well-known lemma of monotonicity of Miranda (see [23, Lemma 3.1]), it follows from (5.11) that‖𝑢‖_𝑊^2,𝑝_{(𝐼1/2(𝑦))}≤𝑐₆(‖𝐿𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}+𝜌₁⁻¹(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}+(𝜌₁⁻¹(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))})^1/2‖𝑢‖_𝑊^1/2_{2,𝑝(𝐼1/2(𝑦))}), (5.12) and then, using Young's inequality, we deduce from (5.12) that‖𝑢‖_𝑊^2,𝑝_{(𝐼1/2(𝑦))}≤𝑐₇(‖𝐿𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}+𝜌₁⁻¹(𝑦)‖𝑢‖_𝐿^𝑝_{(𝐼1(𝑦))}), (5.13)with 𝑐₆,𝑐₇∈𝑅₊ dependent on the same parameters of 𝑐₅.
From (5.13) it follows_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛}(𝑦)‖𝑢‖^𝑝_{𝑊2,𝑝(𝐼1/2(𝑦))}𝑑𝑦≤𝑐₈(_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛}(𝑦)‖𝐿𝑢‖^𝑝_{𝐿𝑝(𝐼1(𝑦))}𝑑𝑦+_𝑅𝑛𝜌₁^{𝑝𝑠−𝑛−𝑝}(𝑦)‖𝑢‖^𝑝_{𝐿𝑝(𝐼1(𝑦))}𝑑𝑦), (5.14) where 𝑐₈∈𝑅₊ depends on the same parameters of 𝑐₇.
If 𝑚≥𝑠−1, since𝐿^𝑝_𝑚(Ω)↪𝐿^𝑝_𝑠−1(Ω), (5.15)from (5.14) and from Lemma 2.1 we have that‖𝑢‖_𝑊𝑠^2,𝑝_(Ω)≤𝑐₉(‖𝐿𝑢‖_𝐿^𝑝_𝑠(Ω)+‖𝑢‖_𝐿^𝑝_𝑠−1(Ω)), (5.16)with 𝑐₉∈𝑅₊ dependent on the same parameters of 𝑐₈ and on 𝑠. Therefore, 𝑢 belongs to 𝑊_𝑠^2,𝑝(Ω).
If 𝑚<𝑠−1, we denote by 𝑘 the positive integer, such that𝑠−𝑚−1≤𝑘<𝑠−𝑚. (5.17)Then, for 𝑖=1,…,𝑘, we have that𝐿^𝑝_𝑠(Ω)↪𝐿^𝑝_𝑚+𝑖(Ω). (5.18)Therefore, using (5.14) and (5.16) with 𝑚+𝑖, 𝑖=1,…,𝑘, instead of 𝑠, we deduce that 𝑢∈𝑊^2,𝑝_𝑚+1(Ω),…,𝑢∈𝑊^2,𝑝_𝑚+𝑘(Ω). On the other hand, we have that𝑊^2,𝑝_𝑚+𝑘(Ω)↪𝐿^𝑝_𝑠−1(Ω) (5.19)and then, since 𝑢∈𝐿^𝑝_𝑠−1(Ω), (5.14) holds. Thus, 𝑢 satisfies (5.16) and then 𝑢∈𝑊_𝑠^2,𝑝(Ω).

Theorem 5.2. If conditions (ℎ₁), (ℎ₂), and (ℎ₃) (with 𝑡₁>𝑛 and 𝑡₂>𝑛/2) hold, and 𝑎≥𝑎₀>0 a.e. in Ω, then the problem𝑢∈𝑊_𝑠^2,𝑝(Ω)∩𝑊^∘_𝑠^1,𝑝(Ω),𝐿𝑢=0, (5.20)admits only the zero solution.

Proof. Fix 𝑢∈𝑊_𝑠^2,𝑝(Ω)∩𝑊^∘_𝑠^1,𝑝(Ω), such that 𝐿𝑢=0. From Lemma 5.1 it follows that 𝑢∈𝑊^2,𝑝(Ω). On the other hand, since 𝑢∈𝑊^1,𝑝(Ω)∩𝑊^∘1,𝑝_loc(Ω), from Lemma 2.2 we have that 𝑢∈𝑊^∘1,𝑝(Ω). Thus, from [13, Theorem 5.2] we deduce that 𝑢=0.