Nonlinear Analysis 70 (2009) 4190–4194

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

A comparison principle for quasilinear operators in unbounded domainsEvgeny GalakhovSteklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street 8, 119991, Moscow, Russia

a r t i c l e i n f o

Article history:Received 8 April 2008Accepted 26 September 2008

MSC:35B0535J60

Keywords:Comparison principleQuasilinear operatorNonlinear capacity

a b s t r a c t

We establish a comparison principle for quasilinear elliptic operators in unboundedcylindrical domains by combining the traditional test function method with nonlinearcapacity techniques. Some applications to monotonicity and symmetry of solutions tononlinear elliptic problems are given.

© 2008 Elsevier Ltd. All rights reserved.

Comparison principles play an important role in the study of solutions to nonlinear elliptic and parabolic differentialequations and inequalities, in particular, for deriving such properties as monotonicity and symmetry, as well as Liouville-type results (see [1–5]). However, most available versions of the comparison principle hold only in bounded domains.Well-known counterexamples show that one cannot extend them to arbitrary unbounded domains, at least for generalnonlinearities. Therefore the problem of finding sufficient conditions for their validity in unbounded domains, both in termsof the geometry of the domain and of the structure of the nonlinearity, is of great interest.To our knowledge, the first result of such a kind in cylindrical domains was obtained by Dancer [4]. It can be formulated

as follows.

Theorem 1. Let N ≥ 1, λ > 0, and define Q = (0, λ)× RN . Then there exists a constant ε(λ) > 0 such that if for a Lipschitzcontinuous function f : Q → R such that

(f (u(x))− f (v(x)))+ ≤ ε(u(x)− v(x))+ in Q (0.1)

one has

−∆u− f (u) ≤ −∆v − f (v) in D(Q ), (0.2)

and, moreover,

u ≤ v on ∂Q = 0, λ × RN , (0.3)

then u ≤ v in Q .

For the proof, a positive function h : (0, λ)× RN → Rwas constructed in such a way that−∆h = εh and h(x)→∞ as|x| → ∞, which can be done for 0 < ε < λ−2π2. If one assumes for a contradiction that Z = x ∈ Q : u(x) < v(x) 6= ∅,this assumption implies that the function (v−u)/h has a nonnegativemaximum in Z , which is impossible. This result easilyextends to more general second-order linear elliptic operators L and domains of the form Q = Ω × RN2 with Ω ⊂ RN1

E-mail address: galakhov@mi.ras.ru.

0362-546X/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.09.008

E. Galakhov / Nonlinear Analysis 70 (2009) 4190–4194 4191

smooth and bounded, N1, N2 ≥ 1 (see [1]). In this case, ε is, in fact, determined by the smallest eigenvalue of the operatorLwith Dirichlet boundary conditions inΩ .However, it is known that such principles do not hold in general for quasilinear operators, such as the p-Laplacian defined

by∆pu := div(|Du|p−2Du), even in boundeddomains. Therefore additional assumptions are required, such as a priori boundsfor Du andDv, and the constant εmust in general depend on these bounds. Under these assumptions, a comparison principlefor the p-Laplacian in bounded domains was established in [2,3].Our goal is to extend the result of [2] to unbounded cylindrical domains. For this purpose, we combine the technique

developed in [2] with the nonlinear capacity method (see [6] and references therein). That is, we use test functions of theform (u − v)α

+ϕR, where ϕR is a mollified ‘‘characteristic’’ function that depends on the parameter R, in order to derive a

priori estimates for integrals of positive functions over the set Z where u > v. These estimates imply that, for R→∞, theintegral asymptotically tends to zero, which is a contradiction unless Z = ∅.The rest of the paper consists of three sections. In Section 1, the assumptions and themain result are formulated. Section 2

is devoted to the proof of this result. In Section 3, some applications are given.

1. Formulation of the problem and auxiliary results

Let Ω ⊂ RN1 be a smooth bounded domain, and define Q = Ω × RN2 . Assume that f : R → R is a locally Lipschitzfunction. Consider functions u, v ∈ C1(Q ) such that

|Du(x)| + |Dv(x)| ≤ M in Q (1.1)

with someM > 0. We shall denote the Lebesgue measure of a set S by |S| and generic positive constants by c1, c2, . . . .Now consider an operator−div A(x,Du)with

A ∈ C0(Q × RN;RN) ∩ C1(Q × RN \ 0;RN), (1.2)

A(x, 0) = 0 ∀x ∈ Q , (1.3)N∑i,j=1

∣∣∣∣∂Aj∂ηi(x, η)

∣∣∣∣ ≤ Γ |η|p−2 ∀x ∈ Q , η ∈ RN \ 0, (1.4)

N∑i,j=1

∂Aj∂ηi

(x, η)ξiξj ≥ γ |η|p−2|ξ |2 ∀x ∈ Q , η ∈ RN \ 0, ξ ∈ RN (1.5)

for suitable constants Γ , γ > 0. A typical example of such an operator is the p-Laplacian where A = A(η) = |η|p−2η.If u, v ∈ W 1,ploc (Q ) ∩ L

∞

loc(Q ) and f ∈ C0(Q × R)we say that (in a weak sense)

− div A(x,Du)+ f (x, u) ≤ −div A(x,Dv)+ f (x, v) (1.6)

if for each nonnegative test function ϕ ∈ W 1,p0 (Q )we have∫Q(A(x,Du(x)) · Dϕ + f (x, u(x))ϕ)dx ≤

∫Q(A(x,Dv(x)) · Dϕ + f (x, v(x))ϕ)dx. (1.7)

For a point x = (x1, ·, xN) ∈ RN , we shall denote the vector consisting of the last N2 coordinates by z(x):

z(x) := (xN1+1, ·, xN) ∈ RN2 .

For S ⊂ RN , we shall denote its sections with respect to the last N2 coordinates by

Sz(x) = y ∈ S : z(y) = z(x).

It is well-known that almost all such sections of a measurable set S are themselves measurable. In the sequel, we take thesupremum of |Sz(x)| over such measurable sections.We shall also use the notationM(S) := supx∈S:u(x)≥v(x)(|Du(x)| + |Dv(x)|), where u and v are C1 functions defined in S.Finally, we shall assume that

u ≤ v on ∂Q = ∂Ω × RN2 . (1.8)

Our aim is to prove the following comparison principle.

4192 E. Galakhov / Nonlinear Analysis 70 (2009) 4190–4194

Theorem 1.1. Let Q be as described, 1 < p < 2 and N1, N2 ≥ 1. Assume that (1.1)–(1.6) and (1.8) hold. Then there existconstants εi(N1,N2, p, |Ω|,M, γ ,Γ ) > 0, i = 1, 2, such that if Q = Q1 ∪ Q2 with Q1, Q2 disjoint, measurable and such that

supx∈Q1:u(x)≥v(x)

(f (u(x))− f (v(x)))+u(x)− v(x)

|Q z(x)1 | ≤ ε1, (1.9)

M(Q2) := supx∈Q2:u(x)≥v(x)

(|Du(x)| + |Dv(x)|) ≤ ε2, (1.10)

then u ≤ v in Q .

Remark 1.1. Conditions (1.2)–(1.5) coincide with assumptions (1-1)–(1-4) in [2] that guarantee the validity of similarcomparison principles in bounded domains.

An important special case of this theorem (with A = ∆p and Q2 = ∅) is

Corollary 1.1. Let Q be as described, N1, N2 ≥ 1, and A = ∆p with 1 < p < 2. Assume that (1.8) holds. Then there exists aconstant ε1(N1,N2, p, |Ω|,M) > 0 such that if

supx∈Q :u(x)≥v(x)

(f (u(x))− f (v(x)))+u(x)− v(x)

|Q z(x)| ≤ ε, (1.11)

then u ≤ v in Q .

In order to prove this theorem, we shall need Lemmas 2-1 and 2-2 from [2]. For the reader’s benefit, we reproduce theirformulation here.

Lemma 1.1. Let assumptions (1.2)–(1.5) hold. Then there exist constants c1, c2, depending on p, γ and Γ , such that for allη, η′ ∈ RN with |η| + |η′| 6= 0 and for all x ∈ Q one has

|A(x, η)− A(x, η′)| ≤ c1(|η| + |η′|)p−2|η − η′|, (1.12)

[A(x, η)− A(x, η′)] · [η − η′] ≥ c2(|η| + |η′|)p−2|η − η′|2, (1.13)

where the dot stands for the scalar product in RN , and consequently, if 1 < p ≤ 2,

|A(x, η)− A(x, η′)| ≤ c1|η − η′|p−1. (1.14)

Lemma 1.2 (A Poincaré-type Inequality). Let Ω be a bounded open set and suppose Ω = A ∪ B, with A, B measurable anddisjoint. If u ∈ W 1,p0 (Ω), 1 < p <∞, then

‖u‖Lp(Ω) ≤ ω−1N

N |Ω|1Np

[|A|

1Np′ ‖Du‖Lp(A) + |B|

1Np′ ‖Du‖Lp(B)

], (1.15)

where p′ = p/(p− 1) and ωN is the measure of the unit sphere in RN .

2. Proof of Theorem 1.1

We introduce a family of test functions ϕR = ξλR ∈ C1(RN2; [0, 1])with λ > 0 large enough and

ξR(x) =1 (|x| ≤ R),0 (|x| ≥ 2R). (2.1)

We also assume that

|DξR(x)| ≤ cR−1 (x ∈ RN2) (2.2)

with c > 0. Now, testing the differential inequality (1.6) with (u− v)α+ϕR and α > 1− 1/p (it is an admissible test function

since (u− v)+ ∈ W1,p0 (Ω) due to (1.8)), we obtain

α

∫Q(|Du|p−2Du− |Dv|p−2Dv,D(u− v)+)(u− v)α−1+ ϕRdx

≤ −

∫Q(|Du|p−2Du− |Dv|p−2Dv,DϕR)(u− v)α+ϕRdx+

∫Q(f (u)− f (v))(u− v)α

+ϕRdx. (2.3)

E. Galakhov / Nonlinear Analysis 70 (2009) 4190–4194 4193

Using inequalities (1.13) and (1.14) from Lemma 1.1 with η = Du, η′ = Dv, from (1.1) and (1.9) we deduce

c1α(Mp−2

∫Q1∩[u≥v]

|Du− Dv|2(u− v)α−1ϕRdx+Mp−2(Q2)∫Q2∩[u≥v]

|Du− Dv|2(u− v)α−1ϕRdx)

≤ c3

∫[u≥v]|Du− Dv|p−1(u− v)α|DϕR|dx+

∫[u≥v]

(f (u)− f (v))(u− v)αϕRdx. (2.4)

In order to estimate the last integral, we make use of (1.9) and of the Poincaré-type inequality (1.15), which is applicablehere due to the fact that u ≤ v on ∂Q and hence (u− v)α

+∈ W 1,p0 (Qz) for each measurable section Qz of Q . This yields∫

[u≥v](f (u)− f (v))(u− v)αϕRdx ≤ ε

∫[u≥v]

(u− v)α+1ϕRdx

≤ c4(α,N1, |Ω|)(ε1

∫Q1∩[u≥v]

|D1(u− v)|2(u− v)α−1ϕRdx+∫Q2∩[u≥v]

|D1(u− v)|2(u− v)α−1ϕRdx), (2.5)

where

D1(u− v) :=(∂(u− v)∂x1

, . . . ,∂(u− v)∂xN1

),

D2(u− v) :=(∂(u− v)∂xN1+1

, . . . ,∂(u− v)∂xN2

)and consequently

|D1(u− v)|2 = |D(u− v)|2 − |D2(u− v)2| ≤ |D(u− v)|2. (2.6)For

ε1 ≤c1αMp−2

2c4and MQ2 ≤ ε2 ≤

(2c4c1α

) 1p−2

,

estimates (2.4)–(2.6) result inc1α2

(Mp−2

∫Q1∩[u≥v]

|Du− Dv|2(u− v)α−1ϕRdx +Mp−2(Q2)∫Q2∩[u≥v]

|Du− Dv|2(u− v)α−1ϕRdx)

≤ c3

∫[u≥v]|Du− Dv|p−1(u− v)α|DϕR|dx. (2.7)

Now we apply to the right-hand side of (2.7) the parametric Young inequality for three factors with exponents

q1 =2p− 1

,

q2 =2(α + 1)

2α − (α − 1)(p− 1), and

q3 =α + 1p− 1

.

We note that all three exponents are larger than 1 due to 1 < p < 2 and α > 1− 1/p > 1− 2/(p− 1). Thus we obtain∫[u≥v]|Du− Dv|p−1(u− v)α|DϕR|dx ≤ δ1

∫[u≥v]|Du− Dv|2(u− v)α−1ϕRdx

+ δ2

∫[u≥v]

(u− v)α+1ϕRdx+ c(δ1, δ2)∫S2R|DϕR|

α+12−p ϕ

1− α+12−pR dx, (2.8)

where SR := x ∈ Q : u(x) ≥ v(x),∑N1+N2i=N1+1

x2i ≤ R2. The second integral on the right-hand side of (2.8) can be estimated

using the Poincaré inequality similarly to (2.5). Thus combining (2.7) and (2.8) with appropriately chosen δ1, δ2 > 0 yields

c1αMp−2

4

∫SR|Du− Dv|2(u− v)α−1dx ≤ c4

∫S2R|DϕR|

α+12−p ϕ

1− α+p2−pR dx. (2.9)

Now take R→∞. Due to (1.5), for α > max1−1/p,N2(2− p)−1 and ϕR = ξλR with λ > (α+1)/(2− p) the right-handside of (2.9) tends to 0. Thus we finally obtain∫

[u≥v]|Du− Dv|2(u− v)α−1dx ≤ 0,

which means u ≤ v in Q . This finishes the proof.

4194 E. Galakhov / Nonlinear Analysis 70 (2009) 4190–4194

Remark 2.1. Restrictions on f , M and/or |Ω| are essential. In fact, the conclusion of the theorem does not hold foru(x1, ·, xN) = u1(x1, ·, xN1)with 1 ≤ N1 < N , u1 being a positive solution of the Dirichlet problem

−∆pu1 = uq1 (x ∈ Ω ⊂ RN1),

u1 = 0 (x ∈ ∂Ω), (2.10)

which is known to exist for any smooth boundedΩ and 1 < q < NpN−p − 1, and v ≡ 0 in Q = Ω × RN−N1 .

3. Some applications

As an application of our results, we prove monotonicity of small solutions to the problem−∆pu = uq in T = (0, h)× RN−1,

u ≥ 0 in T ,u(0, x′) = 0 on RN−1

(3.1)

in the interval [0, h/2], where h > 0 and x′ := (x2, . . . , xN).

Corollary 3.1. Let 1 < p < 2 and q > 1. Then there exists a constant ε0 > 0 such that for each bounded weak solution ofproblem (3.1) with ‖u‖C1(T ) < ε0 there holds

u(x1, x′) ≤ u(y1, x′) ∀x1, y1 : 0 ≤ x1 ≤ y1 ≤ h/2, x′ ∈ RN−1. (3.2)

Proof. It suffices to use the method of moving planes similarly to [2–4] by applying Corollary 1.1 to u and v = uλ, whereλ ∈ (0, h/2) and uλ(x1, x′) = u(2λ− x1, x′) for each x′ ∈ RN−1. One should take into account that in our case the Lipschitzconstant for u does not exceed

q‖u‖q−1L∞(T ) ≤ qεq−10 ,

and

M = supx∈T(|Du(x)| + |Dv(x)|) ≤ cε0.

Moreover, uλ = u for x1 = λ by construction and uλ ≥ u = 0 for x1 = 0 due to the very formulation of problem (3.1). SoCorollary 1.1 with ε1 = qε

q−p+10 does apply to these functions in Q = (0, λ)× RN−1, and we get

u(x1, x′) ≤ u(2λ− x1, x′) ∀0 ≤ x1 ≤ λ ≤ h/2, x′ ∈ RN−1,

which is exactly (3.2) for λ = (x1 + y1)/2.

Remark 3.1. Evidently, for u such that u(0, x′) = u(h, x′) = 0 for all x′ ∈ RN−1, (3.2) implies symmetry with respect to theplane x1 = h/2.

Acknowledgement

This research was supported by the RFBR grant 05-01-00370.

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