On a Liouville phenomenon for entire weak supersolutions of elliptic partial differential equations

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On a Liouville phenomenon for entire weak supersolutions of elliptic partial differential equations

Autour d’un phénomène de Liouville pour les sursolutions entières faibles d’équations aux derivées partielles elliptiques

Vasilii V. Kurta

American Mathematical Society (Mathematical Reviews), 416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA Received 25 October 2004; received in revised form 20 July 2005; accepted 6 December 2005

Available online 7 July 2006 In fond memory of Professor Heinz Bauer

Abstract

We study a new Liouville-type phenomenon for entire weak supersolutions of elliptic partial differential equations of the form A(u)=0 onRⁿ,n2. Typical examples of the operatorA(u)are thep-Laplacian forp >1, the mean curvature operator, and their well-known modifications.

Résumé

Ce travail est consacré à l’étude d’un nouveau phénomène de type de Liouville pour les sursolutions entières faibles d’équa- tions aux derivées partielles elliptiques de la formeA(u)=0 surRⁿ,n2. Des exemples typiques de l’opérateurA(u)sont le p-laplacien pourp >1, l’opérateur de courbure moyenne, et leurs modifications bien connues.

MSC:35J60; 31C45; 31B05; 35R45

Keywords:Elliptic; Entire; Liouville; Partial differential equation; Quasilinear; Supersolution; Weak

1. Introduction

Liouville’s well-known theorem says that any superharmonic function onR²bounded below by a constant is itself a constant. On the other hand it is also well known that forn3 there exist non-constant superharmonic functions on Rⁿ bounded below by a constant. The purpose of this work is to determine forn3 the ‘sharp distance at infinity’

between the non-constant superharmonic functions onRⁿ bounded below by a constant and this constant itself in

E-mail address:[email protected] (V.V. Kurta).

doi:10.1016/j.anihpc.2005.12.001

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the form of a theorem of Liouville type and to characterize basic properties of quasilinear elliptic partial differential operators which make it possible to obtain such a theorem for supersolutions of quasilinear elliptic partial differential equations of the form

A(u)=0 (1)

onRⁿ,n2. Typical examples of the operatorA(u)are thep-Laplacian _p(u):=div

|∇u|^p⁻²∇u

, p >1, (2)

its well-known modification (see, e.g., [8, p. 155]) ˜p(u):=

n

i=1

∂

∂x_i ∂u

∂x_i

^p⁻²∂u

∂x_i

, p >1, (3)

the mean curvature operator Ξ (u):=div

∇u 1+ |∇u|²

, (4)

and its well-known modifications.

Note that a Liouville theorem for solutions of linear uniformly elliptic second-order partial differential equations on Rⁿ,n >2, was first obtained, as a direct consequence of a Harnack inequality, in [1] under some continuity assumptions on the coefficients of the equations and in [12] without continuity assumptions on the coefficients of the equations. In the case of quasilinear uniformly elliptic second-order partial differential equations onRⁿ,n2, a Liouville theorem is a direct consequence of a Harnack inequality first obtained in [14]. Note also that a Liouville theorem for mappings ofRⁿ,n >2, with bounded distortion was first obtained in [13] by using the same Harnack inequality from [14]. Finally, in the case of linear uniformly elliptic second-order partial differential equations onR², a Liouville theorem is a direct consequence of a Harnack inequality first obtained in [7].

2. Definitions

LetA(u)be a differential operator defined formally by A(u)=

n

i=1

d dxi

A_i(x, u,∇u). (5)

Here and in what follows, n 2. We assume that the functions A_i(x, η, ξ ), i =1, . . . , n, satisfy the usual Carathéodory conditions onRⁿ×R¹×Rⁿ; namely, they are continuous inηandξ for almost allx∈Rⁿand measurable inxfor anyη∈R¹andξ∈Rⁿ.

Definition 1.Letα >1 be a given number. The operatorA(u)given by (5) belongs to the classA(α)if for allη∈R¹, allξ, ψ∈Rⁿ, and almost allx∈Rⁿthe following two inequalities hold:

0 n

i=1

ξ_iA_i(x, η, ξ ), (6)

with equality only ifξ=0, and

n

i=1

ψiAi(x, η, ξ )

α

K|ψ|^α n

i=1

ξiAi(x, η, ξ ) α−1

, (7)

withKa certain positive constant.

It is easy to see that condition (7) is fulfilled whenever the inequality n

i=1

A²_i(x, η, ξ ) α/2

K

n

i=1

ξiAi(x, η, ξ ) α−1

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(3)

holds for all η∈R¹, all ξ, ψ ∈Rⁿ, and almost all x∈Rⁿ. Hence, the operatorA(u) given by (5) and satisfying conditions (6) and (8) belongs to the classA(α).

Remark 1.Conditions (7) and (8) on the behavior of the coefficients of partial differential operators were introduced in [10].

It is not difficult to verify that for any givenp >1 the differential operators (2) and (3) as well as the differential operatorA(u)given by (5) and satisfying the well-known growth conditions

n

i=1

A²_i(x, η, ξ ) 1/2

K1|ξ|^p⁻¹ (9)

and

|ξ|^pK2

n

i=1

ξ_iA_i(x, η, ξ ), (10)

withK1,K2positive constants, belong to the classA(α)withα=p.

It is also easy to see that linear divergent elliptic partial differential operators of the form L:=

n

i,j=1

∂

∂x_j

a_ij(x) ∂

∂x_i

(11) witha_ij(x)measurable bounded coefficients and with the (possibly non-uniformly) positive-definite quadratic form

n

i,j=1

a_ij(x)ξ_iξ_j (12)

belong to the classA(α)withα=2 but do not satisfy condition (10) for any fixedp >1.

In connection with this we give another example of an operator that belongs to the class A(α) with a certain α >1 but does not satisfy condition (10). Leta(x, η, ξ )be a positive bounded function that satisfies the Carathéodory conditions onRⁿ×R¹×Rⁿ.It is easy to see that for a givenp >1 the weightedp-Laplacian

¯_p(u):=div

a(x, u,∇u)|∇u|^p⁻²∇u

(13) belongs to the classA(α)withα=pbut does not satisfy condition (10) for any fixedp >1 if the functiona(x, η, ξ ) is only assumed to be positive.

It can happen that an operatorA(u) given by (5) belongs simultaneously to several different classesA(α). For example, the mean curvature operatorΞ (u)given by (4) belongs to the classesA(α)for all 1< α2; similarly its modification forp2,

Ξp(u):=div

|∇u|^p⁻²∇u 1+ |∇u|²

, (14)

belongs to the classesA(α)for allα∈(p−1, p]andp2. Obviously, operators given by (4) and (14) do not satisfy conditions (9)–(10) for any fixedp1.

Definition 2.Letα >1 be a given number, and let the operatorA(u)given by (5) belong to the classA(α). A measurable functionu:Rⁿ→R¹is called an entire weak supersolution of Eq. (1) onRⁿ ifu∈L¹_loc(Rⁿ),|∇u| ∈L^α_loc(Rⁿ), and the integral inequality

Rⁿ

n

i=1

ϕ_x_iA_i(x, u,∇u)dx0 (15)

holds for every non-negative functionϕ∈W^1,α(Rⁿ)with compact support.

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3. Results

Theorem 1.Letn2andα >1be given numbers such thatαn. Let the operatorA(u)given by(5)belong to the classA(α), and letu(x)be an entire weak supersolution of (1)onRⁿbounded below by a constant. Thenu(x)is a constant onRⁿ.

Theorem 2.Letn2andα >1be given numbers such thatn > α. Let the operatorA(u)given by(5)belong to the classA(α), and letu(x)be an entire weak supersolution of (1)onRⁿbounded below by a constantcand such that u∈L^∞_loc(Rⁿ). Then eitheru(x)=conRⁿor the relation

lim inf

r→+∞

sup

r|x|2r

u(x)−c

r^α−ⁿ⁻1^α−ν = +∞ (16)

holds with any fixedν∈(0, α−1).

Theorem 3.Letn2andα >1be given numbers such thatn > α. Let the operatorA(u)given by(5)belong to the classA(α), and letu(x)be an entire weak supersolution of (1)onRⁿbounded below by a constant c. Then either u(x)=conRⁿor the relation

lim inf

r→+∞r⁻^α

r|x|2r

u(x)−cα−1−ν

dx= +∞ (17)

holds with any fixedν∈(0, α−1).

Due to the arbitrariness of the constantcin Theorems 2 and 3, the statements of these theorems can be reformulated in a slightly different form.

Theorem 2.Letn2andα >1be given numbers such thatn > α. Let the operatorA(u)given by(5)belong to the classA(α), and letu(x)be an entire weak supersolution of (1) onRⁿ bounded below by a constant and such thatu∈L^∞_loc(Rⁿ). Then eitheru(x)is a constant onRⁿor relation(16)holds with any fixed real numbercsuch that u(x)conRⁿand any fixedν∈(0, α−1).

Theorem 3.Letn2andα >1be given numbers such thatn > α. Let the operatorA(u)given by(5)belong to the classA(α), and letu(x)be an entire weak supersolution of (1)onRⁿbounded below by a constant. Then either u(x)is a constant onRⁿor relation(17)holds with any fixed real numbercsuch thatu(x)conRⁿand any fixed ν∈(0, α−1).

Remark 2.It is important to note that for any givenn2 andα >1 such thatn > αthe function u(x)=

1+ |x|^α⁻^α¹^α−n_α

(18) is an entire weak supersolution of the equation

_p(u)=0 (19)

withp=αthat is bounded below and is such that relations (16) and (17) hold with any fixedν∈(0, α−1)and, at the same time, the relations

r→+∞lim

sup

r|x|2r

u(x)−0

rⁿ^α⁻−^α1 =C₁ (20)

and

r→+∞lim r⁻^α

r|x|2r

u(x)−0α−1

dx=C₂, (21)

withC₁,C₂certain positive constants, also hold.

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Remark 3.The results of this work were announced in [5]. To prove these results we further develop an approach that was proposed for solving similar problems in [6].

Remark 4.The results of Theorem 1 are new only forα=n. Similar results to those of Theorem 1 for entire weak continuous supersolutions of (1) onRⁿ forα=nwere first obtained in [11]. Forα > n, the results of Theorem 1 for entire weak supersolutions of (1) on Rⁿ, which in this case are continuous onRⁿ by the well-known Sobolev imbedding theory, were also first obtained in [11]. Here, we give a new proof of these results from [11] by developing an approach from [6] which does not explicitly use the continuity of entire weak supersolutions of (1) onRⁿ. Remark 5.In the case whenα=p andA(u)=_p(u), Theorem 1 coincides with well-known Liouville-type theorems for entire superharmonic andp-superharmonic functions locally bounded onRⁿ (see, e.g., [2, p. 68] and [3, p. 179]). Also, in this case, the results of Theorems 2 and 3 correlate well with certain results in the theory of entire superharmonic andp-superharmonic functions (see, e.g., [2, pp. 131, 139] and [3, pp. 133, 135]).

4. Proofs

Proof of Theorem 2. The statement of Theorem 2 follows immediately from Theorem 3. In fact, let n2 and α >1 be given numbers such thatn > α. Let the operatorA(u)given by (5) belong to the classA(α), and letu(x) be an entire weak supersolution of (1) on Rⁿ bounded below by a constantc, i.e.,u(x)conRⁿ, and such that u∈L^∞_loc(Rⁿ). Hence, by Theorem 3, either u(x)=con Rⁿ or relation (17) holds with any fixed ν∈(0, α−1).

Further, via the trivial inequality r⁻^α

r|x|2r

u(x)−cα−1−ν

dxr⁻^α

sup

r|x|2r

u(x)−cα−1−ν

r|x|2r

dx, (22)

which obviously holds for anyr >0, it follows from (17) that lim inf

r→+∞

sup

r|x|2r

u(x)−cα−1−ν

rⁿ⁻^α= +∞. (23)

Then, since sup

r|x|2r

u(x)−cα−1−ν

sup

r|x|2r

u(x)−c^α⁻¹⁻^ν

(24)

and

sup

r|x|2r

u(x)−c^α⁻¹⁻^ν

rⁿ⁻^α= sup

r|x|2r

u(x)−c r^α^n−α−1−ν

α−1−ν

, (25)

the validity of (16) follows immediately from that of (23) and (25). 2

In what follows, a ‘smooth’ function is aC^∞-function onRⁿ,B(r)is an open ball onRⁿof radiusr >0 centered at the origin, andB(r)is the closure ofB(r).

Proof of Theorem 3. Letn2 andα >1 be given numbers such thatn > α. Let the operatorA(u)given by (5) belong to the classA(α), and letu(x)be an entire weak supersolution of (1) onRⁿbounded below by a constantc, i.e.,u(x)conRⁿ. Letrandεbe positive numbers, and letζ:Rⁿ→ [0,1]be a smooth function which equals 1 on B(r)and 0 outsideB(2r). Substituting, without loss of generality,ϕ(x)=(u(x)−c+ε)⁻^νζ^α(x)as a test function in inequality (15), whereν∈(0, α−1)is arbitrary, and integrating by parts, we find

α

B(2r)\B(r)

n

i=1

ζ_x_iA_i(x, u,∇u)(u−c+ε)⁻^νζ^α⁻¹dx

ν

B(2r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹ζ^αdx. (26)

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Estimating the left-hand side of (26) by using condition (7) on the coefficients of the operatorA(u), we have αK^1/α

B(2r)\B(r)

n

i=1

u_x_iA_i(x, u,∇u)

(α−1)/α

|∇ζ|(u−c+ε)⁻^νζ^α⁻¹dx

α

B(2r)\B(r)

n

i=1

. (27) Further, estimating the left-hand side of (27) by Hölder’s inequality, we arrive at

αK^1/α

B(2r)\B(r)

|∇ζ|^α(u−c+ε)^α⁻¹⁻^νdx 1/α

×

B(2r)\B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹ζ^αdx

(α−1)/α

α

B(2r)\B(r)

n

i=1

ζx_iAi(x, u,∇u)(u−c+ε)⁻^νζ^α⁻¹dx

. (28) In turn, (26) and (28) imply the inequality

αK^1/α

B(2r)\B(r)

|∇ζ|^α(u−c+ε)^α⁻¹⁻^νdx 1/α

×

B(2r)\B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹ζ^αdx

(α−1)/α

ν

B(2r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹ζ^αdx (29)

and, therefore, the inequality α^αK

B(2r)\B(r)

|∇ζ|^α(u−c+ε)^α⁻¹⁻^νdxν^α

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx. (30) It is easy to see that the right-hand side of (30) increases monotonically ifε >0 decreases strongly monotonically to zero. Therefore, it follows from (30) that the inequality

α^αK

B(2r)\B(r)

|∇ζ|^α(u−c+ε)⁻^ν⁺^α⁻¹dxν^α

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+δ)⁻^ν⁻¹dx (31) holds with anyδ >0 and anyε∈(0, δ]. Since for any sequenceε_k>0 monotonically decreasing to zero ask→ +∞

the sequence of functions

Φ_k(x):= |∇ζ|^α(u−c+ε_k)^α⁻¹⁻^ν (32) measurable onRⁿconverges a.e. onRⁿto the function

Φ(x):= |∇ζ|^α(u−c)^α⁻¹⁻^ν (33)

measurable onRⁿ, since for sufficiently largek

Φ_k(x)|∇ζ|^α(u−c+1)^α⁻¹⁻^ν (34)

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onRⁿ, and since the function

|∇ζ|^α(u−c+1)^α⁻¹⁻^ν (35)

is locally integrable onRⁿ, then, by Lebesgue’s theorem (see, e.g., [4, p. 303]), forε=ε_k>0 monotonically decreasing to zero we can pass to the limit ask→ +∞on the left-hand side of (31). As a result, we obtain the inequality

α^αK

B(2r)\B(r)

|∇ζ|^α(u−c)^α⁻¹⁻^νdxν^α

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+δ)⁻^ν⁻¹dx, (36) which holds with any δ >0. Then, for any r >0 and any sequence ε_k >0 monotonically decreasing to zero as k→ +∞, it follows from (36), by lettingδ=ε_kand

Ψ_k(x):=

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε_k)⁻^ν⁻¹, (37)

that the sequence of integrals

B(r)

Ψ_k(x)dx (38)

is bounded above by the positive constant c₁=K

α ν

α

B(2r)\B(r)

|∇ζ|^α(u−c)^α⁻¹⁻^νdx, (39)

which does not depend onε_k. Hence, since

Ψ₁(x)Ψ₂(x)· · ·Ψ_k(x)· · · (40)

onRⁿ, then by Beppo Levi’s theorem (see, e.g., [4, p. 305]), for anyr >0 there exists a function Θr :B(r)→R¹ integrable onB(r)and such that the sequence of functionsΨk(x)converges a.e. toΘr(x)onB(r)and

k→+∞lim

B(r)

Ψ_k(x)dx=

B(r)

Θ_r(x)dx. (41)

Further, it is easy to see that the family of functions{Θ_r}r>0uniquely determines a functionΨ:Rⁿ→R¹which is non-negative, measurable, locally integrable onRⁿand is such thatΨ (x)=Θ_r(x)onB(r)for allr >0. Therefore, the sequence of functionsΨk(x)given by (37) converges a.e. toΨ (x)onRⁿfor any sequenceεk>0 monotonically decreasing to zero ask→ +∞. Then, by choosingδ=εkin (36), where the sequenceεk>0 converges monotonically to zero ask→ +∞, and passing to the limit on the right-hand side of (36), we find, due to (41), the inequality

α^αK

B(2r)\B(r)

|∇ζ|^α(u−c)^α⁻¹⁻^νdxν^α

B(r)

Ψ (x)dx. (42)

We divide the rest of the proof into three cases according to the behavior of the right-hand side of (42), which can monotonically approach zero,+∞, or some positive numberI asrstrongly monotonically approaches+∞.

If the right-hand side of (42) approaches zero asr→ +∞, then, due to the non-negativity of the functionΨ (x), we have thatΨ (x)=0 onRⁿ. Further, since by (37) and (40) the inequality

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε_k)⁻^ν⁻¹Ψ (x) (43)

holds onRⁿ for any sequenceε_k>0 monotonically decreasing to zero as k→ +∞, then, again, due to the non- negativity of the left-hand side of (43), we obtain that

n

i=1

ux_iAi(x, u,∇u)(u−c+εk)⁻^ν⁻¹=0 (44)

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onRⁿ. Hence, by condition (6) on the coefficients of the operatorA(u), the supersolutionu(x)=const.onRⁿ, and, therefore, eitheru(x)=conRⁿor relation (17) holds with any fixedν∈(0, α−1).

If the right-hand side of (42) approaches+∞asr→ +∞, then, due to monotonicity, (42) yields that lim inf

r→+∞

B(2r)\B(r)

|∇ζ|^α(u−c)^α⁻¹⁻^νdx= +∞. (45)

Finally, if the right-hand side of (42) monotonically approaches a certain positive numberI asrapproaches+∞, i.e.,

r→+∞lim ν^α

B(r)

Ψ (x)dx=I >0, (46)

we again consider inequality (29), just noting here that, due to monotonicity,

B(2r_k)\B(rk)

Ψ (x)dx→0 (47)

for any sequencer_k>0 such thatr_k→ +∞. First, we have from (29) the inequality αK^1/α

B(2r)\B(r)

|∇ζ|^α(u−c+ε)^α⁻¹⁻^νdx 1/α

×

B(2r)\B(r)

n

i=1

ux_iAi(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx

(α−1)/α

ν

B(r)

n

i=1

ux_iAi(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx. (48)

In (48), letε=ε_k >0 converge monotonically to zero as k→ +∞. Then, by Lebesgue’s theorem (see, e.g., [4, p. 303]), we can pass to the limit on both sides of (48). Namely, we know from the above that for any sequenceε_k>0 monotonically decreasing to zero ask→ +∞the sequences of functionsΦ_k(x)andΨ_k(x)measurable and locally integrable onRⁿ and given, respectively, by (32) and (37), converge a.e. onRⁿ, respectively, to the functionsΦ(x) andΨ (x)measurable and locally integrable onRⁿ. Further, arguing as above and lettingε=εk>0 monotonically decrease to zero ask→ +∞, by Lebesgue’s theorem (see, e.g., [4, p. 303]) we can pass to the limit on both sides of (48). As a result, we arrive at the inequality

αK^1/α

B(2r)\B(r)

|∇ζ|^α(u−c)^α⁻¹⁻^νdx

1/α

B(2r)\B(r)

Ψ (x)dx

(α−1)/α

ν

B(r)

Ψ (x)dx. (49)

In (49), forr=r_k>0 monotonically increasing to+∞, by passing to the limit asr_k→ +∞, we obtain from (46), (47), and (49) that

r_k→+∞lim

B(2r_k)\B(r_k)

|∇ζ|^α(u−c)^α⁻¹⁻^νdx= +∞. (50)

Thus, due to the arbitrariness in the choice of the sequencer_k in (50), we again arrive at relation (45).

Now, without loss of generality, we choose in (45) the function ζ (x) in the formζ (x)=ψ (|x|/(2r)), where ψ:[0,+∞)→ [0,1]is a smooth function that equals 1 on[0,1/2]and 0 on[1,+∞)and is such that the inequality

|∇ζ|c₂r⁻¹ (51)

holds onRⁿ with a certain positive constantc₂for an arbitraryr >0. Relation (17) then follows immediately from (45) and (51). 2

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Proof of Theorem 1. Letn2 andα >1 be given numbers such that αn. Let the operatorA(u)given by (5) belong to the classA(α), and letu(x)be an entire weak supersolution of (1) onRⁿbounded below by a constantc, i.e.,u(x)conRⁿ. Letr,R, andεbe positive numbers such thatR > r, and letζ:Rⁿ→ [0,1]be a smooth function which equals 1 onB(r)and 0 outsideB(R). Substituting, without loss of generality,ϕ(x)=(u(x)−c+ε)⁻^νζ^α(x) as a test function in inequality (15), whereν > α−1 is an arbitrary positive number, and integrating by parts, we have the inequality

α

B(R)\B(r)

n

i=1

ν

B(R)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹ζ^αdx. (52)

Further, we repeat the proof of Theorem 3 word for word from (26) to (30). As a result, we arrive at the inequality α^αK

B(R)\B(r)

|∇ζ|^α(u−c+ε)^α⁻¹⁻^νdxν^α

B(r)

n

i=1

ux_iAi(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx. (53) It follows immediately from (53) that the inequality

α^αε^α⁻¹⁻^νK

B(R)\B(r)

|∇ζ|^αdxν^α

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx (54) holds with any fixedε >0 andν > α−1.

Now, first let α > n. In (54), choosing R =2r and the function ζ (x) in the form ζ (x)=ψ (|x|/R), where ψ:[0,+∞)→ [0,1]is a smooth function that equals 1 on[0,1/2]and 0 on[1,+∞)and is such that the inequality (51) holds onRⁿwith a certain positive constantc2for an arbitraryR >0, we obtain from (51) and (54) the inequality

c3rⁿ⁻^α

B(r)

n

i=1

ux_iAi(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx, (55)

which holds with a certain positive constantc₃that does not depend onr. Passing to the limit asr→ +∞in (55), we find, due to the non-negativity of the integrand, that the equality

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹=0 (56)

holds onRⁿ, and, therefore, by condition (6) on the coefficients of the operatorA(u), thatu(x)=const.onRⁿ. Ifα=n, we choose in (54) the function ζ (x)in the form ζ (x)=ψ (^ln(_ln(R/r)^|^x^|^/r))with arbitrary R > r >1, where ψ: [−∞,+∞)→ [0,1]is a smooth function which equals 1 on[−∞,0]and 0 on[1,+∞). It is not difficult to understand (see, e.g., [9, p. 12]) that the inequality

∇ζ (x) c4

|x|ln(R/r) (57)

holds onRⁿ with a certain positive constantc₄ for arbitraryR > r >1. It then follows from (54) and (57) that the inequality

c₅

B(R)\B(r)

|x|ln(R/r)₋n

dx

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx (58) holds, and, therefore, so does the inequality

c₆

ln(R/r)₋n+1

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx (59)

(10)

with arbitraryR > r >1 and certain positive constantsc5andc6that do not depend on R. Passing to the limit as R→ +∞in (59), we find that the equality

B(r)

n

i=1

u_x_iA_i(x, u,∇u)(u−c+ε)⁻^ν⁻¹dx=0 (60)

holds with an arbitraryr >1. Passing to the limit asr→ +∞in (60), we again obtain, due to the non-negativity of the integrand in (60) and by condition (6), thatu(x)=const.onRⁿ. 2

Acknowledgements

The author is grateful to the referee for useful comments and suggestions for improving the presentation of the results.

References

[1] D. Gilbarg, J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4 (1955/56) 309–

340.

[2] W.K. Hayman, P.B. Kennedy, Subharmonic Functions, Academic Press, 1976 (284 p.).

[3] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, 1993 (363 p.).

[4] A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970 (403 p.).

[5] V.V. Kurta, About a Liouville phenomenon, C. R. Math. Acad. Sci. Paris, Ser. I 338 (2004) 19–22.

[6] V.V. Kurta, Some problems of qualitative theory for nonlinear second-order equations, Doctoral Dissert., Steklov Math. Inst., Moscow, 1994 (323 p.).

[7] L. Lichtenstein, Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Unendliche Folgen positiver Lösungen, Rend. Circ. Mat. Palermo 33 (1912) 201–211.

[8] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969 (554 p.).

[9] V.G. Maz’ya, T.O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston, MA, 1985 (Advanced Publishing Program, 344 p.).

[10] V.M. Miklyukov, Capacity and a generalized maximum principle for quasilinear equations of elliptic type, Dokl. Akad. Nauk SSSR 250 (1980) 1318–1320.

[11] V.M. Miklyukov, Asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion, Mat.

Sb. 111 (153) (1980) 42–66.

[12] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961) 577–591.

[13] Yu.G. Reshetnyak, Mappings with bounded distortion as extremals of integrals of Dirichlet type, Sibirsk. Mat. Zh. 9 (1968) 652–666.

[14] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964) 247–302.