Existence and Nonexistence of Positive Solutions for Singular p-Laplacian Equation in

Volume 2010, Article ID 607453,17pages doi:10.1155/2010/607453

Research Article

Existence and Nonexistence of Positive Solutions for Singular p-Laplacian Equation in

Caisheng Chen, Zhenqi Wang, and Fengping Wang

Department of Mathematics, Hohai University, Nanjing, Jiangsu 210098, China

Correspondence should be addressed to Caisheng Chen,[email protected] Received 15 August 2010; Accepted 10 December 2010

Academic Editor: Zhitao Zhang

Copyrightq2010 Caisheng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence and nonexistence of solutions for the singular quasilinear problem

−div|x|^−ap|∇u|^p−2∇u hxfuλHxgu,x∈^ÊN,ux>0,x∈^ÊN, lim|x| → ∞ux 0, where 1< p < N, 0≤a <N−p/pandfuandgubehave likeu^manduⁿwith 0< m≤p−1< nat the origin. We obtain the existence by the upper and lower solution method and the nonexistence by the test function method.

1. Introduction

In this paper, we study through the upper and lower solution method and the test function method the existence and nonexistence of solution to the singular quasilinear elliptic problem

−div

|x|^−ap|∇u|^p−2∇u

hxfu λHxgu, x∈^ÊN, ux>0, x∈^ÊN, lim

|x| → ∞ux 0 1.1

with 1< p < N, 0 ≤a <N−p/p,λ ≥0.hx, Hx:^ÊN → 0,∞are the locally H ¨older continuous functions, not identically zero andfuandguare locally Lipschitz continuous functions.

The study of this type of equation in1.1is motivated by its various applications, for instance, in fluid mechanics, in Newtonian fluids, in flow through porous media, and in glaciology; see1. The equation in1.1involves singularities not only in the nonlinearities but also in the differential operator.

Many authors studied this kind of problem for the case a 0; see 2–7. In these works, the nonlinearities have sublinear and suplinear growth at infinity, and they behave like a functionu^k k < p−1, ork ≥ p−1at the origin. Roughly speaking, in this case we say that the nonlinearities are concave and convex or “slow diffusion and fast diffusion”;

see8.

Whena 0,fu u^m, andgu uⁿ,m < p−1 < n, by using the lower and upper solution method, Santos in5finds a real numberλ₀ >0, such that the problem1.1has at least one solution if 0≤λ < λ₀.

Fora, λ /0, the existence and multiplicity of solution of singular elliptic equation like 1.1in a bounded domainΩwith the zero Dirichlet data have been widely studied by many authors, for example, the authors9–13and references therein. Assunc¸˜ao et al. in14studied the multiplicity of solution for the singular equations in1.1withhx α|x|^−bm,Hx β|x|^−dq, fu |u|^m−2u, and gu |u|^q−2u in ^ÊN. Similar consideration can be found in 15–20and references therein. We note that the variation method is widely used in the above references.

Recently, Chen et al. in21,22, by using a variational approach, got some existence of solution for1.1withλ 0 andfu u^q,q > p−1. For the caseq < p−1, λ ≥ 0, the problem for the existence of solution for1.1is still open. It seems difficult to consider the caseq < p−1 by variational method.

The main aim of this work is to study the existence and nonexistence of solution for 1.1, wherefuis sublinear andguis suplinear. We will use the upper and lower solution method. To the best of our knowledge, there is little information on upper and lower solution method for the problem1.1. So it is necessary to establish this technique in unbounded domain. To obtain the existence, the assumptionM_∞ <∞see2.17belowis essential. By this, an upper solution for1.1is obtained.

We also obtain a sufficient condition onhx,Hxto guarantee the nonexistence of nontrivial solution for the problem2.21.seeTheorem 2.5below. It must be particularly pointed out that our primary interest is in the mixed case in which 0< m ≤ p−1 < nwith Hxsatisfying

H_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1Htdt _1/p−1

ds <∞, Ht max

|x|tHx, 1.2

whilehxsatisfies

h_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1htdt _1/p−1

ds∞, ht max

|x|thx. 1.3

This paper is organized as follows. InSection 2, we state the main results and present some preliminaries which will be used in what follows. We also introduce the precise hypotheses under which our problem is studied. InSection 3, we give the proof of some lemmas and the existence. The proof of nonexistence is given inSection 4.

2. Preliminaries and Main Results

Let us now introduce some weighted Sobolev spaces and their norms. LetΩbe a bounded domain in^ÊN with smooth boundary∂Ω. Ifr ∈^Ê1 andp≥1, we defineL^pΩ,|x|^−ras being the subspace ofL^pΩof the Lebesgue measurable functionu:Ω → ^Ê1, satisfying

up,r:uL^pΩ,|x|^−r

Ω|x|^−r|u|^pdx 1/p

<∞. 2.1

If 1< p < Nand−∞< a <N−p/p, we defineW^1,pΩ,|x|^−ap resp.,W₀^1,pΩ,|x|^−ap as being the closure ofC^∞Ω resp.,C^∞₀Ωwith respect to the norm defined by

u:

Ω|x|^−ap|∇u|^pdx _1/p

. 2.2

For the weighted Sobolev space W^1,pΩ,|x|^−ap, we have the following compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact theorem.

Theorem 2.1compact imbedding theorem 13. Suppose thatΩ ⊂ ^ÊN is an open bounded domain withC¹ boundary and 0 ∈ Ω, 1 < p < N,−∞ < a < N−p/p. Then, the imbedding W^1,pΩ,|x|^−ap→L^qΩ,|x|^−ris compact if 1≤q < Np/N−p,r <1aqN1−q/p.

We now consider the existence of positive solutions for problem1.1. Our main tool will be the upper and lower solution method. This method, in the bounded domain situation, has been used by many authors, for instance,10,12,13. But for the unbounded domain, we need to establish this method and then to construct an upper solution and a lower solution for1.1. We now give the definitions of upper and lower solutions.

Definition 2.2see10,12. A functionu ∈ W^1,pÊN,|x|^−ap∩L^∞ÊNis said to be a weak lower solution of the equation

−div

|x|^−ap|∇u|^p−2∇u

Fx, u, x∈^ÊN 2.3

if

−div

|x|^−ap∇u^p−2∇u

≤F

x, u , x∈^ÊN 2.4

or

Ê

N|x|^−ap∇u^p−2∇u∇φdx≤

Ê

N

F

x, u φdx, 2.5

for anyφ∈C¹₀^ÊN,φ≥0.

Similarly, a functionu∈W^1,pÊN,|x|^−ap∩L^∞ÊNis said to be a weak upper solution of2.3if

−div

|x|^−ap|∇u|^p−2∇u

≥Fx, u, x∈^ÊN 2.6

or

ÊN

|x|^−ap|∇u|^p−2∇u∇φdx≥

ÊN

Fx, uφdx, 2.7

for anyφ∈C¹₀^ÊNandφ≥0 in^ÊN.

A functionu∈W^1,pÊN,|x|^−ap∩L^∞ÊNis said to be a weak solution of2.3if and only ifuis a weak lower solution and weak upper solution of2.3.

A function v ∈ W^1,pΩ,|x|^−ap ∩ L^∞Ω is said to be less than or equal to w ∈ W^1,pΩ,|x|^−ap∩L^∞Ωon∂Ωif max{0, v−w} ∈W₀^1,pΩ,|x|^−ap.

If 1 < p < N and −∞ < a < N −p/p, we define the weighted Sobolev space W^1,pÊN,|x|^−apas being the closure ofC^∞₀ ^ÊNwith respect to the norm · defined by

u

ÊN

|x|^−ap|∇u|^pdx _1/p

. 2.8

The following lemma will be basic in our approach.

Lemma 2.3. Let Fx, u be Lipschitz continuous and nondecreasing in u and locally H¨older continuous inx. Moreover, assume that there exist the functionsu, u∈W^1,pÊN,|x|^−ap∩L^∞ÊN such that

−div

|x|^−ap∇u^p−2∇u

≤F

x, u , x∈^ÊN,

−div

|x|^−ap|∇u|^p−2∇u

≥Fx, u, x∈^ÊN, ux≤ux, a.e.in ^ÊN.

2.9

Then, there exist a minimal weak solutionV0xand a maximal weak solutionU0xof 2.3 satisfying

ux≤V0x≤U0x≤ux, x∈^ÊN 2.10

andV0x, U0x∈W^1,pÊN,|x|^−ap∩L^∞ÊN.

Proof. Denote Bk {x ∈ ^ÊN | |x| < k},k 1,2, . . .. Letu, ube a pair of upper and lower solutions of2.3withux≤ux, a.e. in^ÊN. We consider the boundary value problem

−div

|x|^−ap|∇u|^p−2∇u

Fx, u, x∈Bk, ux ux, x∈∂Bk.

2.11

By Theorem 1.1 in 10, one concludes that there exists u_kx ∈ W^1,pBk,|x|^−ap∩ L^∞Bkwhich is a weak solution of2.11withux≤ukx≤uxa.e. inBkfork1,2, . . ..

We define its extension by

U_kx

⎧⎨

⎩

u_kx, x∈B_k,

u_kx ux, x∈B^c_k^ÊN\B_k. 2.12

Similarly, letvkxbe a weak solution of the boundary value problem

−div

|x|^−ap|∇vk|^p−2∇vk

Fx, vk, x∈B_k,

vkx ux, x∈∂Bk,

2.13

and its extension is defined by

Vkx

⎧⎨

⎩

vkx, x∈Bk,

v_kx ux, x∈B^c_k. 2.14

Sinceu, u ∈ W^1,pÊN,|x|^−ap∩L^∞ÊN, we haveVkx, Ukx ∈ W^1,pÊN,|x|^−ap∩ L^∞ÊN. By Theorem 2.4 in12, we have

ux≤V_kx≤V_k1x≤U_k1x≤U_kx≤ux, a.e in^ÊN 2.15

fork1,2, . . .. In view of2.15, the pointwise limits V0x lim

k→ ∞Vkx, U0x lim

k→ ∞Ukx 2.16

exist andux≤V₀x≤U₀x≤uxin^ÊN.

Similar to the proof Theorem 1.1 in 10and the proof of Theorem 7.5.1 in23, it is not difficult to get fromTheorem 2.1thatU0xis the maximal weak solution andV0x the minimal solution of2.3, which satisfies2.10andV0, U0∈W^1,pÊN,|x|^−ap∩L^∞ÊN. This ends the proof ofLemma 2.3.

Our main results read as follows.

Theorem 2.4existence. Let 1< p < N, 0≤a <N−p/p. Assume the following.

A1The nonnegative functionsfu, guare Lipschitz continuous and nondecreasing,f0 g0 0. Additionally, sup_t≥0t^−mft<∞and sup_t≥0t⁻ⁿgt<∞with 0< m < p−1<

n.

A2The nonnegative functions hx, Hx are locally H¨older continuous. Let Mr max_|x|r{hx, Hx}. If

M_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1Mtdt1/p−1

ds <∞, 2.17

then there existsλ0 >0, such thatλ∈0, λ0, and the problem1.1admits a weak solutionux∈ W^1,pÊN,|x|^−ap∩L^∞ÊN.

Theorem 2.5nonexistence. Let 1< p < N, 0≤a <N−p/p. Assume that A30< m≤p−1< n;

A4there existα1, α2≥0 such that

qα₁α₂> p−1, α1

n α2

m 1; 2.18

A5the functionshx, Hx>0 in^ÊN satisfy

lim sup

R→ ∞ B1R^{−α1p−1/nq}b1R^{−α2p−1/mq}R^σ1<∞, 2.19 whereσ1N−p1a−Np−1/qand

B1R inf

ΩR

Hx, b1R inf

ΩR

hx, ΩR

x∈^ÊN |R≤ |x| ≤√ 2R

, R≥1. 2.20

Then the problem

−div

|x|^−ap|∇u|^p−2∇u

hxu^mHxuⁿ, x∈^ÊN, ux≥0, x∈^ÊN

2.21

has no nontrivial solutionux∈W^1,pÊN,|x|^−ap.

Remark 2.6. If assumption2.19holds, then

A_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1B1t^λ1b1t^λ2dt 1/p−1

ds∞, 2.22

withλ1α1/n,λ2α2/m.

In fact, for this case, there existt₀≥1 andC₀>0 such that

B1t^λ1b1t^λ2≥C₀t^σ1q/p−1 C₀t−Nq/p−1N−p1a 2.23

fort≥t0. Therefore,

A_∞≥ _∞

t0

s^1−Nap

_s

0

t^N−1B1t^λ1b1t^λ2dt 1/p−1

ds

≥C_{1 ∞}

t0

s1−NapqN−p1a/p−1/p−1ds∞.

2.24

So, condition2.19implies2.22.

3. Proof of Existence

Before proofing the existence, we present some preliminary lemmas which will be useful in what follows.

Lemma 3.1. Suppose thatρx≥0, /≡0 is local H¨older continuous and satisfies

ρ_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1ρtdt 1/p−1

ds <∞. 3.1

Then the problem

−div

|x|^−ap|∇u|^p−2∇u

ρx, x∈^ÊN, ux>0, x∈^ÊN, lim

|x| → ∞ux 0 3.2

has a weak solutionux∈W^1,pÊN,|x|^−ap∩L^∞ÊN, whereρt max_|x|tρx.

Proof. Letρt min_|x|tρx. Thenρt≤ρt. Denote

V|x| Vr _∞

r

s^1−Nap

_s

0

t^N−1ρtdt 1/p−1

ds, r|x| ≥0,

U|x| Ur

_∞

r

s^1−Nap

_s

0

t^N−1ρtdt _1/p−1

ds, r|x| ≥0.

3.3

Obviously, lim_{|x| → ∞}V|x| lim_{|x| → ∞}U|x| 0 andV|x|≤U|x|. It is easy to verify that

−div

|x|^−ap|∇V|^p−2∇V

ρx, x∈^ÊN,

−div

|x|^−ap|∇U|^p−2∇U

ρx, x∈^ÊN.

3.4

This shows that V|x| resp., U|x| is a lower resp., upper solution of 3.2. Then by Lemma 2.3, there exists a weak solution ux for problem 3.2 satisfying ux ∈ W^1,pÊN,|x|^−ap∩L^∞ÊN, and

V|x|≤ux≤U|x|, x∈^ÊN. 3.5

Lemma 3.2. LetN≥3. If 1

_∞

1

t^1apρt_1/p−1

dt <∞, if 1< p≤2, N >2ap, 3.6

2

_∞

0

tNp−21ap/p−1ρtdt <∞, ifp >2, N > p1a, 3.7

one hasρ_∞<∞.

Proof. 1Since 1< p≤2, 1/p−1≥1. By the H ¨older inequality, we obtain

ρ_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1ρtdt _1/p−1

ds

≤ _∞

0

s^{1−Nap/p−1} s

0

t^N−1ρt_1/p−1 dt

_p−1s

0

dt

_2−p1/p−1 ds

_∞

0

s3−Nap−p/p−1_s

0

t^N−1ρt_1/p−1 dt ds

_∞

0

t^N−1/p−1

ρt ^1/p−1

_∞

t

s3−Nap−p/p−1ds dt p−1

N−2−ap _∞

0

t^1apρt_1/p−1

dt <∞.

3.8

2 Ifp > 2 and N > pa1, we takep N−N−pa1/p−1and then p∈p1a, N.

Note that

₁

0

s^1−Nap

_s

0

t^N−1ρtdt _1/p−1

ds≤ ₁

0

ρtdt

_1/p−1

<∞, _∞

1

s^1−Nap

_s

0

t^N−1ρtdt 1/p−1

ds≤ _∞

1

s^{ap1−p/p−1} _s

0

t^p−1ρtdt 1/p−1

ds

≤

p−1 ² p−2 N−pa1

_∞

0

tNp−21ap/p−1ρtdt _1/p−1

<∞.

3.9

This impliesρ_∞<∞and ends the proof ofLemma 3.2.

Corollary 3.3. Ifρt max_|x|tρxsatisfies the conditions inLemma 3.2, then the problem3.2 admits a solutionux∈W^1,pÊN,|x|^−ap∩L^∞ÊN.

Lemma 3.4. Suppose that ft ≥ 0 is nondecreasing and sup_t≥0t^−mft < ∞with m < p−1.

Additionally, let the functionhx≥0 be locally H¨older continuous and satisfy

h_{∞ ∞}

0

s^1−Nap

_s

0

t^N−1htdt _1/p−1

ds <∞, 3.10

whereht max_|x|thx. Then the problem

−div

|x|^−ap|∇u|^p−2∇u

hxfu, x∈^ÊN, ux>0, x∈^ÊN, lim

|x| → ∞ux 0 3.11

has a weak solutionux∈W^1,pÊN,|x|^−ap∩L^∞ÊN. Proof. We first consider the problem

−div

|x|^−ap|∇u|^p−2∇u

hx, x∈^ÊN, ux>0, x∈^ÊN, lim

|x| → ∞ux 0. 3.12

ByLemma 3.1, there is a solutionwhxfor3.12satisfyingwhx∈W^1,pÊN,|x|^−ap∩ L^∞ÊN. In order to get the existence of solution for3.11, we chose a pair of upper-lower solution of the equation in3.11by means ofw_hx.

Lett >0. It is easy to verify thatuhtwhis an upper solution of

−div

|x|^−ap|∇u|^p−2∇u

hxfu, x∈^ÊN 3.13

if and only if

−div

|x|^−ap|∇uh|^p−2∇uh

≥hxfuh, x∈^ÊN 3.14

or

t^p−1≥ftwh, x∈^ÊN. 3.15

By the assumption on fu, we know that there existsc₀ > 0, such thatft ≤ c₀t^m. So, c0t^mwh^m_∞≥ftwh_∞≥ftwh. Then we taket0 c0wh^m_∞^1/p−1−mso thatuhtwht >

t₀is an upper solution of3.13.

We now construct a lower solution of3.13. Consider the boundary value problem

−div

|x|^−ap|∇v|^p−2∇v

hxfv, x∈B_k, v >0, x∈Bk, v0, x∈∂Bk

3.16

fork1,2, . . ..

By Theorem 3.1 in12, there exists a solutionvk ∈W^1,pBk,|x|^−ap∩L^∞Bkfor3.16.

We define an extension byvkx 0 for|x| ≥k. Then, by Theorem 2.4 in12and D´ıaz-Sa´a’s inequality in24, we get

v1x≤v2x≤ · · · ≤vkx≤v_k1x≤ · · · ≤uhx, x∈Bk. 3.17

Settingvx limk→ ∞vkxand performing some standard computations, we see thatv ∈ W^1,pÊN,|x|^−ap∩L^∞ÊN,

−div

|x|^−ap|∇v|^p−2∇v

hxfv, x∈^ÊN, vx>0, x∈^ÊN, lim

|x| → ∞vx 0, 3.18

andvx≤uhxin^ÊN. Then, our result follows fromLemma 2.3.

We now give the proof ofTheorem 2.4.

Proof ofTheorem 2.4. Letu_M∈W^1,pÊN,|x|^−ap∩L^∞ÊNbe a solution of the problem

−div

|x|^−ap|∇u|^p−2∇u

Mx, x∈^ÊN, u >0, x∈^ÊN, lim

|x| → ∞ux 0, 3.19

whereMx max{hx, Hx}. We see thatw tuM t > 0is an upper solution of the equation

−div

|x|^−ap|∇u|^p−2∇u

Mx

fu λgu , x∈^ÊN 3.20

if and only if

−div

|x|^−ap|∇w|^p−2∇w

≥Mx

fw λgw , x∈^ÊN 3.21

or

t^p−1≥ftuM λgtuM, x∈^ÊN. 3.22

Since

sup

t≥0 t^−mft<∞, sup

t≥0 t⁻ⁿgt<∞, 3.23

we have a constantc₀ >0, such that

ft≤c0t^m, gt≤c0tⁿ, ∀t≥0. 3.24

Denote

φt t^p−1−c0t^muM^m_∞

c₀tⁿuMⁿ_∞ . 3.25

Sincem < p−1 < n, we have lim_t→0φt −∞, limt→ ∞φt 0 and there existt0 > 0, such thatφt > 0 for 0≤ t < t₀ andφt <0 fort > t₀. Thenφt0 max_t>0φt. A simple computation shows that

t0

c0n−m n−p1uM^m_∞

_1/p−1−m

. 3.26

Thus

λ0φt0 c^{m−n/p−1−m}₀ uMm−np−1/p−1−m

∞

p−1−m n−p1

n−m n−p1

_{m−n/p−1−m}

>0.

3.27

Hence, for any 0< λ < λ₀, there exists a uniquet_λ>0, such thatλφtλ. That is

t^p−1_λ c0t^m_λuM^m_∞c0λtⁿ_λuMⁿ_∞≥ftλuM λgtλuM. 3.28

Now definingwt_λu_M, we get

−div

|x|^−ap|∇w|^p−2∇w t^p−1_λ

−div

|x|^−ap|∇uM|^p−2∇uM

Mxt^p−1_λ Mx

t^m_λuM^m_∞λtⁿ_λuMⁿ_∞

≥Mx

fw λgw .

3.29

This shows thatwis an upper solution of3.20. Noting that

Mx

fw λgw ≥hxfw λHxgw, 3.30

we know thatwis an upper solution of1.1. Letvbe a solution of3.11. Obviously,vis a lower solution of1.1. We now show thatvx≤wxin^ÊN.

Sinceφt<0 fort > t₀andφt → 0 ast → ∞, then for anyλ∈0, λ0, there exist t_λ>0, such thatλφtλ. Without loss of generality, lett_λ> t₀.

From the proof ofLemma 3.4and the definition ofuMx, we haveuhx twhx ≤ tuMxfort > t0. Further, by3.17, we getvkx ≤ tλuMx wx. Lettingk → ∞, we obtainvx≤wxinR^N.

ByLemma 2.3, there exists a solutionu∈W^1,pÊN,|x|^−ap∩L^∞ÊNfor the problem 1.1. We then complete the proof ofTheorem 2.4.

Remark 3.5. The nonlinear term Fx, u hxfu λHxgu can be regarded as a perturbation of the nonlinear termhxfu.

4. Proof of Nonexistence

In order to prove the nonexistence of nontrivial solution of the problem2.21, we use the test function method, which has been used in25and references therein. Some modification has been made in our proof. The proof is based on argument by contradiction which involves a priori estimate for a nonnegative solution of 2.21by carefully choosing the special test function and scaling argument.

Proof ofTheorem 2.5. Letφ0s∈C₀¹0,∞be defined by

φ0s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1, 0≤s <1,

l−k⁻¹

l2−s^k−k2−s^l

, 1≤s≤2,

0, s >2,

4.1

and putφx φ0R⁻²|x|², by which the parametersl > k > 2 will be determined later. It is not difficult to verify that 0≤φ₀s≤1 and|φ₀s| ≤β₀φ^1−1/k₀ s, whereβ₀ kl/l−k^1/k.

Suppose thatuxis a solution to problem2.21. Without loss of generality, we can assume thatux >0 in^ÊN otherwise, we consideru uand let ↓0. Letα <0 be a parameterαwill also be chosen below.

By the Young inequality, we get

hxu^mHxuⁿ≥H^α1/nxh^α2/mxu^q ≡H^λ1xh^λ2xu^q, 4.2

whereα1,α2, andqsatisfy2.18andλ1α1/n,λ2α2/m.

Multiplying the equation in2.21byu^αφand integrating by parts, we obtain

ÊN

H^λ1h^λ2u^qαφdx≤α

ÊN

|x|^−apu^α−1|∇u|^pφdx

ÊN

|x|^−apu^α|∇u|^p−1∇φdx. 4.3

Then applying the Young inequality with parameterε >0, we have

ÊN

H^λ1h^λ2u^qαφdxβε

ÊN

|x|^−ap|∇u|^pu^α−1φdx≤Cε

ÊN

|x|^−apu^pα−1∇φ^pφ^1−pdx, 4.4 whereβε|α| −ε >0.

Similarly, let us multiply the equation in2.21byφand integrate by parts:

ÊN

H^λ1h^λ2u^qφdx≤

ΩR

|x|^−ap|∇u|^p−1∇φdx

≤

ΩR

|x|^−ap|∇u|^pu^α−1φdx

_p−1/p

ΩR

|x|^−ap∇φ^pφ^1−pu^1−αp−1dx _1/p

. 4.5

By4.4,

Ê

N|x|^−ap|∇u|^pu^α−1φdx≤C

ΩR

|x|^−ap∇φ^pφ^1−pu^pα−1dx. 4.6

Now, we apply the H ¨older inequality to the integral on the right-hand side of4.6:

ΩR

|x|^−ap∇φ^pφ^1−pu^pα−1dx

≤

ΩR

H^λ1h^λ2u^qφdx

_1/λ

ΩR

|x|^−apλ∇φ^pλφ^1−pλ

H^λ1h^λ2_1−λ dx

_1/λ 4.7

withλq/pα−1>1,λq/q−p−α1andΩR{x∈^ÊN |R≤ |x| ≤√ 2R}.

Sinceq > p−1, we choseα <0 so small thatq >p−11−α. Then, we have

ΩR

|x|^−ap∇φ^pφ^1−pu^1−αp−1dx

≤

ΩR

H^λ1h^λ2u^qφdx

1/μ

ΩR

|x|^−apμ|∇φ|^pμφ^1−pμ

H^λ1h^λ2_1−μ dx

1/μ 4.8

withμq/1−αp−1>1,μq/q−1−αp−1.

Sinceφx φ0R⁻²|x|²,|∇φx| ≤C0R⁻¹φ₀^1−1/k|ξ| C0R⁻¹φ₀^1−1/k withxRξ. Then we get

ΩR

|x|^−apλ∇φ^pλφ^1−pλ

H^λ1h^λ2_1−λ dx

≤CR^N−1apλB1R^λ11−λb1R^λ21−λ

Ω1

φ₀^1−1/kpλ|ξ|φ^1−pλ₀ |ξ|dξ,

ΩR

|x|^−apμ∇xφ^pμφ^1−pμH^1−μdx

≤CR^N−1apμB1R^λ11−μb1R^λ21−μ

Ω1

φ^1−1/kpμ₀ |ξ|φ^1−pμ₀ |ξ|dξ,

4.9

whereB1R inf_ΩRHxandb1R inf_ΩRhx.

Letk > max{pλ, pμ}. Then,

Ω1

φ^1−1/kpλ₀ |ξ|φ^1−pλ₀ |ξ|dξ≤

Ω1

φ₀|ξ|dξ≤ |Ω1|. 4.10

Similarly,

Ω1

φ^1−1/kpμ₀ |ξ|φ^1−pμ₀ |ξ|dξ≤ |Ω1|. 4.11

Then it follows from4.5–4.11that

ÊN

H^λ1h^λ2u^qφdx _1−s

≤CR^σ1B1R^σ2b1R^σ3 4.12

withs p−1/pλ1/pμ p−1/q <1 and

σ₁ p−1 pλ

N−1apλ 1 pμ

N−1apμ N−p1a− N p−1

q ,

σ2 λ1

p−1 pλ

1−λ λ1

pμ

1−μ −λ1

p−1

q ,

σ3 λ₂ p−1 pλ

1−λ λ2

pμ

1−μ −λ₂ p−1

q .

4.13

If lim sup_{R→ ∞}R^σ1B1R^σ2b1R^σ3 0, it follows from4.12that

ÊN

H^λ1h^λ2u^qdx0. 4.14

This implies thatux 0, a.e. in^ÊN. That is,uis a trivial solution for2.21.

If lim sup_{R→ ∞}R^σ1B1R^σ2b1R^σ3 C₁<∞, then4.12gives that

ÊN

H^λ1h^λ2u^qdx <∞,

Rlim→ ∞

ΩR

H^λ1h^λ2u^qdx0.

4.15

By4.5, we derive

BR

H^λ1h^λ2u^qdx≤

B2R

H^λ1h^λ2u^qφdx

≤

ΩR

|x|^−ap|∇u|^pu^α−1φdx

_p−1/p

ΩR

|x|^−ap∇φ^pφ^1−pu^1−αp−1dx _1/p

. 4.16

Reasoning as in the first part of the proof, we infer that

BR

H^λ1h^λ2u^qdx≤CR^σ1B1R^σ2b1R^σ3

ΩR

H^λ1h^λ2u^qφdx

_p−1/q

≤CC1

ΩR

H^λ1h^λ2u^qφdx

_p−1/q .

4.17

LettingR → ∞in4.17, we obtain4.14. Thus,u0, a.e. in^ÊN. Then the proof of Theorem 2.5is completed.

Acknowledgments

The authors wish to express their gratitude to the referees for useful comments and suggestions. The work was supported by the Fundamental Research Funds for the Central Universities Grant no. 2010B17914 and Science Funds of Hohai University Grants no.

2008430211 and 2008408306.

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