Existence of Solutions for the -Laplacian Problem with Singular Term

Volume 2010, Article ID 584843,20pages doi:10.1155/2010/584843

Research Article

Existence of Solutions for the

p

x

-Laplacian

Problem with Singular Term

Fu Yongqiang and Yu Mei

Department of Mathematics, Harbin Institute of Technology, Harbin 15000, China

Correspondence should be addressed to Yu Mei,[email protected]

Received 19 August 2009; Accepted 7 March 2010

Academic Editor: Raul F. Man´asevich

Copyrightq2010 F. Yongqiang and Y. Mei. 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 following px-Laplacian problem with singular term: −div|∇u|px−2_∇u

|u|px−2_{u λ|u|}αx−2_{ufx, u,x ∈ Ω,u 0,x ∈ ∂Ω, whereΩ ⊂ R}N_{is a bounded domain,}

1< p−≤px≤p< N. We obtain the existence of solutions inW₀1,pxΩ.

1. Introduction

After Kov´aˇcik and R´akosn´ık first discussed theLpxΩspaces andWk,pxΩspaces in1, a lot of research has been done concerning these kinds of variable exponent spaces, for example, see2–5for the properties of such spaces and6–9for the applications of variable exponent spaces on partial differential equations. Especially in W1,pxΩ spaces, there are a lot of studies on px-Laplacian problems; see 8,9. In the recent years, the theory of problems withpx-growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluidssee10–14.

In this paper, we study the existence of the weak solutions for the following px -Laplacian problem:

−div|∇u|px−2∇u|u|px−2uλ|u|αx−2ufx, u, x∈Ω,

u0, x∈∂Ω,

1.1

Let PΩ be the set of all Lebesgue measurable functionsp : Ω → 1,∞. For all

px∈PΩ, we denotep sup_x∈Ωpx,p− inf_x∈Ωpx, and denote byp1xp2xthe fact that inf{p2x−p1x}>0.

We impose the following condition onf:

Ffx, t bxgx, t, 0< bx∈Lrx_{Ω, 1rx∈CΩand forgx, t∈CΩ×}

R, there existM >0 such that|gx, t| ≤c1c2|t|qx−1, whenever|t| ≥M.

A typical example of 1.1 is the following problem involving subcritical Sobolev-Hardy exponents of the form

−div|∇u|px−2∇u|u|px−2uλ|u|αx−2u λ

|x|sx|u|

qx−2_{u, x∈Ω,}

u0, x∈∂Ω,

1.2

where 0 < λ ∈ R,sx, qx ∈ CΩ, 0 ≤ s− ≤ s < N, 1 qx < N−sx/Np∗x,

for allx∈Ω. In fact, takebx λ/ |x|sx_{,gx, t |t|}qx−2_{t, andrx≡1/21N/s, then} it is easy to verify thatFis satisfied.

Our object is to obtain the existence of solutions in the following four cases:

1αx> px, qx> px; 2αx< px, qx> px; 3αx< px, qx< px; 4αx> px, qx< px.

Whenbx 1, the solution of thep-Laplacian equations without singularity has been studied by many researchers. The study of problem1.1with variable exponents is a new topic.

The paper is organized as follows. InSection 2, we present some necessary preliminary knowledge of variable exponent Lebesgue and Sobolev spaces. InSection 3, we prove our main results.

2. Preliminaries

In this section we first recall some facts on variable exponent Lebesgue spaceLpx_Ωand variable exponent Sobolev spaceW1,px_{Ω, whereΩ⊂R}N_{is an open set; see1–4,8,15for} the details.

Letpx∈PΩand

u_pinf

λ >0 :

Ω

u

λ

pxdx≤1

. 2.1

For a givenpx∈PΩ,we define the conjugate functionpxas

p px

px−1 . 2.2

Theorem 2.1. Letpx∈PΩ. Then the inequality

Ω

_{fx·gxdx≤rpf}

pgp 2.3

holds for everyf∈LpxΩandg ∈LpxΩwith the constantrpdepending onpxandΩonly.

Theorem 2.2. The dual space ofLpx_ΩisLpx_{Ωif and only ifp∈L}∞_{Ω. The spaceL}px_Ω is reflexive if and only if

1< p−≤p<∞. 2.4

Theorem 2.3. Suppose that px satisfies 2.4. Let measΩ < ∞, p1x, p2x ∈ PΩ, then necessary and sufficient condition forLp2xΩ ⊂ _Lp1xΩ_{is that for almost allx} ∈ Ω _{one has}

p1x≤p2x, and in this case, the imbedding is continuous.

Theorem 2.4. Suppose thatpxsatisfies2.4. Letρu _Ω|ux|pxdx. Ifu, uk ∈ LpxΩ, then

1u_p<11;>1if and only ifρu<11;>1, 2ifu_p>1, thenupp−≤ρu≤ up

p,

3ifu_p<1, thenupp≤ρu≤ up − p,

4limk→ ∞ukp0if and only iflimk→ ∞ρuk 0, 5ukp → ∞if and only ifρuk → ∞.

We assume thatkis a given positive integer.

Given a multi-indexα α1, . . . , αn∈Nn, we set|α|α1· · ·αnandDαD₁α1· · ·Dαnn, whereDi∂/∂xiis the generalized derivative operator.

The generalized Sobolev spaceWk,px_{Ω is the class of functionsf onΩsuch that}

Dα_{f ∈L}px_{for every multi-indexαwith|α| ≤k, endowed with the norm}

f_k,p

|α|≤k

Dαf_p. 2.5

By W₀k,pxΩ we denote the subspace of Wk,px_{Ω which is the closure of C}∞

0 Ω with respect to the norm2.5.

In this paper we use the following equivalent norm onW1,px_Ω:

u_1,p inf

λ >0 :

Ω

∇_λupxu λ

pxdx≤1

. 2.6

Theorem 2.5. The spacesWk,px_ΩandWk,px

0 Ωare separable reflexive Banach spaces, ifpx satisfies2.4.

Theorem 2.6. Suppose thatpxsatisfies2.4. LetIu _Ω|∇ux|px_|ux|px_{dx. Ifu, u} k∈

W1,px_{Ω, then}

1u_1,p<11;>1if and only ifIu<11;>1,

2ifu_1,p>1, thenu_1,pp− ≤Iu≤ up_1,p,

3ifu_1,p<1, thenu_1,pp ≤Iu≤ up_1,p−,

4limk→ ∞uk1,p 0if and only iflimk→ ∞Iuk 0, 5uk1,p → ∞if and only ifIuk → ∞.

We denote the dual space ofW₀k,pxΩbyW−k,px_{Ω, then we have the following.}

Theorem 2.7. Letp∈PΩ∩L∞Ω. Then for everyG∈W−k,pxΩthere exists a unique system of functions{gα∈Lp

_x

Ω:|α| ≤k}such that

Gf

|α|≤k

ΩD α_fxg

αxdx, f∈W₀k,pxΩ. 2.7

The norm ofW−k,px_{Ωis defined as}

G_−k,psup

_G

f

f_k,p :f∈W

k,px

0 Ω\ {0}

. 2.8

Theorem 2.8. LetΩbe a domain inRn_{with cone property. Ifp :Ω → Ris Lipschitz continuous} and 1 < p− ≤ p < N/k,qx : Ω → R is measurable and satisfiespx ≤ qx ≤ p∗x :

Npx/N−kpxa.e.x∈Ω, then there is a continuous embeddingWk,px_Ω→Lqx_Ω.

Theorem 2.9. LetΩbe a bounded domain. Ifpx∈L∞Ωandu∈W₀1,pxΩ, then

u_p≤C∇u_p, 2.9

whereCis a constant depending onΩ.

Next let us consider the weighted variable exponent Lebesgue space. Letax∈PΩ, andax>0 forx∈Ω. Define

Lp_ax_xΩ

u∈PΩ:

Ωax|ux|

with the norm

|u|_Lpx

axΩup,a inf

λ >0 :

Ωax

u_λxpxdx≤1

, 2.11

thenLp_ax_xΩis a Banach space.

Theorem 2.10. Suppose that px satisfies 2.4. Let ρu _Ωax|ux|px_{dx. If u,u} k ∈

Lp_ax_xΩ, then

1foru /0,up,aλif and only ifρu/λ 1, 2u_p,a <11;>1if and only ifρu<11;>1,

3ifu_p,a >1, thenupp,a− ≤ρu≤ up

p,a,

4ifu_p,a <1, thenupp,a ≤ρu≤ up − p,a,

5limk→ ∞ukp,a0if and only if limk→ ∞ρuk 0, 6ukp,a → ∞if and only ifρuk → ∞.

Theorem 2.11. Assume that the boundary of Ω possesses the cone property and px ∈ CΩ. Suppose that 0 < ax ∈ Lrx_{Ω, and 1 rx ∈ CΩ, for x ∈ Ω. If qx ∈ CΩ} and 1 ≤ qx < rx − 1/rxp∗x, for allx ∈ Ω, then there is a compact embedding

W1,px_Ω→Lqx axΩ.

Theorem 2.12. LetΩ⊂Rn_{be a measurable subset. Letg:Ω×R → Rbe a Caracheodory function} and satisfies

_{gx, u≤ax b|u|}p1x/p2x _{for anyx}∈Ω_{, t}∈R_{, 2.12}

where pix ≥ 1, i 1,2,ax ∈ Lp2xΩ,ax ≥ 0,b ≥ 0 is a constant, then the Nemytsky operator fromLp1xΩ_toLp2xΩ_{defined byN}

gux gx, uxis a continuous and bounded operator.

3. Existence and Multiplicity of Solutions

Let

Iu

Ω 1

px

|∇u|px|u|px−λ|u|

αx

αx −Fx, udx,

Fx, t _t

0

fx, sds, Gx, t _t

0

gx, sds.

The critical points ofIu, that is,

0Iuϕ

Ω|∇u|

px−2_∇u∇ϕ|u|px−2_uϕ−λ|u|αx−2_{uϕ−fx, uϕ dx 3.2}

for allϕ ∈W₀1,pxΩ, are weak solutions of problem1.1. So we need only to consider the existence of nontrivial critical points ofIu.

Denote by c, ci, C, Ci, K, and Ki the generic positive constants. Denote by|Ω|the Lebesgue measure ofΩ.

To study the existence of solutions for problem1.1in the first case, we additionally impose the following conditions.

A-1αx, qx∈CΩandpxαx< p∗x,pxqx<rx−1/rxp∗x.

B-1There exists a functionμx∈C1_{Ω, such thatpxμxand 0< μxGx, t≤}

tgx, tforx∈Ω,|t| ≥M.

C-1there exist δ >0 such that|gx, t| ≤c3|t|qx−1_{forx∈Ω,|t| ≤δ, whereqx∈CΩ} andpxqx<rx−1/rxp∗x.

D-1gx,−t −gx, t, for allx∈Ω, t∈R.

Theorem 3.1. Under assumptions (F) and (A-1)–(C-1), problem1.1admits a nontrivial solution.

Proof. First we show that anyP S_c sequence is bounded. Let{un} ⊂ W₀1,pxΩand c ∈

R, such thatIun → cand Iun → 0 in W−1,p _x

Ω. By A-1and B-1, inf{1/px−

1/αx} a1 > 0, and inf{1/px−1/μx} a2 > 0. Let a0 min{a1, a2}and lx 1/px−a0/2, then lx > 1/αx,lx > 1/μx,|∇lx| < C, 1/px−lx a0/2 and

lx−1/αx≥a0/2. LetΩ1{x∈Ω:|ux| ≥M}andΩ2 Ω\Ω1. Then

Iun− Iun, lxun

Ω

1

px −lx

|∇un|px|un|pxdx

Ωλ

lx− 1 αx

|un|αxdx

Ω1

lxfx, unun−Fx, undx−

Ω2

Fx, undx

Ω2

lxfx, unundx−

Ω|∇un|

px−2_∇un∇l·u ndx.

3.3

ByB-1, we get

Ω1

lxfx, unun−Fx, undx≥

Ω1

lx− 1 μx

ByF, we getgx, t∈CΩ×−M, M, so there existK >0 such that|gx, t| ≤Kon Ω×−M, M. Notefx, t bxgx, t, so we have

Ω2

Fx, undx

≤

Ω2

MKbxdx <∞,

Ω2

fx, unundx

≤

Ω2

MKbxdx <∞.

3.5

By Young’s inequality, forε1∈0,1, we get

Ω|∇un|

px−2_∇un∇l·u

ndx≤Cε1

Ω|∇un|

px_dxCε1−p 1

Ω|un|

px_{dx. 3.6}

Takeε1sufficiently small so thata0/2> Cε1.

Note thatpxαx, by Young’s inequality again and forε2∈0,1, we get

Ω|∇un|

px−2_∇un∇l·u ndx

≤Cε1

Ω|∇un|

px_dxCε1−p 1 ε2

Ω|un|

αx_dxCε1−p

1 ε

−p/α−p− 2 |Ω|.

3.7

Takeε2sufficiently small so thatλa0/2≥Cε1−p

1 ε2. From the above remark, we have

Iun−

Iun, lxun

≥

Ω

_a0

2 −Cε1

|∇un|px|un|pxdx−C1. 3.8

Aslxunp ≤lunp,lx∇unp ≤l∇unpand∇lx·unp ≤Cunp, we have

lxun1,p≤4Clun1,p. SinceIun → candIun → 0, byTheorem 2.6we have

co1 4Clun_1,p ≥a0

2 −Cε1

unp_1,p− −C1, 3.9

whenun1,p ≥ 1 andnis sufficiently large. Then it is easy to see that{un}is bounded in

Note that

Ω

|∇un|px−2∇un− |∇u|px−2∇u

∇un− ∇udx

≤Iun, un−uIu, un−u

Ω

λ|un|αx−2un− |u|αx−2u

un−udx

Ω

_{fx, un−fx, u}

un−udx

I1I2I3I4.

3.10

Because {un} is bounded in W₀1,pxΩ, there exists a subsequence {un} still denoted by {un}, such that un u weakly in W₀1,pxΩ. By Theorem 2.11, there are compact embeddingsW1,px_{Ω → L}αx_{Ωand W}1,px_{Ω → L}qx

bxΩ, then un → u inLαxΩ andLq_bx_xΩ. So we get

I3

Ω

λ|un|αx−2un− |u|αx−2u

un−udx

≤2λ|un|αx−1

αun−uα2λ

|u|αx−1

αun−uα.

3.11

HenceI3 → 0 asn → ∞. ByF, we have

Ω

_{fx, uun−udx≤}

Ω2

C2bx|un−u|dx

Ω1

bx|un−u||u|qx−1dx

≤

ΩC2bx|un−u|dx

Ωbx|un−u||u|

qx−1_dx,

3.12

and similarly for everyn,

Ω

fx, unun−udx≤

ΩC2bx|un−u|dx

Ωbx|un−u||un|

qx−1_{dx. 3.13}

Since

Ωbx|un−u||u|

qx−1_dx

Ωbx

1/qx_bx1/qx_|u

n−u||u|qx−1dx

≤2bx1/qx|u|qx−1

q

bx1/qx|un−u| q

and un → uin L1_bxΩ andLq_bx_xΩ, we obtain _Ωbx|un−u|dx → 0 and _Ωbx|un −

u||u|qx−1_{dx → 0. Similarly,}

Ωbx|un−u||un|

qx−1_dx≤2bx1/qx_|un|qx−1 q

bx1/qx|un−u|

q. 3.15

Becausebx1/qx|un|qx−1_{qis bounded, we get}

Ωbx|un−uun|qx−1dx → 0, asn → ∞. From the above remark, we concludeI4 _Ω|fx, un−fx, uun−u|dx → 0, asn → ∞. ThusI1I2I3I4 → 0, asn → ∞. Then we get _Ω|∇un|px−2∇un−|∇u|px−2∇u∇un− ∇udx → 0. As in the proof of Theorem 3.1 in6,7, we divideΩinto the following two parts:

Ω3

x∈Ω: 1< px<2, Ω4

x∈Ω:px≥2. 3.16

OnΩ3, we have

Ω3

|∇un− ∇u|pxdx≤

Ω3

K1

|∇un|px−2∇un− |∇u|px−2∇u

∇un− ∇u

px/2

×|∇un|px|∇u|px2−px/2dx

≤2K1

|∇un|px−2∇un− |∇u|px−2∇u

∇un− ∇u

px/2

2/px,Ω3

×|∇un|px|∇u|px2−px/2

2/2−px,Ω3

.

3.17

Then _Ω

3|∇un− ∇u|

px_{dx → 0,asn → ∞.}

OnΩ4, we have

Ω4

|∇un− ∇u|pxdx≤K2

Ω4

|∇un|px−2∇un− |∇u|px−2∇u

∇un− ∇udx, 3.18

so _Ω

4|∇un− ∇u|

px_{dx → 0, asn → ∞.}

Thus we get _Ω|∇un− ∇u|pxdx → 0. Thenun → uinW₀1,pxΩ. FromFand B-1 we haveGx, t ≥ c4|t|μx _−c

5, for allx ∈ Ω,t ∈ R. So we get

ε, and |αx−αx0| < ε, for x ∈ Bx0, δ. Letϕ ∈ C∞₀Bx0, δ, such that ϕx 1 for

x∈Bx0, δ/2,ϕx 0 forx∈Ω\Bx0, δ, and 0≤ϕx≤1 inBx0, δ. Then we have

Isϕ≤

Ωs

px∇ϕ px

px

ϕpx px

−λsαxϕ

αx

αx −c4bxs

μx_ϕμx

c5bxdx

≤

Bx0,δ

spx∇ϕ

px

px

ϕpx px

−λsαxϕ

αx

αx dxc5

Bx0,δ

bxdx

≤spB

Bx0,δ

_∇ϕpx

px

_ϕpx

px

dx−sα−B

Bx0,δ

λϕ

αx

αx dx

c5

Bx0,δ

bxdx,

3.19

wherep_B< α−_B. So ifsis sufficiently large, we obtainIsϕ<0.

FromFandC-1, we have|Gx, t| ≤c3|t|qxcδ|t|qx, then|Fx, t| ≤c3bx|t|qx

cδbx|t|qx_{. So we get}

Iu≥ 1 p

Ω

|∇u|px|u|pxdx− λ α−

Ω|u| αx_−c

3bx|u|qx−cδbx|u|qxdx. 3.20

Let θx min{αx,qx, qx}, then θx ∈ CΩ. By Theorem 2.11, uα ≤ c6u1,p, uq,b ≤ c7u1,p,anduq,b ≤c8u1,p. Whenu1,p is sufficiently small,uα <1,uq,b <1 and uq,b < 1. For any x ∈ Ω, as px, θx ∈ CΩ, for any ε > 0, we can find

QRx {y y1, . . . , yn : |yk −xk| < R, k 1, . . . , n}such that |py−px| < ε and |θy−θx|< εwhenevery∈QRx∩Ω. Takeε 1/4θx−px, then sup_y∈QRx∩Ωpy< inf_y∈QRx∩Ωθy.{QRx}_x∈Ωis an open covering ofΩ. AsΩis compact, we can pick a finite subcovering {QRkxk}

m

k1 for Ω from the covering {QRx}_x∈Ω. If QRkxk/⊂Ω we define u 0 on QRkxk\Ω. We can use all the hyperplanes, for each of which there exists at

least one hypersurface of someQRkxklying on it, to divide

m

k1QRkxkinto finite open

hypercubes{Qi}Q_i1which mutually have no common points. It is obvious thatΩ ⊆ Q_i1Qi and for eachQithere exists at least oneQRkxksuch thatQi⊆QRkxk. LetΩiQi∩Ω, then p_i sup_y∈Ω

ipy<infy∈Ωiθy θ−i and we have

Iu≥

Q

i1

1 p Ωi

|∇u|px|u|pxdx− λ α−

Ωi

|u|αx−c3bx|u|qx−cδbx|u|qxdx

≥Q i1

1

pu

pi

1,p,Ωi− λ α−c

α−i

6 u α−i

1,p,Ωi−c3c

q−i

7 u

qi−

1,p,Ωi−cδc

q−i

8 u q−i

1,p,Ωi

.

Ifu1,pk≥ u1,p,Ωiis sufficiently small such that

1

pu

p_i 1,p,Ωi−

λ α−c

α−_i 6 u

α−_i

1,p,Ωi−c3c

q−_i 7 u

q−_i

1,p,Ωi−cδc

q_i− 8 u

q−_i

1,p,Ωi >0, 3.22

we haveIu>0.

The mountain pass theorem guarantees thatIhas a nontrivial critical pointu.

Since W₀1,pxΩ is a separable and reflexive Banach space, there exist {en}∞_n1 ⊂ W₀1,pxΩand{e∗_n}∞_n1⊂W−1,px_{Ωsuch that}

ei, e∗j

⎧ ⎨ ⎩

1, ij,

0, i /j,

W₀1,pxΩ span{en:n1,2, . . .},

W−1,pxΩ span{e∗n:n1,2, . . .}.

3.23

Fork1,2, . . ., denoteXkspan{ek},Yk⊕kj1Xj,Zk⊕∞_jkXj.

Theorem 3.2. Under assumptions (F), (A-1)–(D-1), problem1.1admits a sequence of solutions {un} ∈W₀1,pxΩsuch thatIun → ∞.

Proof. Letϕu _Ωλ|u|αx_{/αx Fx, udx. We first show thatϕuis weakly strongly} continuous. Let un u weakly in W01,pxΩ. By the compact embeddingW

1,px

0 Ω →

Lαx_{Ω, we have}

Ω|un−u|αxdx → 0 andun → ua.e. onΩ. By the inequality|un|αx ≤ 2α−1_|u

n−u|αx|u|αxand the Vitali Theorem, we get _Ω|un|αxdx → _Ω|u|αxdx. Note that

Ω|Fx, un−Fx, u|dx

Ωbx|Gx, un−Gx, u|dx ≤2bx_rGx, un−Gx, ur.

3.24

When|u| ≥M,

|Gx, u| ≤c1 |

u| μxc2

|u|qx μx ≤

cq₁x qx

|u|qx

μxqxqxc2

|u|qx μx

≤C3

1|u|qxC3

1|u|rxqx/rx.

3.25

When|u| ≤M,Gx, uis bounded. So we get

|Gx, u| ≤C4

By the compact embeddingW₀1,pxΩ → Lrxqx_{Ω, we get u}

n → uinLr _xqx

Ω. So byTheorem 2.12we obtain|Gx, un−Gx, u|r → 0, that is, _Ω|Fx, u_n−Fx, u|dx → 0. Hence we obtain that ϕu is weakly strongly continuous. By Proposition 3.5 in8,βk

βkr supu∈Zk,u1,p≤r|ϕu| → 0 ask → ∞forr >0. For alln, there exists a positive integer knsuch thatβkn≤1 for allk≥kn. Assumekn < kn1for eachn. Define{rk:k 1,2, . . .}in the following way:

rk

⎧ ⎨ ⎩

n, kn≤k < kn1,

1, 1≤k < k1.

3.27

Note thatrk → ∞ask → ∞. Hence foru∈Zkwithu1,prk, we get

Iu

Ω 1

px

|∇u|px|u|pxdx−ϕu≥ 1 pu

p−

1,p−1. 3.28

So

inf u∈Zk,u1,prk

Iu−→ ∞ ask−→ ∞. _3.29

Note thatFx, u bxGx, u≥c4bx|u|μx−c5bxandpx, αx∈CΩ. Since the dimension ofYkis finite, any two norms onYk are equivalent, thenc9|u|1,p ≤ |u|α ≤c10|u|1,p. If|u|1,p ≤ 1, it is immediate that|u|α ≤ c10. If|u|1,p > 1/c9, then|u|α > 1. As in the proof of

Theorem 3.1we can find hypercubes{Qi}Q_i1which mutually have no common points such thatΩ⊆Q_i1Qiandp_i supy∈Ωipy<infy∈Ωiαy α−i, whereΩi Qi∩Ω. Then we need only to consider the case:|u|1,p,Ωi >max{1,1/c9}for everyi. We have

Iu≤

Ω

|∇u|px

px

|u|px px

−λ|u|

αx

αx −c4bx|u|

μx_c

5bxdx

≤

Ω

|∇u|px

px

|u|px px

−λ|u|

αx

αx dxc5

Ωbxdx

≤Q i1

1

p−u

p_i 1,p,Ωi−

λ αc

α−_i 9 u

α−_i 1,p,Ωi

c5

Ωbxdx

Q i1

λ αc

α−_i 9

⎛

⎝ α−

p−λcα−i

9 upi

1,p,Ωi− u

α−_i 1,p,Ωi

⎞

⎠_c5

Ωbxdx.

3.30

Let c0 α/p−λcα −

i

9 , fit tα −

i − c₀tpi. Let f_is_i mint_≥0f_it < 0 and f_it_i 0.

Denotet "Q_i1u1,p,_Ωi. Lettbe sufficiently large such thatt >max{Q, Qti, Q2c01/α −

i1,2, . . . , Q}. There at least exists onei0 such thatu1,p,Ωi₀ ≥1/Q "Q

i1u1,p,Ωi t/Q.

We have

Q

i1

fi

u_1,p,Ωi

Q

i1 upi

1,p,Ωi

uα−i−pi

1,p,Ωi −c0

i /i0

upi

1,p,Ωi

uα−i−pi

1,p,Ωi −c0

up i₀

1,p,Ωi0

uα

−

i₀−pi₀

1,p,Ωi0 −c0

≥ i /i0

fisi

t Q

p_i

0t

Q α−_i

0−p

i0

−c0

≥Q i1

fisi c0

Qt,

3.31

and"Q_i1fiu1,p,Ωi → ∞ast → ∞. Hence we obtain thatIu → −∞asu1,p → ∞. Thus

for eachk, there existsρk > rksuch thatIu < 0 foru ∈ YK∩Sρk. FromTheorem 3.1Iu

satisfiesP S_ccondition. In view ofD-1, by Fountain Theoremsee16, we conclude the result.

In the second case, we additionally impose the following condition: A-2αx, qx∈CΩand 1≤αxpxqx<rx−1/rxp∗x.

Theorem 3.3. Under assumptions (F), (A-2), (B-1), and (C-1) there existλ∗ > 0 such that when

λ∈0, λ∗, problem1.1admits a nontrivial solution.

Proof. It is obvious thatα−< p−. Letε0>0 be such thatα−ε0< p−. Sinceαx∈CΩ, there existsδ > 0 andx0 ∈ Ωsuch that|αx−α−|< ε0, for allx∈ Bx0, δ. Thusαx ≤ α−ε0 for allx ∈ Bx0, δ. Let ϕ be as defined in Theorem 3.1. ByC-1,|Gx, t| ≤ c3|t|qx, and |Fx, t| ≤c3bx|t|qx, whent < δ. Then for anyt∈0,1andt < δ, we have

Itϕ

Ωt

px∇ϕ px

px

ϕpx px

−λtαxϕ αx

αx −F

x, tϕdx

≤tp−

Bx0,δ

_∇ϕpx

px

ϕpx px

dx−tα

Bx0,δ

λϕ

αx

αx dx

tq−

Bx0,δ

c3bxϕqxdx

≤tp−

Bx0,δ

_∇ϕpx

px

ϕpx px

dx−tα−ε0

Bx0,δ

λϕ

αx

αx dx

tq−

Bx0,δ

c3bxϕqxdx.

3.32

FromFandC-1, we have|Gx, t| ≤c3|t|qx_cδ|t|qx_{and|Fx, t| ≤c}

3bx|t|qx

cδbx|t|qx_{. By Theorems2.8and2.11, there exist positive constantsk}

1, k2, k3 such that |u|α ≤ k1|u|1,p, |u|q,b ≤ k2|u|1,p, |u|q,b ≤ k3|u|1,p. When u1,p is sufficiently small, we have

uα < 1,uq,b <1, anduq,b < 1. As in the proof ofTheorem 3.1we can find hypercubes

{Qi}Q_i1which mutually have no common points such thatΩ ⊆ Q_i1Qi,p_i sup_y∈Ωipy <

infy∈Ωiqy q−i andpi <qi−, whereΩiQi∩Ω. Then

Iu≥

Q

i1

Ωi

1

p

|∇u|px|u|px−c3bx|u|qx−cδbx|u|qxdx

− λ

α−

Ω|u| αx_dx

≥Q i1

1

pu

p_i

1,p,Ωi−c3k

q−_i 3 u

q−_i

1,p,Ωi−cδk

q−_i 2 u

q−_i 1,p,Ωi

− λ

α−k

α− 1 uα

− 1,p.

3.33

Sinceu1,p≥ u1,p,Ωi, for alli, whenu1,p ρ,u1,p,Ωi ρi< ρ. Fixρsuch that1/pρ

p_i i −

c3k q−_i 3 ρ

q−_i

i −cδk q−_i 2 ρ

q−_i

i >0. Then we have

Iu≥

N

i1

1

pρ

p_i i −c3k

q−_i 3 ρ

q−_i

i −cδk q−_i 2 ρ

q_i− i

− λ

α−k

α− 1 ρα

− l1

ρ−λl2

ρ. 3.34

Let λ∗ l1ρ/2l2ρ. When λ ∈ 0, λ∗,Iu ≥ l1/2 > 0. As in the proof of Theorem 2.1 in17, denoteBρ0 {u ∈ W₀1,pxΩ : u1,p < ρ}, we have inf∂Bρ0Iu > 0 and−∞ < c inf_B

ρ0Iu < 0. Let 0 < ε < inf∂Bρ0Iu−infBρ0Iu. Applying Ekeland’s variational

principle to the functionalIu:Bρ0 → R, we finduε∈Bρ0such thatIuε<inf_Bρ0Iu

ε,Iuε< Iuεu−uε1,p,uε/u∈Bρ0, andIuε−1,p≤ε. Thus we get a sequence{un} ⊂

Bρ0such thatIun → candIun → 0. It is clear that{un}is bounded inW₀1,pxΩ. As in the proof ofTheorem 3.1, we get a subsequence of{un}, still denoted by {un}, such that un → uinW₀1,pxΩ. SoIu c <0 andIu 0.

Theorem 3.4. Under assumptions (F), (A-2), and (B-1)–(D-1), problem 1.1 has a sequence of solutions{un} ∈W₀1,pxΩsuch thatIun → ∞.

Proof. First we show that anyP S_csequence is bounded. Let{un} ∈W₀1,pxΩandc∈R, such thatIun → candIun → 0 in W−1,p

_x

Ω. ByB-1, there exist σ > 0 such that inf{1/px−1σ/μx}μ1>0. FromF,A-2, andB-1–D-1, we have

Ω1

1

μxfx, unun−Fx, undx >0,

Ω2

_μ1_xfx, unun−Fx, un

dx < C1,

Ω 1

μxfx, unundxC2≥

ΩFx, undx≥

Ωc4bx|un|

μx_dx−

Ωc5bxdx.

Thus we have

Iun

Ω 1

px

|∇un|px|un|px−λ|un|

αx

αx −Fx, undx

Ω

1

px−

1σ μx

|∇un|px|un|px−λ|un|

αx

αx −Fx, undx

Ω 1σ

μxλ|un|

αx_dx Ω

1σ

μxfx, unun

1σ μx

|∇un|px|un|px

− 1σ

μxλ|un|

αx₋1σ

μxfx, unundx

≥

Ω

1

px−

1σ μx

|∇un|px|un|pxdx−

Ω|∇un|

px−2_∇un∇

1σ μx

undx

−

Ωλ |un|αx

αx dx

Ωσc4bx|un|

μx_dx−

Ωσc5bxdx

# Iun,

1σ μxun

$

−C3.

3.36

By Young’s inequality, forε1, ε2, ε3 ∈0,1, we get

Ω|∇un|

px−2_∇un∇

1σ μx

·undx

≤C1ε1

Ω|∇un|

px_dxC 1ε1−p

1 ε2

Ω|un|

μx_dxC 1ε1−p

1 ε

−p/μ−p− 2 |Ω|,

Ω |un|αx

αx dx≤ε3

Ω|un|

px_dxε−α/p−α− 3 |Ω|.

3.37

Takeε1,ε2,andε3sufficiently small so thatμ1−C1ε1>0,μ1−λε3 ≥0 andσc4bx−C1ε1−p

1 ε2≥ 0, then

Iun−

# Iun,

1σ μxun

$

≥

ΩC4|∇un|

px_dx−C

5. 3.38

Therefore by Theorems2.6and2.9, we get that{un}is bounded inW₀1,pxΩ. Then as in the proof ofTheorem 3.1{un}possesses a convergent subsequence{un}still denoted by{un}. ByTheorem 3.2, we can also have

inf u∈Zk,u1,prk

As in the proof ofTheorem 3.1we can find hypercubes{Qi}Q_i1 which mutually have no common points such thatΩ ⊆ Q_i1Qi andp_i sup_y∈Ωipy < infy∈Ωiμy μ−i, where ΩiQi∩Ω. Since the dimension ofYkis finite, any two norms onYkare equivalent. Then we need only to consider the cases|u|1,p,_Ωi >1 anduμ,b,_Ωi >1 for everyi. We have

Iu≤

Ω

|∇u|px

px

|u|px px

−λ|u|

αx

αx −c4bx|u|

μx_c

5bxdx

≤

Ω

|∇u|px

px

|u|px px

−c4bx|u|μxdxc5

Ωbxdx

≤Q i1

1

pu

p_i

1,p,Ωi−C4u

μ−_i 1,p,Ωi

c5

Ωbxdx.

3.40

HenceIu → −∞asu1,p → ∞. As in the proof ofTheorem 3.2, we complete the proof.

In the third case, we additionally impose the following condition:

A-3αx, qx∈CΩ, and 1≤αx, qxpx,

B-3there existθ >0 such thatGx, t≥0 fort∈0, θ.

Theorem 3.5. Under assumptions (F), (A-3), and (B-3), problem1.1admits a nontrivial solution.

Proof. By Young’s inequality, forε∈0,1, we get|un|αx_{/αx≤ε|un|}px_ε−α/p−α−

. ByF, we haveFx, u≤Cbx|u|qx_C

1bx. Thus

Iu

Ω

|∇u|px

px

|u|px px

−λ|u|

αx

αx −Fx, udx

≥

Ω

|∇u|px

px

|u|px px

dx−λε

Ω|u| px_dx

−C

Ωbx|u|

qx_dx−λε−α/p−α−

|Ω| −C1

Ωbxdx

≥

Ω

1

p −λε

|∇u|px|u|pxdx−C

Ωbx|u|

qx_dx−C 2.

3.41

Takeεsufficiently small so that 1/p−λε >0. FromTheorem 2.11,uq,b≤cu1,p. Ifu1,p ≤

Theorem 3.1we can find hypercubes{Qi}Q_i1which mutually have no common points such thatΩ⊆Q_i1Qiandq_isup_y∈Ωiqy<infy∈Ωipy p−i, whereΩiQi∩Ω. Then we have

Iu≥

M

i1

c1up−i

1,p,Ωi−c2u

q_i 1,p,Ωi

J j1

c1up

−

j

1,p,Ωj−c3u

q−_j 1,p,Ωj

C3

≥M i1

c1

up−i

1,p,Ωi−c4u

q_i 1,p,Ωi

C4,

3.42

where _Ω

ibx|u|

qx_{dx >1, i1,2, . . . M, and}

Ωjbx|u|

qx_{dx≤1, i1,2, . . . J,MJ Q.} As in the proof ofTheorem 3.2, we obtain thatIuis coercive, that is,Iu → ∞asu1,p →

∞. ThusI has a critical pointusuch thatIu inf_u∈W1,px

0 ΩIuand further uis a weak

solution of1.1.

Next we show thatuis nontrivial. LetBx0, δbe the same as that inTheorem 3.3. By B-3,Fx, tϕ≥0. Then

Itϕ

Ωt

px∇ϕ px

px

ϕpx px

−λtαxϕ αx

αx −F

x, tϕdx

≤tp−

Ω

_∇ϕpx

px

ϕpx px

dx−tα

Ω0

λϕ

αx

αx dx

≤tp−

Ω

_∇ϕpx

px

_ϕpx

px

dx−tα−ε0

Ω0

λϕ

αx

αx dx.

3.43

Iftis sufficiently small,Itϕ<0.

In the fourth case, we additionally impose the following condition:

A-4αx, qx∈CΩand 1≤qxpxαx< p∗x.

Theorem 3.6. Under assumptions (F), (A-4), and (D-1), problem1.1admits a sequence of solutions {un} ∈W₀1,pxΩsuch thatIun → ∞.

We can get

Iun− Iun, lxun

Ω 1

px

|∇un|px|un|px−λ|un|

αx

αx −Fx, undx

Ω

1

px−lx

|∇un|px|un|px−λ|un|

αx

αx −Fx, undx

Ωlxλ|un|

αx_dx

Ωlxfx, unun− |∇un|

px−2_{∇un∇lx·u} ndx

≥

Ω

a

2

|∇un|px|un|pxdx−

Ω|∇un|

px−2_{∇un∇lx·u} ndx

Ω

lx− 1 αx

λ|un|αxlxfx, unun−Fx, un

dx.

3.44

By Young’s inequality, forε1, ε2∈0,1, we get

Ω|∇un|

px−2_{∇un∇lx·u} ndx

≤Cε1

Ω|∇un|

px_dxCε1−p 1 ε2

Ω|un|

αx_dxCε1−p

1 ε

−p/α−p− 2 |Ω|.

3.45

Takeε1andε2sufficiently small so thatCε1< a/2 andCε1−p

1 ε2 ≤a/2. Then

∞> Iun− Iun, lxun

≥

Ω

_a

2 −Cε1

|∇un|px|un|px−lxfx, unun|Fx, un|

dx−C1

≥

Ω

_a

2 −Cε1

|∇un|px|un|pxdx−C2

Ωbx|un|

qx_dx−C 3.

3.46

As in the proof ofTheorem 3.5,Iun− Iun, lxun → ∞, whenun1,p → ∞. Thus, we conclude that {un} is bounded in W₀1,pxΩ. Then as in the proof ofTheorem 3.1{un}

possesses a convergent subsequence{un} still denoted by{un}. ByTheorem 3.2, we can also get

inf u∈Zk,u1,prk

Iu−→ ∞ ask−→ ∞. _3.47

ΩiQi∩Ω. Since the dimension ofYkis finite, any two norms onYkare equivalent. Then we need only to consider the cases|u|1,p,_Ωi >1|u|α,_Ωi >1 and|u|q,b,_Ωi >1 for everyi. We have

Iu≤

Ω

|∇u|px

px

|u|px px

−λ|u|

αx

αx dxC4

Ωbx|un|

qx_dxC 5

≤Q i1

c0

upi

1,p,Ωiu

q_i 1,p,Ωi

−cuα−i

1,p,Ωi

C6.

3.48

Hence we obtainIu → −∞asu1,p → ∞. As in the proof ofTheorem 3.2, we complete the proof.

Acknowledgments

This work is supported by the Science Research Foundation in Harbin Institute of Technology HITC200702, The Natural Science Foundation of Heilongjiang ProvinceA2007-04, and the Program of Excellent Team in Harbin Institute of Technology.

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