B. Cekic and R. A. Mashiyev

Volume 2010, Article ID 120646,7pages doi:10.1155/2010/120646

Research Article

Existence and Localization Results for px -Laplacian via Topological Methods

B. Cekic and R. A. Mashiyev

Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

Correspondence should be addressed to B. Cekic,[email protected] Received 23 February 2010; Revised 16 April 2010; Accepted 20 June 2010 Academic Editor: J. Mawhin

Copyrightq2010 B. Cekic and R. A. Mashiyev. 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 show the existence of a week solution inW₀^1,pxΩto a Dirichlet problem for−Δpxufx, u inΩ, and its localization. This approach is based on the nonlinear alternative of Leray-Schauder.

1. Introduction

In this work, we consider the boundary value problem

−Δpxufx, u inΩ,

u0 on∂Ω, P

whereΩ⊂R^N, N ≥2,is a nonempty bounded open set with smooth boundary∂Ω,Δpxu div|∇u|^px−2∇uis the so-calledpx-Laplacian operator, andCAR:f : Ω×R → Ris a Caratheodory function which satisfies the growth condition

fx, s≤ax C|s|^qx/qx for a.e. x∈Ωand alls∈R, 1.1 withCconst. >0, 1/qx 1/qx 1 for a.e.x∈Ω, anda∈L^qxΩ,ax≥ 0 for a.e.

x∈Ω.

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spacesL^pxΩ,W^1,pxΩ, andW₀^1,pxΩ. In that context, we refer to1,2 for the fundamental properties of these spaces.

Set

L^∞Ω

p:p∈L^∞Ω,ess inf

x∈Ω px>1

. 1.2

Forp ∈L^∞Ω,letp1 : ess inf_x∈Ωpx ≤ px ≤ p2 : ess sup_x∈Ωpx < ∞, for a.e.

x∈Ω.

Let us define byUΩthe set of all measurable real functions defined onΩ. For any p∈L^∞Ω,we define the variable exponent Lebesgue space by

L^pxΩ

u∈ UΩ:ρ_pxu

Ω|ux|^pxdx <∞

. 1.3

We define a norm, the so-called Luxemburg norm, on this space by the formula

u_pxinf

δ >0 :ρpx

u δ

≤1

, 1.4

andL^pxΩ, · _pxbecomes a Banach space.

The variable exponent Sobolev spaceW^1,pxΩis

W^1,pxΩ

u∈L^pxΩ: ∂u

∂xi ∈L^pxΩ, i1, . . . , N

1.5

and we define on this space the norm

uu_px∇u_px 1.6

for allu∈W^1,pxΩ.The spaceW₀^1,pxΩis the closure ofC^∞₀ ΩinW^1,pxΩ.

Proposition 1.1 see 1, 2 . If p ∈ L^∞Ω, then the spaces L^pxΩ, W^1,pxΩ, and W₀^1,pxΩare separable and reflexive Banach spaces.

Proposition 1.2see1,2 . Ifu∈L^pxΩandp2<∞,then we have i u_px <11; >1⇔ρ_pxu<11; >1,

ii u_px >1⇒ u^p_px¹ ≤ρpxu≤ u^p_px² , iii u_px <1⇒ u^p_px² ≤ρpxu≤ u^p_px¹ , iv u_px a >0⇔ρpxu/a 1.

Proposition 1.3see3 . Assume thatΩis bounded and smooth. Denote byCΩ {h∈CΩ: h1>1}.

iLetp, q∈CΩ. If

qx< p^∗x

⎧⎪

⎪⎨

⎪⎪

⎩

Npx

N−px ifpx< N,

∞ ifpx≥N,

1.7

thenW₀^1,pxΩ, · is compactly imbedded inL^qxΩ.

ii(Poincar´e inequality, see [1, Theorem 2.7]). Ifp ∈CΩ, then there is a constantC >0 such that

u_px≤C|∇u|_px, ∀u∈W₀^1,pxΩ. 1.8

Consequently, u_1,px |∇u|_px and uare equivalent norms onW₀^1,pxΩ. In what follows,W₀^1,pxΩ, withp ∈ CΩ, will be considered as endowed with the norm u_1,px.

Lemma 1.4. Assume thatr ∈L^∞Ωandp∈CΩ.If|u|^rx∈L^pxΩ, then we have min

u^r_rxpx¹ ,u^r_rxpx²

≤|u|^rx

px≤max

u^r_rxpx¹ ,u^r_rxpx²

. 1.9

Proof. ByProposition 1.2iv, we have

1

Ω

|u|^rx |u|^rx

px

px

dx

Ω

|u|

u_rxpx

rxpx u^rxpx_rxpx |u|^rxpx

px

dx

≤

Ω

|u|

u_rxpx

rxpxmax

u^r_rxpx^1px ,u^r_rxpx^2px |u|^rxpx

px

dx.

1.10

By the mean value theorem, there existsξ∈Ωsuch that

1≤ max

u^r_rxpx^1pξ ,u^r_rxpx^2pξ |u|^rxpξ

px

Ω

|u|

u_rxpx

rxpx

dx 1.11

and we have

|u|^rx

px≤max

u^r_rxpx¹ ,u^r_rxpx²

. 1.12

Similarly

1≥ min

u^r_rxpx^1pξ ,u^r_rxpx^2pξ |u|^rxpξ

px

dx,

|u|^rx

px≥min

u^r_rxpx¹ ,u^r_rxpx² .

1.13

Remark 1.5. Ifrx rconst., then

|u|^r_pxu^r_rpx. 1.14

For simplicity of notation, we write

XW₀^1,pxΩ, X^∗

W₀^1,pxΩ_∗

, Y L^qxΩ, Y^∗L^qxΩ,

·_X·_1,px, ·_Y ·_qx.

1.15

In 4 , a topological method, based on the fundamental properties of the Leray- Schauder degree, is used in proving the existence of a week solution inX to the Dirichlet problemPthat is an adaptation of that used by Dinca et al. for Dirichlet problems with classicalp-Laplacian5 . In this work, we use the nonlinear alternative of Leray-Schauder and give the existence of a solution and its localization. This method is used for finding solutions in H ¨older spaces, while in6 , solutions are found in Sobolev spaces.

Let us recall some results borrowed from Dinca 4 about px-Laplacian and Nemytskii operatorNf. Firstly, since qx < px < p^∗xfor all x ∈ Ω, X is compactly embedded inY. Denote byithe compact injection ofXinYand byi^∗ : Y^∗ → X^∗,i^∗υυ◦i for allυ∈Y^∗, its adjoint.

Since the Caratheodory function f satisfies CAR, the Nemytskii operator Nf

generated by f, Nfux fx, ux, is well defined from Y into Y^∗, continuous, and bounded3, Proposition 2.2 . In order to prove that problem Phas a weak solution in Xit is sufficient to prove that the equation

−Δ_pxu i^∗Nfi

u 1.16

has a solution inX.

Indeed, ifu∈Xsatisfies1.16then, for allυ∈X, one has −Δpxu, υ

X,X^∗ i^∗Nfi

u, υ

X,X^∗

Nfiu, iυ

Y,Y^∗ 1.17

which rewrites as

Ω|∇u|^px∇u∇υdx

Ωfυdx 1.18 and tells us thatuis a weak solution inXto problemP.

Since−Δ_pxis a homeomorphism ofXontoX^∗,1.16may be equivalently written as u

−Δpx₋₁ i^∗Nfi

u. 1.19

Thus, proving that problemPhas a weak solution inXreduces to proving that the compact operator

K

−Δ_px−1 i^∗Nfi

:X → X 1.20

has a fixed point.

Theorem 1.6Alternative of Leray-Schauder,7 . LetB0, R denote the closed ball in a Banach spaceE,{u∈E:u ≤R},and letK:B0, R → Ebe a compact operator. Then either

ithe equationλKuuhas a solution inB0, R forλ1 or

iithere exists an elementu∈EwithuRsatisfyingλKuufor someλ∈0,1.

2. Main Results

In this work, we present new existence and localization results forX-solutions to problemP, underCARcondition onf.Our approach is based on regularity results for the solutions of Dirichlet problems and again on the nonlinear alternative of Leray-Schauder.

We start with an existence and localization principle for problemP.

Theorem 2.1. Assume that there is a constantR >0,independent ofλ >0, withu_X/Rfor any solutionu∈Xto

−Δpxuλfx, u inΩ,

u0 on∂Ω Pλ

and for eachλ∈0,1. Then the Dirichlet problemPhas at least one solutionu∈Xwithu_X≤R.

Proof. By 3, Theorem 3.1 , −Δpx is a homeomorphism of X onto X^∗. We will apply Theorem 2.1toEXand to operator K:X → X,

Ku

−Δpx₋₁ i^∗Nfi

u, 2.1

wherei^∗Nfi:X → X^∗ is given byNfux fx, ux. Notice that, according to a well- known regularity result4 , the operator−Δpx⁻¹fromXtoXis well defined, continuous, and order preserving. Consequently,Kis a compact operator. On the other hand, it is clear that the fixed points ofKare the solutions of problemP. Now the conclusion follows from Theorem 1.6since conditioniiis excluded by hypothesis.

Theorem 2.2immediately yields the following existence and localization result.

Theorem 2.2. LetΩ⊂R^N, N≥2, be a smooth bounded domain and letp, q∈CΩbe such that qx< pxfor allx∈Ω. Assume thatf :Ω×R → Ris a Caratheodory function which satisfies the growth condition (CAR).

Suppose, in addition, that

Ci^∗_Y^∗_→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

<1, 2.2

whereCis the constant appearing in condition (CAR). LetR≥1 be a constant such that

R≥

⎛

⎜⎝ i^∗_Y∗→X^∗a_Y∗

1−Ci^∗_Y∗→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

⎞

⎟⎠

1/p1−1

. 2.3

Then the Dirichlet problemPhas at least a solution inXwithu_X≤R.

Proof. Letu ∈ X be a solution of problem Pλwithu_X R ≥ 1, corresponding to some λ∈0,1. Then by Propositions1.2,1.3, andLemma 1.4, we obtain

u^p_X¹≤

Ω|∇u|^pxdxλ i^∗Nfi

u, u

X,X^∗λ

Nfiu, iu

Y,Y^∗

≤λi^∗_Y∗→X^∗Nfiu

Y^∗u_X

≤λi^∗_Y∗→X^∗u_X

a_Y∗Cmax

iu^q_Y¹⁻¹,iu^q_Y²⁻¹

≤λi^∗_Y∗→X^∗u_X

a_Y∗Cu^q_X²⁻¹max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

≤λi^∗_Y∗→X^∗u_X

a_Y∗Cu^p_X¹⁻¹max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y .

2.4

Therefore, we have

u^p_X¹⁻¹≤ λi^∗_Y∗→X^∗a_Y∗

1−λCi^∗_Y∗→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y. 2.5

Substitutingu_XRin the above inequality, we obtain

R≤

⎛

⎜⎝ λi^∗_Y^∗_→X^∗a_Y^∗ 1−λCi^∗_Y∗→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

⎞

⎟⎠

1/p1−1

, 2.6

which, taking into account2.3andλ∈0,1,gives

R≤λ^1/p1−1

⎛

⎜⎝ i^∗_Y∗→X^∗a_Y∗

1−Cλi^∗_Y∗→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

⎞

⎟⎠

1/p1−1

≤λ^1/p1−1

⎛

⎜⎝ i^∗_Y∗→X^∗a_Y∗

1−Ci^∗_Y∗→X^∗max

i^q_X¹⁻¹_→Y,i^q_X²⁻¹_→Y

⎞

⎟⎠

1/p1−1

≤λ^1/p1−1R < R,

2.7

a contradiction.Theorem 2.1applies.

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

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