Báo cáo hóa học: " Research Article Existence and Localization Results for p x -Laplacian via Topological Methods" doc

We show the existence of a week solution in W01,pxΩ to a Dirichlet problem for −Δpx u fx, u inΩ, and its localization.. This approach is based on the nonlinear alternative of Leray-Scha

Trang 1

Volume 2010, Article ID 120646, 7 pages

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,bilalcekic@gmail.com

Received 23 February 2010; Revised 16 April 2010; Accepted 20 June 2010

Academic Editor: J Mawhin

Copyrightq 2010 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 in W01,pxΩ to a Dirichlet problem for −Δpx u fx, 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

−Δpxu f x, u in Ω,

whereΩ ⊂ RN , N ≥ 2, is a nonempty bounded open set with smooth boundary ∂Ω, Δ px u

div|∇u|px−2 ∇u is the so-called px-Laplacian operator, and CAR: f : Ω × R → R is a

Caratheodory function which satisfies the growth condition

f x, s ≤ ax C|s| qx/qx for a.e x ∈ Ω and all s ∈ R, 1.1

with C const > 0, 1/qx 1/qx 1 for a.e x ∈ Ω, and a ∈ L qx Ω, ax ≥ 0 for a.e.

x ∈ Ω.

We recall in what follows some definitions and basic properties of variable exponent

Lebesgue and Sobolev spaces L px Ω, W 1,px Ω, and W 1,px

0 Ω In that context, we refer

to1,2 for the fundamental properties of these spaces

Trang 2

L∞Ω

p : p ∈ L∞Ω, ess inf

x∈Ω p x > 1

For p ∈ L∞Ω, let p1 : ess infx∈Ωpx ≤ px ≤ p2 : ess supx∈Ω px < ∞, for a.e.

x ∈ Ω.

Let us define byUΩ the set of all measurable real functions defined on Ω For any

p ∈ L∞Ω, we define the variable exponent Lebesgue space by

L pxΩ

u ∈ U Ω : ρpxu 

Ω|ux| px dx < ∞

. 1.3

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

px inf

δ > 0 : ρ px

u δ

≤ 1

andL px

px becomes a Banach space

The variable exponent Sobolev space W 1,pxΩ is

W 1,pxΩ

u ∈ L pxΩ : ∂u

∂x i ∈ L px Ω, i 1, , N

1.5

and we define on this space the norm

for all u ∈ W 1,px Ω The space W 1,px

0 Ω is the closure of C∞

0 Ω in W 1,pxΩ

Proposition 1.1 see 1, 2 If p ∈ L∞

Ω, then the spaces L px Ω, W 1,px Ω, and

W01,px Ω are separable and reflexive Banach spaces.

Proposition 1.2 see 1,2 If u ∈ L px Ω and p2< ∞, then we have

px < 1 1; > 1 ⇔ ρ pxu < 1 1; > 1,

px

p1

px ≤ ρpx p2

px ,

px

p2

px ≤ ρpx p1

px ,

px a > 0 ⇔ ρpxu/a 1.

Trang 3

Proposition 1.3 see 3 Assume that Ω is bounded and smooth Denote by CΩ {h ∈ CΩ :

h1> 1}.

i Let p, q ∈ CΩ If

q x < p∗x 

⎧

⎪

Np x

N − p x if p x < N,

∞ if p x ≥ N,

1.7

then W 1,px

ii (Poincar´e inequality, see [1 , Theorem 2.7]) If p ∈ CΩ, then there is a constant C > 0

such that

px px , ∀u ∈ W 1,px

what follows, W01,px Ω, with p ∈ CΩ, will be considered as endowed with the norm

1,px

Lemma 1.4 Assume that r ∈ L∞

Ω and p ∈ CΩ If |u| rx ∈ L px Ω, then we have

min

r1

r xpx , r2

r xpx ≤|u| rx

px≤ max r1

r xpx , r2

Proof ByProposition 1.2iv, we have

1

Ω

|u| rx

px

dx

Ω

rxpx |u|

rxpx rxpx

r xpx

|u| rxpx

p x

dx

≤

Ω

rxpx |u|

rxpxmax r

1px

r xpx , r2px

r xpx

|u| rxpx

p x

dx.

1.10

By the mean value theorem, there exists ξ ∈ Ω such that

1≤ max

r

1pξ

r xpx , r2pξ

r xpx

|u| rxpξ

p x

Ω

|u| rxpx

rxpx

dx 1.11

Trang 4

and we have

|u| rx

px≤ max r1

r xpx , r2

Similarly

1≥ min

r

1pξ

rxpx , r2pξ

rxpx

|u| rxpξ

p x

dx,

|u| rx

px≥ min r1

r xpx , r2

r xpx

1.13

Remark 1.5 If rx r const., then

|u| r

px

r

rp x 1.14 For simplicity of notation, we write

X W01,px Ω, X∗W01,pxΩ∗, Y L qx Ω, Y∗ L qx Ω,

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 in X to the Dirichlet

problemP that is an adaptation of that used by Dinca et al for Dirichlet problems with

classical p-Laplacian 5 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 operator Nf Firstly, since qx < px < p∗x for all x ∈ Ω, X is compactly embedded in Y Denote by i the compact injection of X in Y and by i∗ : Y∗ → X∗, i∗υ υ ◦ i

for all υ ∈ Y∗, its adjoint

Since the Caratheodory function f satisfies CAR, the Nemytskii operator Nf generated by f, Nf ux fx, ux, is well defined from Y into Y∗, continuous, and bounded3, Proposition 2.2 In order to prove that problem P has a weak solution in

X it is suﬃcient to prove that the equation

−Δpxu

i∗N f i

has a solution in X.

Trang 5

Indeed, if u ∈ X satisfies 1.16 then, for all υ ∈ X, one has

−Δpxu, υ

X,X∗ i∗N f i

u, υ

X,X∗N f iu, iυY,Y∗ 1.17 which rewrites as

Ω|∇u| px ∇u∇υdx 

and tells us that u is a weak solution in X to problem P.

Since−Δpxis a homeomorphism of X onto X∗, 1.16 may be equivalently written as

u 

−Δpx−1i∗N f i

Thus, proving that problemP has a weak solution in X reduces to proving that the

compact operator

K 

has a fixed point

Theorem 1.6 Alternative of Leray-Schauder, 7 Let B0, R denote the closed ball in a Banach

i the equation λKu u has a solution in B0, R for λ 1 or

2 Main Results

In this work, we present new existence and localization results for X-solutions to problem P, underCAR condition on f 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

X / R for any

solution u ∈ X to

−Δpxu λf x, u in Ω,

Proof By 3, Theorem 3.1 , −Δpx is a homeomorphism of X onto X∗ We will apply

Theorem 2.1to E X and to operator K : X → X,

Ku 

Trang 6

where i∗N f i : X → X∗ is given byNf ux fx, ux Notice that, according to a

well-known regularity result4 , the operator −Δpx−1from X to X is well defined, continuous, and order preserving Consequently, K is a compact operator On the other hand, it is clear that the fixed points of K are the solutions of problem P Now the conclusion follows from

Theorem 1.6since conditionii is excluded by hypothesis

Theorem 2.2immediately yields the following existence and localization result

Theorem 2.2 Let Ω ⊂ R N , N ≥ 2, be a smooth bounded domain and let p, q ∈ CΩ be such that

qx < px for all x ∈ Ω Assume that f : Ω × R → R is a Caratheodory function which satisfies

the growth condition (CAR).

Suppose, in addition, that

C ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1 < 1, 2.2

where C is the constant appearing in condition (CAR) Let R ≥ 1 be a constant such that

R ≥

⎛

⎜ ∗ Y∗→ X∗ Y∗

1 ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1

⎞

⎟

1/p1−1

Proof Let u ∈ X be a solution of problem P λ X R ≥ 1, corresponding to some

λ ∈ 0, 1 Then by Propositions1.2,1.3, andLemma 1.4, we obtain

p1

X ≤

Ω|∇u| px

dx λ

i∗N f i

u, u

X,X∗ λN f iu, iuY,Y∗

∗

Y∗→ X∗N f iu

Y∗ X

∗

Y∗→ X∗ X

Y∗ C max q1−1

Y , q Y2−1

∗

Y∗→ X∗ X

Y∗ q2−1

X max q1−1

X → Y , q X → Y2−1

∗

Y∗→ X∗ X

Y∗ p1−1

X max q1−1

X → Y , q X → Y2−1 .

2.4

Therefore, we have

p1−1

X ≤ λ ∗ Y∗→ X∗ Y∗

1 ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1 2.5

Trang 7

Substituting X R in the above inequality, we obtain

R ≤

⎛

⎜ λ ∗ Y∗→ X∗ Y∗

1 ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1

⎞

⎟

1/p1−1

, 2.6

which, taking into account2.3 and λ ∈ 0, 1, gives

R ≤ λ 1/p1−1

⎛

⎜ ∗ Y∗→ X∗ Y∗

1 ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1

⎞

⎟

1/p1−1

≤ λ 1/p1−1

⎛

⎜ ∗ Y∗→ X∗ Y∗

1 ∗ Y∗→ X∗max q1−1

X → Y , q X → Y2−1

⎞

⎟

1/p1−1

≤ λ 1/p1−1 R < R,

2.7

a contradiction.Theorem 2.1applies

Acknowledgment

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

References

1 X Fan and D Zhao, “On the spaces L px Ω and W m,px Ω,” Journal of Mathematical Analysis and

Applications, vol 263, no 2, pp 424–446, 2001.

2 O Kov´aˇcik and J R´akosn´ık, “On spaces L px and W k,px ,” Czechoslovak Mathematical Journal, vol.

41116, no 4, pp 592–618, 1991

3 X.-L Fan and Q.-H Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear

Analysis: Theory, Methods & Applications, vol 52, no 8, pp 1843–1852, 2003.

4 G Dinca, “A fixed point method for the px-Laplacian,” Comptes Rendus Math´ematique, vol 347, no.

13-14, pp 757–762, 2009

5 G Dinca, P Jebelean, and J Mawhin, “Variational and topological methods for Dirichlet problems with

p-Laplacian,” Portugaliae Mathematica, vol 58, no 3, pp 339–378, 2001.

6 D O’Regan and R Precup, Theorems of Leray-Schauder Type and Applications, vol 3 of Series in

Mathematical Analysis and Applications, Gordon and Breach, Amsterdam, The Netherlands, 2001.

7 J Dugundji and A Granas, Fixed Point Theory I, vol 61 of Monografie Matematyczne, PWN-Polish

Scientific, Warsaw, Poland, 1982

p x

p x

dx

Ω

r x p x ... r x p x |u| 

r x p x r x p x

r x p x

 

|u| r x p x ... r x p x

p x

dx

≤

Ω

r x p x |u|

Định dạng
Số trang	7
Dung lượng	474,04 KB