báo cáo hóa học:" Existence and multiplicity of positive solutions for a class of p(x)-Kirchho type equations" ppt

China ∗Corresponding author: daiguowei@nwnu.edu.cn Email address: RM: mary@nwnu.edu.cnCG: gaokuguo@163.com Abstract In this article, we study the existence and multiplicity of positive s

This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted

PDF and full text (HTML) versions will be made available soon

Existence and multiplicity of positive solutions for a class of p(x)-Kirchho type

equations

Boundary Value Problems 2012, 2012:16 doi:10.1186/1687-2770-2012-16

Ruyun Ma (mary@nwnu.edu.cn)Guowei Dai (daiguowei2009@126.com)Chenghua Gao (gaokuguo@163.com)

ISSN 1687-2770

Article type Research

Submission date 24 September 2011

Acceptance date 13 February 2012

Publication date 13 February 2012

Article URL http://www.boundaryvalueproblems.com/content/2012/1/16

This peer-reviewed article was published immediately upon acceptance It can be downloaded,

printed and distributed freely for any purposes (see copyright notice below)

For information about publishing your research in Boundary Value Problems go to

http://www.boundaryvalueproblems.com/authors/instructions/

For information about other SpringerOpen publications go to

http://www.springeropen.com

Boundary Value Problems

© 2012 Ma et al ; licensee Springer.

Existence and multiplicity of positive solutions for a

class of p(x)-Kirchhoff type equations

Ruyun Ma, Guowei Dai∗ and Chenghua Gao

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P R China

∗Corresponding author: daiguowei@nwnu.edu.cn

Email address:

RM: mary@nwnu.edu.cnCG: gaokuguo@163.com

Abstract

In this article, we study the existence and multiplicity of positive solutions for the

Neumann boundary value problems involving the p(x)-Kirchhoff of the form

Using the sub-supersolution method and the variational method, under appropriate

assumptions on f and M , we prove that there exists λ ∗ > 0 such that the problem

has at least two positive solutions if λ > λ ∗ , at least one positive solution if λ = λ ∗

and no positive solution if λ < λ ∗ To prove these results we establish a special strong

comparison principle for the Neumann problem

Keywords: p(x)-Kirchhoff; positive solution; sub-supersolution method; comparison

dx and satisfies the following condition:

(M0) M(t) : [0, +∞) → (m0, +∞) is a continuous and increasing function with m0 > 0.

The operator −div(|∇u| p(x)−2 ∇u) := −∆ p(x) u is said to be the p(x)-Laplacian, and

be-comes p-Laplacian when p(x) ≡ p (a constant) The p(x)-Laplacian possesses more cated nonlinearities than the p-Laplacian; for example, it is inhomogeneous The study of

compli-various mathematical problems with variable exponent growth condition has been receivedconsiderable attention in recent years These problems are interesting in applications andraise many difficult mathematical problems One of the most studied models leading toproblem of this type is the model of motion of electrorheological fluids, which are character-ized by their ability to drastically change the mechanical properties under the influence of

an exterior electromagnetic field [1–3] Problems with variable exponent growth conditionsalso appear in the mathematical modeling of stationary thermo-rheological viscous flows ofnon-Newtonian fluids and in the mathematical description of the processes filtration of anideal barotropic gas through a porous medium [4,5] Another field of application of equationswith variable exponent growth conditions is image processing [6] The variable nonlinearity

is used to outline the borders of the true image and to eliminate possible noise We referthe reader to [7–11] for an overview of and references on this subject, and to [12–16] for thestudy of the variable exponent equations and the corresponding variational problems.The problem

³

P λ f1

´

is a generalization of the stationary problem of a model introduced

by Kirchhoff [17] More precisely, Kirchhoff proposed a model given by the equation

where ρ, ρ0, h, E, L are constants, which extends the classical D’Alembert’s wave equation,

by considering the effect of the changing in the length of the string during the vibration Adistinguishing feature of Equation (1.2) is that the equation contains a nonlocal coefficient

no longer a pointwise identity The equation

systems where u describes a process which depends on the average of itself, such as the

pop-ulation density [23–26] The study of Kirchhoff type equations has already been extended to

the case involving the p-Laplacian (for details, see [27–29]) and p(x)-Laplacian (see [30–33]) Many authors have studied the Neumann problems involving the p-Laplacian, see e.g.,

[34–36] and the references therein In [34,35] the authors have studied the problem

³

P λ f1

´in

the cases of p(x) ≡ p = 2, M(t) ≡ 1 and of p(x) ≡ p > 1, M(t) ≡ 1, respectively In [36], Fan and Deng studied the Neumann problems with p(x)-Laplacian, with the nonlinear potential

f (x, u) under appropriate assumptions By using the sub-supersolution method and variation

method, the authors get the multiplicity of positive solutions of

³

P λ f1

´

with M(t) ≡ 1 The aim of the present paper is to generalize the main results of [34–36] to the p(x)-Kirchhoff

case For simplicity we shall restrict to the 0-Neumann boundary value problems, but themethods used in this article are also suitable for the inhomogeneous Neumann boundaryvalue problems

In this article we use the following notations:

The main results of this article are the following theorems Throughout the article we

always suppose that the condition (M0) holds

Theorem 1.1 Suppose that f satisfies the following conditions:

f (x, t) ≥ 0, f (x, t) 6≡ 0 ∀x ∈ Ω, ∀t ≥ 0 (1.4)

and

for each x ∈ Ω, f (x, t) is nondecreasing with respect to t ≥ 0. (1.5)

Then Λ 6= ∅, λ ∗ ≥ 0 and (λ ∗ , +∞) ⊂ Λ Moreover, for every λ > λ ∗ problem

³

P λ f

´

has a minimal positive solution u λ in [0, w1], where w1 is the unique solution of (P0

λ ) and u λ1 < u λ2

if λ ∗ < λ2 < λ1

Theorem 1.2 Under the assumptions of Theorem 1.1, also suppose that there exist positive

constants M, c1 and c2 such that

has at least two positive solutions u λ and v λ , where u λ

is a local minimizer of the energy functional and u λ ≤ v λ

Theorem 1.3 (1) Suppose that f satisfies (1.4),

So the conditions (M0) and (1.7) are satisfied.

The underlying idea for proving Theorems 1.1–1.3 is similar to the one of [36] The specialfeatures of this class of problems considered in the present article are that they involve the

nonlocal coefficient M(t) To prove Theorems 1.1–1.3, we use the results of [37] on the global

C 1,α regularity of the weak solutions for the p(x)-Laplacian equations The main method

used in this article is the sub-supersolution method for the Neumann problems involving the

p(x)-Kirchhoff A main difficulty for proving Theorem 1.1 is that a special strong comparison

principle is required It is well known that, when p 6= 2, the strong comparison principles for the p-Laplacian equations are very complicated (see e.g [38–41]) In [13, 42, 43] the required

strong comparison principles for the Dirichlet problems have be established, however, theycannot be applied to the Neumann problems To prove Theorem 1.1, we establish a specialstrong comparison principle for the Neumann problem

³

P λ f

´(see Lemma 4.6 in Section 4),which is also valid for the inhomogeneous Neumann boundary value problems

In Section 2, we give some preliminary knowledge In Section 3, we establish a generalprinciple of sub-supersolution method for the problem ³P λ f´based on the regularity results.

In Section 4, we give the proof of Theorems 1.1–1.3

, we need some theories on W 1,p(x)(Ω) which we call

variable exponent Sobolev space Firstly we state some basic properties of spaces W 1,p(x)(Ω)which will be used later (for details, see [17]) Denote by S(Ω) the set of all measurable real

functions defined on Ω Two functions in S(Ω) are considered as the same element of S(Ω)

when they are equal almost everywhere

|u| L p(x)(Ω) = |u| p(x) = inf

W 1,p(x)(Ω) =©u ∈ L p(x) (Ω) : |∇u| ∈ L p(x)(Ω)ªwith the norm

kuk = kuk W 1,p(x)(Ω) = |u| L p(x)(Ω)+ |∇u| L p(x)(Ω).

Denote by W01,p(x) (Ω) the closure of C ∞

0 (Ω) in W 1,p(x) (Ω) The spaces L p(x) (Ω) , W 1,p(x)(Ω)

and W01,p(x) (Ω) are all separable Banach spaces When p − > 1 these spaces are reflexive.

Let λ > 0 Define for u ∈ W 1,p(x) (Ω) ,

Then kuk λ is a norm on W 1,p(x) (Ω) equivalent to kuk W 1,p(x)(Ω).

By the definition of kuk λ we have the following

Proposition 2.1 [11, 14] Put ρ λ (u) = RΩ

Proposition 2.2 [14] If u, u k ∈ W 1,p(x) (Ω), k = 1, 2, , then the following statements are

equivalent each other:

Proposition 2.3 [14] Let p ∈ C(Ω) If q ∈ C(Ω) satisfies the condition

1 ≤ q(x) < p ∗ (x), ∀x ∈ Ω, (2.1)

then there is a compact embedding W 1,p(x) (Ω) ,→ L q(x) (Ω).

Proposition 2.4 [14] The conjugate space of L p(x) (Ω) is L q(x) (Ω), where 1

¡div|∇u| p(x)−2 ∇u − λ|u| p(x)−2 u¢,

where λ > 0 is a parameter Denotes

Φ(u) := c M



Z

For simplicity we write X = W 1,p(x) (Ω), denote by u n * u and u n → u the weak

convergence and strong convergence of sequence {u n } in X, respectively It is obvious that

the functional Φ is a Gˆateaux differentiable whose Gˆateaux derivative at the point u ∈ X is

the functional Φ0 (u) ∈ X ∗, given by

hΦ 0 (u), vi = M



Z

Ω

¡

|∇u| p(x)−2 ∇u∇v + λ|u| p(x)−2 uv¢ dx,

(2.3)

where h·, ·i is the duality pairing between X and X ∗ Therefore, the p(x)-Kirchhoff–Laplace

operator is the derivative operator of Φ in the weak sense We have the following propertiesabout the derivative operator of Φ

Proposition 2.5 If (M0) holds, then

(i) Φ0 : X → X ∗ is a continuous, bounded and strictly monotone operator;

(ii) Φ0 is a mapping of type (S+), i.e., if u n * u in X and lim

n→+∞ hΦ 0 (u n )−Φ 0 (u), u n −ui ≤

0, then u n → u in X;

(iii) Φ0 (u) : X → X ∗ is a homeomorphism;

(iv) Φ is weakly lower semicontinuous.

Proof Applying the similar method to prove [15, Theorem 2.1], with obvious changes,

we can obtain the conclusions of this proposition

based on the regularity results and the comparison principle

Definition 3.1 u ∈ X is called a weak solution of the problem

³

P λ f

´

if for all v ∈ X,

P λ f

´

Proposition 3.1 (1) If f satisfies (1.6), then u ∈ L ∞ (Ω) for every weak solution u of

If the function p is log-H¨older continuous

on Ω, i.e., there is a positive constant H such that

|p(x) − p(y)| ≤ H

− log |x − y| f or x, y ∈ Ω with |x − y| ≤

1

2, (3.2)

then u ∈ C 0,α (Ω) for some α ∈ (0, 1).

(3) If in (2), the condition (3.2) is replaced by that p is H¨older continuous on Ω, then

u ∈ C 1,α (Ω) for some α ∈ (0, 1).

For u, v ∈ S(Ω), we write u ≤ v if u(x) ≤ v(x) for a.e x ∈ Ω In view of (M0), applyingTheorem 1.1 of [16], we have the following strong maximum principle

Proposition 3.2 Suppose that p(x) ∈ C+(Ω) ∩ C1(Ω), u ∈ X, u ≥ 0 and u 6≡ 0 in Ω If

−M (t)¡div¡|∇u| p(x)−2 ∇u¢− d(x)|u| p(x)−2 u¢≥ 0,

has a unique solution u ∈ X.

Proof According to Propositions 2.3 and 2.4, (f, v) :=RΩf (x)v dx (for any v ∈ X) defines

a continuous linear functional on X Since Φ 0 is a homeomorphism,

³

P λ f

´has a uniquesolution

Let q ∈ C(Ω) satisfy (2.1) For h ∈ L q(x)−1 q(x) (Ω), we denote by K(h) = K λ (h) = u the

unique solution of (3.3λ ) K = K λ is called the solution operator for (3.3λ) From the

regularity results and the embedding theorems we can obtain the properties of the solution

operator K as follows ¤

Proposition 3.3 (1) The mapping K : L q(x)−1 q(x) (Ω) → X is continuous and bounded

More-over, the mapping K : L q(x)−1 q(x) (Ω) ,→ L q(x) (Ω) is completely continuous since the embedding

X ,→ L q(x) (Ω) is compact.

(2) If p is log-H¨older continuous on Ω, then the mapping K : L ∞ (Ω) → C 0,α (Ω) is

bounded, and hence the mapping K : L ∞ (Ω) → C(Ω) is completely continuous.

(3) If p is H¨older continuous on Ω, then the mapping K : L ∞ (Ω) → C 1,α (Ω) is bounded,

and hence the mapping K : L ∞ (Ω) → C1(Ω) is completely continuous.

Using the similar proof to [36], we have

Proposition 3.4 If h ∈ L q(x)−1 q(x) (Ω) and h ≥ 0, where q ∈ C(Ω) satisfies (2.1), then

Proof Taking ϕ = (u − v)+ as a test function in (3.4), we have

hΦ 0 (u) − Φ 0 (v), ϕi = M



Z

Ω

12

Ω

12

It follows from Theorem 3.2 that the solution operator K is increasing under the condition

(M0), that is, K(u) ≤ K(v) if u ≤ v.

In this article we will use the following sub-supersolution principle, the proof of which

is based on the well known fixed point theorem for the increasing operator on the order

interval (see e.g., [45]) and is similar to that given in [12] for Dirichlet problems involving

the p(x)-Laplacian.

Theorem 3.3 (A sub-supersolution principle) Suppose that u0,v0 ∈ X ∩ L ∞ (Ω), u0 and

v0 are a subsolution and a supersolution of

has a minimal solution u ∗ and a maximal solution v ∗ in the order interval

[u0, v0], i.e., u0 ≤ u ∗ ≤ v ∗ ≤ v0 and if u is any solution of

³

P λ f

´

such that u0 ≤ u ≤ v0, then u ∗ ≤ u ≤ v ∗

The energy functional corresponding to

³

P λ f

´is

tional in the C1-topology is also a local minimizer in the H1-topology Such lemma have

been extended to the case of the p-Laplacian equations (see [43, 49]) and also to the case

of the p(x)-Laplacian equations (see [12, Theorem 3.1]) In [50], Fan extended the Brezis– Nirenberg type theorem to the case of the p(x)-Kirchhoff [50, Theorem 1.1] The Theorem

1.1 of [50] concerns with the Dirichlet problems, but the method for proving the theorem isalso valid for the Neumann problems Thus we have the following

Theorem 3.4 Let λ > 0 and (1.6) holds If u ∈ C1(Ω) is a local minimizer of J λ in the

C1(Ω)-topology, then u is also a local minimizer of J λ in the X-topology.

4 Proof of theorems

In this section we shall prove Theorems 1.1–1.3 Since only the positive solutions areconsidered, without loss of generality, we can assume that

f (x, t) = f (x, 0) for t < 0 and x ∈ Ω, otherwise we may replace f (x, t) by f(+)(x, t), where

The proof of Theorem 1.1 consists of the following several Lemmata 4.1–4.6

Lemma 4.1 Let (1.4) hold Then λ > 0 if λ ∈ Λ.

Proof Let λ ∈ Λ and u be a positive solution of



Z

Ω

λ p(x) |u|

which implies λ > 0 because the value of the right side in (4.1) is positive ¤

Lemma 4.2 Let (1.4) and (1.5) hold Then Λ 6= ∅.

Proof By Theorem 3.1, Propositions 3.4 and 3.3 (3), the problem

3.4, u λ1 > 0 on Ω So λ1 ∈ Λ and Λ 6= ∅ ¤

Lemma 4.3 Let (1.4) and (1.5) hold If λ0 ∈ Λ, then λ ∈ Λ for all λ > λ0.

Proof Let λ0 ∈ Λ and λ > λ0 Let u λ0 be a positive solution of

³

P λ f0

´ Then, we have

³

P λ f

´ By Theorem 3.3,

Proof The proof is similar to [36, Lemma 3.4], we omit it here ¤

Lemma 4.5 Let (1.4) and (1.5) hold Let λ1, λ2 ∈ Λ and λ2 < λ < λ1 Suppose that u λ1and u λ2 are the positive solutions of

of the restriction of J λ to the order interval [u λ1, u λ2]∩X.

It is easy to see that the global minimum of eJ on X is achieved at some v λ ∈ X Thus v λ is

a solution of the following problem

constant c such that J λ (u) = e J λ (u) + c for u ∈ [u λ1, u λ2] ∩ X Hence v λ is a global minimizer

of J λ | [u λ1 ,u λ2 ]∩X ¤

A key lemma of this paper is the following strong comparison principle

Lemma 4.6 (A strong comparison principle) Let (1.4) and (1.5) hold Let λ1, λ2 ∈ Λ and λ2 < λ1 Suppose that u λ1 and u λ2 are the positive solutions of (1.1 λ1) and (1.1 λ2) respectively Then u λ1 < u λ2 on Ω.

Proof Since u λ1, u λ2 ∈ C1(Ω) and u λ1 > 0 on Ω, in view of Lemma 4.4, there exist two

positive constants b1 ≤ 1 and b2 such that

b1 ≤ u λ1 ≤ u λ2 ≤ b2 on Ω.

For ε ∈ (0, b1

2), setting v ε = u λ2 − ε, then