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

Ruyun Ma, Guowei Dai* and Chenghua Gao

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

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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/16


Received:24 September 2011
Accepted:13 February 2012
Published:13 February 2012

© 2012 Ma et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

- M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x ( div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u ) = f ( x , u ) in Ω , u v = 0 on Ω .

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.

2000 Mathematical Subject Classification: 35D05; 35D10; 35J60.

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

1 Introduction

In this article we study the following problem

- M ( t ) ( div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u ) = f ( x , u ) in Ω , u v = 0 on Ω , ( P λ f )

where Ω is a bounded domain of ℝN with smooth boundary Ω and N ≥ 1, u v is the outer unit normal derivative, λ ∈ ℝ is a parameter, p = p ( x ) C 1 ( Ω ¯ ) with 1 < p-: = infΩ p(x) ≤ p+ := supΩ p(x) < +∞, f C ( Ω ¯ × , ) , M(t) is a function with Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x 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)-2u) := -Δp(x)u is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1-3]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [4,5]. Another field of application of equations with 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 refer the reader to [7-11] for an overview of and references on this subject, and to [12-16] for the study of the variable exponent equations and the corresponding variational problems.

The problem P λ 1 f is a generalization of the stationary problem of a model introduced by Kirchhoff [17]. More precisely, Kirchhoff proposed a model given by the equation

ρ 2 u t 2 - ρ 0 h + E 2 L 0 L u x 2 d x 2 u x 2 = 0 , (1.2)

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. A distinguishing feature of Equation (1.2) is that the equation contains a nonlocal coefficient ρ 0 h + E 2 L 0 L u x 2 d x which depends on the average 1 L 0 L u x 2 d x , and hence the equation is no longer a pointwise identity. The equation

- ( a + b Ω u 2 d x ) Δ u = f ( x , u ) in Ω , u = 0 on Ω (1.3)

is related to the stationary analogue of the Equation (1.2). Equation (1.3) received much attention only after Lions [18] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [19-22]. Moreover, nonlocal boundary value problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population 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 P λ 1 f 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 P λ 1 f 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 the methods used in this article are also suitable for the inhomogeneous Neumann boundary value problems.

In this article we use the following notations:

F ( x , t ) = 0 t f ( x , s ) d s ,

Λ = {λ ∈ ℝ: there exists at least a positive solution of P λ f },

λ * = inf Λ .

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 ) 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 , λ* ≥ 0 and *, +∞) ⊂ Λ. Moreover, for every λ > λ* problem P λ f has a minimal positive solution uλ in [0,w1], where w1 is the unique solution of P λ 0 and u λ 1 < u λ 2 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

f ( x , t ) c 1 + c 2 t q ( x ) - 1 , x Ω , t M , (1.6)

where q C ( Ω ¯ ) and 1 ≤ q(x) < p*(x) for x Ω ¯ , μ ∈ (0,1) such that

M ^ ( t ) ( 1 - μ ) M ( t ) t , (1.7)

where M ^ ( t ) = 0 t M ( τ ) d τ and M1 > 0, θ > p + 1 - μ such that

0 < θ F ( x , t ) t f ( x , t ) , x Ω , t M 1 . (1.8)

Then for each λ ∈ (λ*, +∞), P λ f has at least two positive solutions uλ and vλ, where uλ is a local minimizer of the energy functional and uλ vλ.

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

f ( x , 0 ) f ( x , t ) for t > 0 and x Ω (1.9)

and the following conditions:

f ( x , t ) c 3 + c 4 t r ( x ) - 1 , x Ω , t M 2 , (1.10)

where M2, c3 and c4 are positive constants, r C ( Ω ¯ ) and 1 ≤ r(x) < p(x) for x Ω ¯ . Then λ* = 0.

(2) If f satisfies (1.4)-(1.8), then λ* ∈ Λ.

Example 1.1. Let M(t) = a + bt, where a and b are positive constants. It is clear that

M ( t ) a > 0 .

Taking μ = 1 2 , we have

M ^ ( t ) = 0 t M ( s ) d s = a t + 1 2 b t 2 1 2 ( a + b t ) t = ( 1 - μ ) M ( t ) t .

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 special features 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 C1,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 ≠ 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, they cannot be applied to the Neumann problems. To prove Theorem 1.1, we establish a special strong comparison principle for the Neumann problem 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 general principle of sub-supersolution method for the problem P λ f based on the regularity results. In Section 4, we give the proof of Theorems 1.1-1.3.

2 Preliminaries

In order to discuss problem P λ f , we need some theories on W1,p(x) (Ω) which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces W1,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.

Write

C + ( Ω ¯ ) = { h : h C ( Ω ¯ ) , h ( x ) > 1 for any x Ω ¯ }

and

L p ( x ) ( Ω ) = u S ( Ω ) : Ω u ( x ) p ( x ) d x < +

with the norm

u L p ( x ) ( Ω ) = u p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 ,

and

W 1 , p ( x ) ( Ω ) = { u L p ( x ) ( Ω ) : u L p ( x ) ( Ω ) }

with the norm

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

Denote by W 0 1 , p ( x ) ( Ω ) the closure of C 0 ( Ω ) in W1,p(x) (Ω). The spaces Lp(x) (Ω), W1,p(x) (Ω) and W 0 1 , p ( x ) ( Ω ) are all separable Banach spaces. When p- > 1 these spaces are reflexive.

Let λ > 0. Define for u W1,p(x) (Ω),

u λ = inf σ > 0 : Ω u σ p ( x ) + λ u σ p ( x ) d x 1 .

Then ||u||λ is a norm on W1,p(x) (Ω) equivalent to u W 1 , p ( x ) ( Ω ) .

By the definition of ||u||λ we have the following

Proposition 2.1. [11,14]Put ρ λ ( u ) = Ω u p ( x ) + λ u p ( x ) d x for λ > 0 and u W1,p(x) (Ω). We have:

(1) u λ 1 u λ p - ρ λ ( u ) u λ p + ;

(2) u λ 1 u λ p + ρ λ ( u ) u λ p - ;

(3) lim k + u k λ = 0 lim k + ρ λ ( u k ) = 0 ( a s k + ) ;

(4) lim k + u k λ = + lim k + ρ λ ( u k ) = + ( a s k + ) .

Proposition 2.2. [14]If u, uk W1,p(x) (Ω), k = 1,2,..., then the following statements are equivalent each other:

(i) lim k + u k - u λ = 0 ;

(ii) lim k + ρ λ ( u k - u ) = 0 ;

(iii) uk u in measure in Ω and lim k + ρ λ ( u k ) = ρ ( u ) .

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 W1,p(x) (Ω) ↪ Lq(x) (Ω).

Proposition 2.4. [14]The conjugate space of Lp(x) (Ω) is Lq(x) (Ω), where 1 q ( x ) + 1 p ( x ) = 1 . For any u Lp(x) (Ω) and v Lq(x) (Ω), we have the following Hölder-type inequality

Ω u v d x 1 p - + 1 q - u p ( x ) v q ( x ) .

Now, we discuss the properties of p(x)-Kirchhoff-Laplace operator

Φ K ( u ) : = - M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x ( div u p ( x ) - 2 u - λ u p ( x ) - 2 u ) ,

where λ > 0 is a parameter. Denotes

Φ ( u ) : M ^ Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x . (2.2)

For simplicity we write X = W1,p(x) (Ω), denote by un u and un u the weak convergence and strong convergence of sequence {un} in X, respectively. It is obvious that the functional Φ is a Gâteaux differentiable whose Gâteaux derivative at the point u X is the functional Φ'(u) ∈ X*, given by

Φ ( u ) , v = M Ω 1 p ( x ) u p ( x ) + λ u p ( x ) d x Ω ( u p ( x ) - 2 u v + λ u p ( x ) - 2 u v ) d x , (2.3)

where 〈·, ·〉 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 properties about the derivative operator of Φ.

Proposition 2.5. If (M0) holds, then

(i) Φ': X X* is a continuous, bounded and strictly monotone operator;

(ii) Φ' is a mapping of type (S+), i.e., if un u in X and lim n + ¯ Φ ( u n ) - Φ ( u ) , u n - u 0 , then un u in X;

(iii) Φ'(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.

3 Sub-supersolution principle

In this section we give a general principle of sub-supersolution method for the problem P λ f based on the regularity results and the comparison principle.

Definition 3.1. u X is called a weak solution of the problem P λ f if for all v X,

M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x Ω ( u p ( x ) - 2 u v + λ u p ( x ) - 2 u v ) d x = Ω f ( x , u ) v d x .

In this article, we need the global regularity results for the weak solution of P λ f . Applying Theorems 4.1 and 4.4 of [44] and Theorem 1.3 of [37], we can easily get the following results involving of the regularity of weak solutions of P λ f .

Proposition 3.1. (1) If f satisfies (1.6), then u L(Ω) for every weak solution u of P λ f .

(2) Let u X L(Ω) be a solution of P λ f . If the function p is log-Hölder continuous on Ω ¯ , i.e., there is a positive constant H such that

p ( x ) - p ( y ) H - log x - y f o r x , y Ω ¯ w i t h 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 Hölder 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), applying Theorem 1.1 of [16], we have the following strong maximum principle.

Proposition 3.2. Suppose that p ( x ) C + ( Ω ¯ ) C 1 ( Ω ¯ ) , u X, u ≥ 0 and u 0 in Ω. If

- M ( t ) ( div ( u p ( x ) - 2 u ) - d ( x ) u p ( x ) - 2 u ) 0 ,

where t = Ω 1 p ( x ) u p ( x ) + 1 p ( x ) d ( x ) u p ( x ) d x , M(t) ≥ m0 > 0, 0 ≤ d(x) ∈ L(Ω), q ( x ) C ( Ω ¯ ) with p(x) ≤ q(x) ≤ p* (x), then u > 0 in Ω.

Definition 3.2. u X is called a subsolution (resp. supersolution) of P λ f if for all v X with v ≥ 0, u ≤ 0 (resp. ≥) on Ω and

M Ω 1 p ( x ) | u | p ( x ) + 1 p ( x ) λ | u | p ( x ) d x Ω u p ( x ) - 2 u v + λ u p ( x ) - 2 u v d x ( resp. ) Ω f ( x , u ) v d x .

Theorem 3.1. Let λ > 0 and q C ( Ω ¯ ) satisfies (2.1). Then for each h L q ( x ) q ( x ) - 1 ( Ω ) , the problem

- M Ω 1 p ( x ) u p ( x ) + 1 p ( x ) λ u p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = h ( x ) in Ω u v = 0 on Ω (3.3λ)

has a unique solution u X.

Proof. According to Propositions 2.3 and 2.4, ( f , v ) : = Ω f ( x ) v d x (for any v X) defines a continuous linear functional on X. Since Φ' is a homeomorphism, P λ f has a unique solution.

Let q C ( Ω ¯ ) satisfy (2.1). For h L q ( x ) q ( x ) - 1 ( Ω ) , we denote by K(h) = Kλ(h) = u the unique solution of (3.3λ). K = Kλ 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 ) q ( x ) - 1 ( Ω ) X is continuous and bounded. Moreover, the mapping K : L q ( x ) q ( x ) - 1 ( Ω ) L q ( x ) ( Ω ) is completely continuous since the embedding X Lq(x) (Ω) is compact.

(2) If p is log-Hölder 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 Hölder continuous on Ω ¯ , then the mapping K : L ( Ω ) C 1 , α ( Ω ¯ ) is bounded, and hence the mapping K : L ( Ω ) C 1 ( Ω ¯ ) is completely continuous.

Using the similar proof to [36], we have

Proposition 3.4. If h L q ( x ) q ( x ) - 1 ( Ω ) and h ≥ 0, where q C ( Ω ¯ ) satisfies (2.1), then K(h) ≥ 0. If p C1(Ω), h L(Ω) and h ≥ 0, then K(h) > 0 on Ω ¯ .

Now we give a comparison principle as follows.

Theorem 3.2. Let u, v X, φ W 0 1 , p ( x ) ( Ω ) . If

M I 0 ( u ) Ω u p ( x ) - 2 u φ + λ u p ( x ) - 2 u φ d x M I 0 ( v ) Ω v p ( x ) - 2 v φ + λ v p ( x ) - 2 v φ d x (3.4)

with φ ≥ 0 and u v on ∂Ω, I 0 ( u ) : = Ω 1 p ( x ) u p ( x ) + 1 p ( x ) λ u p ( x ) d x , then u v in Ω.

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

Φ ( u ) - Φ ( v ) , φ = M Ω u p ( x ) + λ u p ( x ) p ( x ) d x Ω u p ( x ) - 2 u φ + λ u p ( x ) - 2 u φ d x - M Ω v p ( x ) + λ v p ( x ) p ( x ) d x Ω v p ( x ) - 2 v φ + λ v p ( x ) - 2 v φ d x 0 .

Using the similar proof to Theorem 2.1 of [15] with obvious changes, we can show that

Φ ( u ) - Φ ( v ) , φ m 0 Ω 1 2 ( u p ( x ) - 2 - v p ( x ) - 2 ) ( u 2 - v 2 ) d x + m 0 λ Ω 1 2 ( u p ( x ) - 2 - v p ( x ) - 2 ) ( u 2 - v 2 ) d x 0 .

Therefore, we get 〈Φ'(u) - Φ'(v), φ〉 = 0. Proposition 2.5 implies that φ ≡ 0 or u v in Ω. It follows that u v in Ω.

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 P λ f respectively, and u0 v0. If f satisfies the condition:

f ( x , t ) is nondecreasing in t [ inf u 0 ( x ) , sup v 0 ( x ) ] , (3.5)

then P λ f 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 P λ f such that u0 u v0, then u* u v*.

The energy functional corresponding to P λ f is

J λ ( u ) = Φ ( u ) - Ω F ( x , u ) d x , u X . (3.6)

The critical points of Jλ are just the solutions of P λ f . Many authors, for example, Chang [46], Brezis and Nirenberg [47] and Ambrosetti et al. [48], have combined the sub-supersolution method with the variational method and studied successfully the semilinear elliptic problems, where a key lemma is that a local minimizer of the associated energy functional 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 is also valid for the Neumann problems. Thus we have the following

Theorem 3.4. Let λ > 0 and (1.6) holds. If u C 1 ( Ω ¯ ) is a local minimizer of Jλ in the C 1 ( Ω ¯ ) -topology, then u is also a local minimizer of Jλ in the X-topology.

4 Proof of theorems

In this section we shall prove Theorems 1.1-1.3. Since only the positive solutions are considered, 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

f ( + ) ( x , t ) = f ( x , t ) if t 0 , f ( x , 0 ) if t < 0 .

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 P λ f . Taking v ≡ 1 as a test function in Definition 3.1. (1) yields

M Ω λ p ( x ) u p ( x ) d x λ Ω u p ( x ) - 1 d x = Ω f ( x , u ) d x , (4.1)

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 .

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

- M Ω 1 p ( x ) ( u p ( x ) + u p ( x ) ) d x div ( u p ( x ) - 2 u ) - u p ( x ) - 2 u = 0 in Ω u ν = 0 on Ω (4.2)

has a unique positive solution w 1 C 1 ( Ω ¯ ) and w1(x) ≥ ε > 0 for x Ω ¯ . We can assume ε ≤ 1. Put d = sup { f ( x , w 1 ( x ) ) : x Ω ¯ } , M 3 = d m 0 ε p + - 1 and λ1 = 1 + M3. Then

- M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x Δ p ( x ) w 1 - λ 1 w 1 p ( x ) - 1 = - M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x Δ p ( x ) w 1 - w 1 p ( x ) - 1 + M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x M 3 w 1 p ( x ) - 1 m 0 M 3 ε p + - 1 = d f ( x , w 1 ( x ) ) .

This shows that w1 is a supersolution of the problem P λ 1 f . Obviously 0 is a subsolution of P λ 1 f . By Theorem 3.3, P λ 1 f has a solution u λ 1 such that 0 u λ 1 w 1 . By Proposition 3.4, u λ 1 > 0 on Ω ¯ . So λ1 ∈ Λ and Λ .

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 λ 0 be a positive solution of P λ 0 f . Then, we have

- Δ p ( x ) u λ 0 + λ u λ 0 p ( x ) - 1 - Δ p ( x ) u λ 0 + λ 0 u λ 0 p ( x ) - 1 = f ( x , u λ 0 ) M Ω 1 p ( x ) ( u λ 0 p ( x ) + λ 0 u λ 0 p ( x ) ) d x f ( x , u λ 0 ) M Ω 1 p ( x ) ( u λ 0 p ( x ) + λ u λ 0 p ( x ) ) d x

thanks to (M0). This shows that u λ 0 is a supersolution of P λ f . We know that 0 is a subsolution of P λ f By Theorem 3.3, P λ f has a solution uλ such that 0 u λ u λ 0 . By Proposition 3.4, uλ > 0 on Ω ¯ . Thus λ ∈ Λ.

Lemma 4.4. Let (1.4) and (1.5) hold. Then for every λ > λ*, there exists a minimal positive solution uλ of P λ f such that u λ 1 u λ 2 if λ* < λ2 < λ1.

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 λ 1 and u λ 2 are the positive solutions of P λ 1 f and P λ 2 f respectively and u λ 1 u λ 2 . Then there exists a positive solution vλ of P λ f such that u λ 1 v λ u λ 2 and vλ is a global minimizer of the restriction of Jλ to the order interval u λ 1 , u λ 2 X .

Proof. Define f ̃ : Ω ¯ × by

f ̃ ( x , t ) = f ( x , u λ 1 ( x ) ) , if t < u λ 1 ( x ) f ( x , t ) , if u λ 1 ( x ) t u λ 2 ( x ) f ( x , u λ 2 ( x ) ) , if t > u λ 2 ( x ) .

Define F ̃ ( x , t ) = 0 t f ̃ ( x , s ) d s and for all u X,

J ̃ λ ( u ) = M ^ Ω u p ( x ) + λ u p ( x ) p ( x ) d x - Ω F ̃ ( x , u ) d x .

It is easy to see that the global minimum of J ̃ on X is achieved at some vλ X. Thus vλ is a solution of the following problem

- M Ω u p ( x ) + λ u p ( x ) p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = f ̃ ( x , u ) in Ω u ν = 0 on Ω (4.3)

and v λ C 1 ( Ω ¯ ) . Noting that

f x , u λ 1 = f ̃ x , u λ 1 f ̃ ( x , v λ ) f ¯ ( x , u λ 2 ) = f ( x , u λ 2 )

and λ2 < λ < λ1, since K is increasing operator, we obtain that u λ 1 v λ u λ 2 . So f ̃ ( x , v λ ) = f ( x , v λ ) , and vλ is a positive solution of P λ f . It is easy to see that there exists a constant c such that J λ ( u ) J ̃ λ ( u ) + c for u u λ 1 , u λ 2 X . Hence vλ is a global minimizer of J λ | u λ 1 , u λ 2 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 λ 1 and u λ 2 are the positive solutions of ( 1 . 1 λ 1 ) and ( 1 . 1 λ 2 ) respectively. Then u λ 1 < u λ 2 on Ω ¯ .

Proof. Since u λ 1 , u λ 2 C 1 ( Ω ¯ ) and u λ 1 > 0 on Ω ¯ , in view of Lemma 4.4, there exist two positive constants b1 ≤ 1 and b2 such that

b 1 u λ 1 u λ 2 b 2 on Ω ¯ .

For ε 0 , b 1 2 , setting v ε = u λ 2 - ε , then

- M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( v ε p ( x ) - 2 v ε ) - λ 1 v ε p ( x ) - 1 ) = - M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( u λ 2 p ( x ) - 2 u λ 2 ) - λ 2 v ε p ( x ) - 1 + ( λ 2 - λ 1 ) v ε p ( x ) - 1 ) = - M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( u λ 2 p ( x ) - 2 u λ 2 ) - λ 2 u λ 2 p ( x ) - 1 ) + M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( λ 1 - λ 2 ) v ε p ( x ) - 1 - λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 f ( x , u λ 2 ) M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x M Ω u λ 2 p ( x ) + λ 2 u λ 2 p ( x ) p ( x ) d x + M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( λ 1 - λ 2 ) b 1 2 p + - 1 - λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 .

Taking an ε > 0 sufficiently small such that

λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 < ( λ 1 - λ 2 ) b 1 2 p + - 1 for x Ω ¯

and

λ 1 Ω 1 p ( x ) v ε p ( x ) d x λ 2 Ω 1 p ( x ) u λ 2 p ( x ) d x ,

then

- M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( d i v ( v ε p ( x ) - 2 v ε ) - λ 1 v ε p ( x ) - 1 ) = g ( x ) f ( x , u λ 2 ) ,

consequently, vε is a solution of the problem

- M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x div( v ε p ( x ) - 2 v ε ) -  λ 1 v ε p ( x ) - 1 = g ( x ) in Ω u ν = 0 on Ω ,

where g ( x ) f ( x , u λ 2 ) . With other words, v ε = K λ 1 ( g ) , where K λ 1 is the solution operator of ( 3 . 1 λ 1 ) . Since u λ 1 = K λ 1 ( h ) , where h ( x ) = f ( x , u λ 1 ) f ( x , u λ 2 ) g ( x ) , noting that K λ 1 is increasing, we have v ε u λ 1 , that is, u λ 2 - ε u λ 1 on Ω ¯ .

The proof of Theorem 1.1 is complete. Let us now turn to the proof of Theorem 1.2.

Proof of Theorem 1.2. Let (1.4)-(1.8) hold. Let λ > λ*. Take λ1, λ2 ∈ Λ such that λ2 < λ < λ1 and let u λ 1 u λ u λ 2 be as in Lemma 4.5.

We claim that uλ is a local minimizer of Jλ in the X-topology.

Indeed, Lemma 4.6 implies that u λ 1 < u λ < u λ 2 on Ω ¯ . It follows that there is a C0-neighborhood U of uλ such that U [ u λ 1 , u λ 2 ] , consequently uλ is a local minimizer of Jλ in the C0-topology, and of course, also in the C1-topology. By Theorem 3.4, uλ is also a local minimizer of Jλ in the X-topology.

Define

f ̃ λ ( x , t ) = f ( x , t ) , if t > u λ ( x ) , f ( x , u λ ( x ) ) , if t u λ ( x ) ,

and F ̃ λ ( x , t ) = 0 t f ̃ λ ( x , s ) d s . Consider the problem

- M Ω u p ( x ) + λ u p ( x ) p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = f ̃ λ ( x , u ) in Ω u ν = 0 on Ω

and denote by J ̃ λ the energy functional corresponding to (4.4λ). By the definition of f ̃ λ , we have f ̃ λ ( x , u ( x ) ) f ( x , u λ ( x ) ) for every u X. Hence, for each solution u of (4.4λ), we have that u uλ, consequently f ̃ λ ( x , u ) = f ( x , u ) and u is also a solution of P λ f . It is easy to see that u λ 1 and u λ 2 are a subsolution and a supersolution of (4.4λ) respectively. By Theorems 3.3 and 1.2, there exists u λ * [ u λ 1 , u λ 2 ] C 1 ( Ω ¯ ) such that u λ * is a solution of (4.4λ) and is a local minimizer of J ̃ λ in the C1-topology. As was noted above, we know that u λ * u λ and u λ * is also a solution of P λ f . If u λ * u λ , then the assertion of Theorem 1.2 already holds, hence we can assume that u λ * = u λ . Now uλ is a local minimizer of J ̃ λ in the C1-topology, and so also in the X-topology. We can assume that uλ is a strictly local minimizer of J ̃ λ in the X-topology, otherwise we have obtained the assertion of Theorem 1.2. It is easy to verify that, under the assumptions of Theorem 1.3, J ̃ λ C 1 ( X , ) and J ̃ λ satisfies the (P.S.) condition (see e.g., [30]). It follows from the condition (1.7) and (1.8) that { J ̃ λ ( u ) : u X } = - (see e.g., [30]). Using the mountain pass lemma (see [51]), we know that (4.4λ) has a solution vλ such that vλ uλ. vλ, as a solution of (4.4λ), must satisfy vλ uλ, and vλ is also a solution of P λ f . The proof of Theorem 1.2 is complete.

Proof of Theorem 1.3. (1) Let f satisfy (1.4), (1.9), and (1.10). For given any λ > 0, consider the energy functional Jλ defined by (3.3). By (1.10) and noting that r(x) < p(x) for x Ω ¯ , there is a positive constant M4 such that

F ( x , t ) λ m 0 2 p + t p ( x ) , x Ω , t M 4 . (4.5)

For u X with ||u||λ ≥ 1, we have that

J λ ( u ) m 0 p + Ω p ( x ) + λ u p ( x ) d x - λ m 0 2 p + Ω u p ( x ) d x - c 5 m 0 p + Ω u p ( x ) d x + λ m 0 2 p + Ω u p ( x ) d x - c 5 m 0 2 p + u λ p - - c 5 ,

where c5 is a positive constant. This shows that Jλ(u) → +∞ as ||u||λ → +∞, that is, Jλ is coercive. In view of Proposition 2.5. (iv), the condition (1.10) also implies that Jλ is weakly sequentially lower semi-continuous. Thus Jλ has a global minimizer u0. Put v0(x) = |u0(x)| for x Ω ¯ . It is easy to see that Jλ(v0) ≤ Jλ(u0), consequently, v0 is a global minimizer of Jλ and is a positive solution of P λ f . This shows that λ ∈ Λ for all λ > 0. Hence λ* = 0 and the statement (1) is proved.

To prove Theorem 1.3. (2) we give the following lemma.

Lemma 4.7. Let (1.4) and (1.5) hold. Then for each λ > λ*, P λ f has a positive solution uλ such that Jλ(uλ) ≤ 0.

Proof. Let λ > λ*. Take λ2 ∈ (λ*, λ) and let u λ 2 be a positive solution of P λ 2 f . then u λ 2 is a supersolution of P λ f . We know that 0 is a subsolution of P λ f . Analogous to the proof of Lemma 4.5, we can prove that P λ f has a positive solution u λ [ 0 , u λ 2 ] such that J λ ( u λ ) = inf { J λ ( u ) : u [ 0 , u λ 2 ] } . So Jλ(uλ) ≤ Jλ(0) = 0.

Proof of Theorem 1.3. (2). Let (1.4)-(1.8) hold. Let λn > λ* and λn → λ* as n → +∞. By Lemma 4.7, for each n, P λ n f has a positive solution u λ n such that J λ n ( u λ n ) 0 , that is

M ^ Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω F ( x , u λ n ) d x .

Since u λ n is a solution of P λ n f , we have that

M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω u λ n p ( x ) + λ n u λ n p ( x ) d x = Ω f ( x , u λ n ) u λ n d x .

It follows from (1.8) that there exists a positive constant c6 such that

Ω F ( x , u λ n ) d x c 6 + 1 θ Ω f ( x , u λ n ) u λ n d x .

Thus, using condition (1.7), we have that

1 - μ p + M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω ( u λ n p ( x ) + λ n u λ n p ( x ) ) d x c 6 + 1 θ Ω f ( x , u λ n ) u λ n d x c 6 + 1 θ M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω ( u λ n p ( x ) + λ n u λ n p ( x ) ) d x ,

and consequently,

m 0 1 - μ p + - 1 θ u λ n λ n p - c 7 ,

where the positive constant c7 is independent of n. This shows that { u λ n λ n } is bounded. Noting that λn → λ* > 0, we have that { u λ n } is bounded. Without loss of generality, we can assume that u λ n u * in X and u λ n ( x ) u * ( x ) for a.e. x ∈ Ω. By (1.6) and the L(Ω)-regularity results of [44], the boundedness of { u λ n } implies the boundedness of u λ n L ( Ω ) . By the C 1 , α ( Ω ¯ ) -regularity results of [37], the boundedness of u λ n L ( Ω ) implies the boundedness of u λ n C 1 , α ( Ω ¯ ) , where α ∈ (0, 1) is a constant. Thus we have u λ n u * in C 1 ( Ω ¯ ) . For every v X, since u λ n is a solution of P λ n f , we have that, for each n,

M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω u λ n p ( x ) - 2 u λ n v + λ n u λ n p ( x ) - 2 u λ n v d x = Ω f ( x , u λ n ) v d x

Passing the limit of above equality as n → +∞, yields

M Ω u * p ( x ) + λ * u * p ( x ) p ( x ) d x Ω u * p ( x ) - 2 u * v + λ * u * p ( x ) - 2 u * v d x = Ω f ( x , u * ) v d x ,

which shows that u* is a solution of P λ * f . Obviously u* ≥ 0 and u * 0 . Hence u* is a positive solution of P λ * f and λ* ∈ Λ.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030 and 10971087), NWNU-LKQN-10-21.

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