Open Access

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

Boundary Value Problems20122012:16

DOI: 10.1186/1687-2770-2012-16

Received: 24 September 2011

Accepted: 13 February 2012

Published: 13 February 2012

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 N with smooth boundary Ω and N ≥ 1, u v is the outer unit normal derivative, λ 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 [13]. 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 [711] for an overview of and references on this subject, and to [1216] 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 [1922]. 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 [2326]. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [2729]) and p(x)-Laplacian (see [3033]).

Many authors have studied the Neumann problems involving the p-Laplacian, see e.g., [3436] 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 [3436] 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 : 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 *, +∞), P λ f has at least two positive solutions uλ and vλ, where uλ is a local minimizer of the energy functional and uλvλ.

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,αregularity 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. [3841]). 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, u k W1,p(x)(Ω), k = 1,2,..., then the following statements are equivalent each other:
  1. (i)

    lim k + u k - u λ = 0 ;

     
  2. (ii)

    lim k + ρ λ ( u k - u ) = 0 ;

     
  3. (iii)

    u k 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 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 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
  1. (i)

    Φ': XX* is a continuous, bounded and strictly monotone operator;

     
  2. (ii)

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

     
  3. (iii)

    Φ'(u): XX* is a homeomorphism;

     
  4. (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 .
  1. (2)
    Let u XL (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).
  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 uv 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.
  1. (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.

     
  2. (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 uv on ∂ Ω, I 0 ( u ) : = Ω 1 p ( x ) u p ( x ) + 1 p ( x ) λ u p ( x ) d x , then uv 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 uv in Ω. It follows that uv 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 uv.

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 XL(Ω), u0 and v0 are a subsolution and a supersolution of P λ f respectively, and u0v0. 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., u0u*v*v0 and if u is any solution of P λ f such that u0uv0, then u*uv*.

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, v ε 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 uuλ, 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 vλuλ. vλ, as a solution of (4.4λ), must satisfy vλuλ, 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) 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 λ n > λ* and λ n → 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 λ n → 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.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

References

  1. Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.Google Scholar
  2. Mihăilescu M, Rădulescu V: A mulyiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc R Soc A 2006, 462: 2625–2641. 10.1098/rspa.2005.1633View ArticleGoogle Scholar
  3. Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv 1987, 9: 33–66.View ArticleGoogle Scholar
  4. Antontsev SN, Shmarev SI: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal TMA 2005, 60: 515–545.MathSciNetView ArticleGoogle Scholar
  5. Antontsev SN, Rodrigues JF: On stationary thermo-rheological viscous flows. Ann Univ Ferrara Sez Sci Mat 2006, 52: 19–36. 10.1007/s11565-006-0002-9MathSciNetView ArticleGoogle Scholar
  6. Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math 2006, 66(4):1383–1406. 10.1137/050624522MathSciNetView ArticleGoogle Scholar
  7. Harjulehto P, Hästö P: An overview of variable exponent Lebesgue and Sobolev spaces. In Future Trends in Geometric Function Theory. Edited by: Herron, D. RNC Workshop, Jyväskylä; 2003:85–93.Google Scholar
  8. Samko S: On a progress in the theory of Lebesgue spaces with variable exponent maximal and singular operators. Integr Trans Spec Funct 2005, 16: 461–482. 10.1080/10652460412331320322MathSciNetView ArticleGoogle Scholar
  9. Zhikov VV, Kozlov SM, Oleinik OA: Homogenization of Differential Operators and Integral Functionals (Translated from the Russian by Yosifian, GA). Springer-Verlag, Berlin; 1994.Google Scholar
  10. Zhikov VV: On some variational problems. Russ J Math Phys 1997, 5: 105–116.MathSciNetGoogle Scholar
  11. Kováčik O, Rákosník J: On spaces Lp(x)(Ω) and Wk,p(x)(Ω). Czechoslovak Math J 1991, 41(116):592–618.MathSciNetGoogle Scholar
  12. Fan XL: On the sub-supersolution methods for p ( x )-Laplacian equations. J Math Anal Appl 2007, 330: 665–682. 10.1016/j.jmaa.2006.07.093MathSciNetView ArticleGoogle Scholar
  13. Fan XL, Zhang QH: Existence of solutions for p ( x )-Laplacian Dirichlet problems. Non-linear Anal 2003, 52: 1843–1852.MathSciNetView ArticleGoogle Scholar
  14. Fan XL, Zhao D: On the Spaces Lp(x)and Wm,p(x). J Math Anal Appl 2001, 263: 424–446. 10.1006/jmaa.2000.7617MathSciNetView ArticleGoogle Scholar
  15. Dai G, Ma R: Solutions for a p ( x )-Kirchhoff type equation with Neumann boundary data. Nonlinear Anal Real World Appl 2011, 12: 2666–2680. 10.1016/j.nonrwa.2011.03.013MathSciNetView ArticleGoogle Scholar
  16. Fan XL, Zhao YZ, Zhang QH: A strong maximum principle for p ( x )-Laplace equations. Chin J Contemp Math 2003, 24(3):277–282.MathSciNetGoogle Scholar
  17. Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.Google Scholar
  18. Lions JL: On some equations in boundary value problems of mathematical physics. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc Internat Sympos Inst Mat Univ Fed Rio de Janeiro Rio de Janeiro 1977). North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam; 1978:284–346.View ArticleGoogle Scholar
  19. Arosio A, Pannizi S: On the well-posedness of the Kirchhoff string. Trans Am Math Soc 1996, 348: 305–330. 10.1090/S0002-9947-96-01532-2View ArticleGoogle Scholar
  20. Cavalcante MM, Cavalcante VN, Soriano JA: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv Diff Equ 2001, 6: 701–730.Google Scholar
  21. D'Ancona P, Spagnolo S: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math 1992, 108: 247–262. 10.1007/BF02100605MathSciNetView ArticleGoogle Scholar
  22. He X, Zou W: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal 2009, 70: 1407–1414. 10.1016/j.na.2008.02.021MathSciNetView ArticleGoogle Scholar
  23. Chipot M, Rodrigues JF: On a class of nonlocal nonlinear elliptic problems. RAIRO Model Math Anal 1992, 26: 447–467.MathSciNetGoogle Scholar
  24. Chipot M, Lovat B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal 1997, 30: 4619–4627. 10.1016/S0362-546X(97)00169-7MathSciNetView ArticleGoogle Scholar
  25. Alves CO, Corrêa FJSA: On existence of solutions for a class of problem involving a nonlinear operator. Comm Appl Nonlinear Anal 2001, 8: 43–56.MathSciNetGoogle Scholar
  26. Corrêa FJSA, Menezes SDB, Ferreira J: On a class of problems involving a nonlocal operator. Appl Math Comput 2004, 147: 475–489. 10.1016/S0096-3003(02)00740-3MathSciNetView ArticleGoogle Scholar
  27. Corrêa FJSA, Figueiredo GM: On a elliptic equation of p -kirchhoff type via variational methods. Bull Aust Math Soc 2006, 74: 263–277. 10.1017/S000497270003570XView ArticleGoogle Scholar
  28. Dreher M: The Kirchhoff equation for the p -Laplacian. Rend Semin Mat Univ Politec Torino 2006, 64: 217–238.MathSciNetGoogle Scholar
  29. Dreher M: The ware equation for the p -Laplacian. Hokkaido Math J 2007, 36: 21–52.MathSciNetView ArticleGoogle Scholar
  30. Dai G, Hao R: Existence of solutions for a p ( x )-Kirchhoff-type equation. J Math Anal Appl 2009, 359: 275–284. 10.1016/j.jmaa.2009.05.031MathSciNetView ArticleGoogle Scholar
  31. Fan XL: On nonlocal p ( x )-Laplacian Dirichlet problems. Nonlinear Anal 2010, 72: 3314–3323. 10.1016/j.na.2009.12.012MathSciNetView ArticleGoogle Scholar
  32. Dai G, Liu D: Infinitely many positive solutions for a p ( x )-Kirchhoff-type equation. J Math Anal Appl 2009, 359: 704–710. 10.1016/j.jmaa.2009.06.012MathSciNetView ArticleGoogle Scholar
  33. Dai G, Wei J: Infinitely many non-negative solutions for a p ( x )-Kirchhoff-type problem with Dirichlet boundary condition. Nonlinear Anal 2010, 73: 3420–3430. 10.1016/j.na.2010.07.029MathSciNetView ArticleGoogle Scholar
  34. Deng YB, Peng SJ: Existence of multiple positive solutions for inhomogeneous Neumann problem. J Math Anal Appl 2002, 271: 155–174. 10.1016/S0022-247X(02)00106-3MathSciNetView ArticleGoogle Scholar
  35. Abreu EAM, Marcos do ÓJ, Medeiros ES: Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems. Nonlinear Anal 2005, 60: 1443–1471. 10.1016/j.na.2004.09.058MathSciNetView ArticleGoogle Scholar
  36. Fan XL, Deng SG: Multiplicity of positive solutions for a class of inhomogeneous Neumann problems involving the p ( x )-Laplacian. Nonlinear Diff Equ Appl (NoDEA) 16(2):255–271.
  37. Fan XL: Global C1, αregularity for variable exponent elliptic equations in divergence form. J Diff Equ 2007, 235: 397–417. 10.1016/j.jde.2007.01.008View ArticleGoogle Scholar
  38. Damascelli L, Sciunzi B: Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m -Laplace equations. Calc var PDE 2005, 25: 139–159.MathSciNetView ArticleGoogle Scholar
  39. Damascelli L, Sciunzi B: Regularity, monotonicity and symmetry of positive solutions of m -Laplace equations. J Diff Equ 2004, 206: 483–515. 10.1016/j.jde.2004.05.012MathSciNetView ArticleGoogle Scholar
  40. Gilbarg D, Trudinger NS: Elliptic partial differential equations of second order. 2nd edition. Springer, Berlin; 1983.View ArticleGoogle Scholar
  41. Pucci P, Serrin J: The strong maximum principle revisited. J Diff Equ 2004, 196: 1–66. 10.1016/j.jde.2003.05.001MathSciNetView ArticleGoogle Scholar
  42. Guedda M, Veron L: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal 1989, 13: 879–902. 10.1016/0362-546X(89)90020-5MathSciNetView ArticleGoogle Scholar
  43. Guo ZM, Zhang ZT: W1, pversus C1local minimizers and multiplicity results for quasilinear elliptic equations. J Math Anal Appl 2003, 286: 32–50. 10.1016/S0022-247X(03)00282-8MathSciNetView ArticleGoogle Scholar
  44. Fan XL, Zhao D: A class of De Giorgi type and Hölder continuity. Nonlinear Anal 1996, 36: 295–318.MathSciNetView ArticleGoogle Scholar
  45. Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev 1976, 18: 620–709. 10.1137/1018114MathSciNetView ArticleGoogle Scholar
  46. Chang KC: A variant of mountain pass lemma. Scientia Sinica Ser A 1983, 26: 1241–1255.Google Scholar
  47. Brezis H, Nirenberg L: H1versus C1local minimizers. C R Acad Sci Paris Ser I Math 1993, 317: 465–472.MathSciNetGoogle Scholar
  48. Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal 1994, 122: 519–543. 10.1006/jfan.1994.1078MathSciNetView ArticleGoogle Scholar
  49. Azorero JG, Manfredi JJ, Alonso IP: Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations. Commun Contemp Math 2000, 2: 385–404.MathSciNetGoogle Scholar
  50. Fan XL: A Brezis-Nirenberg type theorem on local minimizers for p ( x )-Kirchhoff Dirichlet problems and applications. Diff Equ Appl 2010, 2(4):537–551.Google Scholar
  51. Ambrosetti A, Rabinowitz P: Dual variational methods in critical point theory and applications. J Funct Anal 1973, 14: 349–381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar

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