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 ℝ^N 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^-: = inf_Ω p(x) ≤ p⁺ := sup_Ω p(x) < +∞, f ∈ C ( Ω ¯ × ℝ , ℝ ) , 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)-2∇u) := -Δ_p(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 123. 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 45. 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 7891011 for an overview of and references on this subject, and to 1213141516 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 ,

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

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 19202122. 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 23242526. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see 272829) and p(x)-Laplacian (see 30313233).

Many authors have studied the Neumann problems involving the p-Laplacian, see e.g., 343536 and the references therein. In 3435 the authors have studied the problem Pλ1f 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 Pλ1f with M(t) ≡ 1. The aim of the present paper is to generalize the main results of 343536 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

and

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

Then Λ ≠ 0̸ , λ_* ≥ 0 and (λ_*, +∞) ⊂ Λ. Moreover, for every λ > λ_* problem Pλf has a minimal positive solution u_λ 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 ,

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 ,

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 .

Then for each λ ∈ (λ_*, +∞), Pλf 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),

f ( x , 0 ) ≤ f ( x , t ) for t > 0 and x ∈ Ω

and the following conditions:

f ( x , t ) ≤ c 3 + c 4 t r ( x ) - 1 , ∀ x ∈ Ω , ∀ t ≥ M 2 ,

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 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 ≠ 2, the strong comparison principles for the p-Laplacian equations are very complicated (see e.g. 38394041). In 134243 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 Pλf (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 Pλf based on the regularity results. In Section 4, we give the proof of Theorems 1.1-1.3.

In order to discuss problem Pλf , 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.

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 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),

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 W^1,p(x) (Ω) equivalent to u W 1 , p ( x ) ( Ω ) .

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

Proposition 2.1. 1114 Put ρ λ ( u ) = ∫ Ω ∇ u p ( x ) + λ u p ( x ) d x for λ > 0 and u ∈ W^1,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 ∈ W^1,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) 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 ¯ ,

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 q ( x ) + 1 p ( x ) = 1 . For any u ∈ L^p(x) (Ω) and v ∈ L^q(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 .

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 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 ,

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 u_n ⇀ u in X and lim n → + ∞ ¯ Φ ′ ( u n ) - Φ ′ ( u ) , u n - u ≤ 0 , then u_n → 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.

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

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

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

(2) Let u ∈ X ∩ L^∞ (Ω) be a solution of Pλf . 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 ,

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 Pλf 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

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, Pλf has a unique solution.

Let q∈C(Ω¯) satisfy (2.1). For h∈Lq(x)q(x)-1(Ω) , we denote by K(h) = K_λ(h) = u the unique solution of (3.3_λ). K = K_λ is called the solution operator for (3.3_λ). 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 ↪ L^q(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∈Lq(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

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 Pλf respectively, and u₀ ≤ v⁰. If f satisfies the condition:

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

then Pλf 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 Pλf such that u₀ ≤ u ≤ v⁰, then u_* ≤ u ≤ v*.

The energy functional corresponding to Pλf is

J λ ( u ) = Φ ( u ) - ∫ Ω F ( x , u ) d x , ∀ u ∈ X .

The critical points of J_λ are just the solutions of Pλf . 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 4349) 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 J_λ in the C 1 ( Ω ¯ ) -topology, then u is also a local minimizer of J_λ in the X-topology.

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 Pλf . 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 ,

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≠0̸ .

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

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 Pλ1f . Obviously 0 is a subsolution of Pλ1f . By Theorem 3.3, Pλ1f 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≠0̸ .

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 uλ0 is a supersolution of Pλf . We know that 0 is a subsolution of Pλf By Theorem 3.3, Pλf has a solution u_λ such that 0 ≤ u λ ≤ u λ 0 . By Proposition 3.4, u_λ > 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 u_λ of Pλf 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 uλ1 and u λ 2 are the positive solutions of Pλ1f and P λ 2 f respectively and uλ1≤uλ2 . Then there exists a positive solution v_λ of Pλf such that u λ 1 ≤ v λ ≤ u λ 2 and v_λ is a global minimizer of the restriction of J_λ to the order interval u λ 1 , u λ 2 ∩ X .

Proof. Define f ̃ : Ω ¯ × ℝ → ℝ 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 v_λ ∈ X. Thus v_λ 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

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 uλ1≤vλ≤uλ2 . So f ̃ ( x , v λ ) = f ( x , v λ ) , and v_λ is a positive solution of Pλf . 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 v_λ 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 uλ1 and uλ2 are the positive solutions of ( 1 . 1 λ 1 ) and ( 1 . 1 λ 2 ) respectively. Then uλ1<uλ2 on Ω¯ .

Proof. Since u λ 1 , u λ 2 ∈ C 1 ( Ω ¯ ) and uλ1>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 .