Existence of positive solutions of nonlocal p ( x )-Kirchhoff hyperbolic systems via sub-super solutions concept

1 Introduction

Elliptic differential equations (PDEs) are of crucial importance in modelization and description of a wide variety of phenomena such as fluid dynamics, quantum physics, sound, heat, electrostatics, diffusion, gravitation, chemistry, biology, simulation of airplane, calculator charts and time prediction. PDEs are equations involving functions of several variables and their derivatives and model multidimensional systems generalizing ODEs (ordinary differential equations), which deal with functions of a single variable and their derivatives. In contrary to ODEs, there is no general result such as the Picard-Lindelöf theorem for PDEs to settle the existence and uniqueness of solutions. Malgrange-Ehrenpreis theorem states that linear partial differential equations with constant coefficients always have at least one solution. Another powerful and general result in case of polynomial coefficients is the Cauchy-Kovalevskaya theorem ensuring the existence and uniqueness of a locally analytic solution for PDEs with coefficients that are analytic in the unknown function and its derivatives. Otherwise, the existence of solutions is not guaranteed at all for non-analytic coefficients even if they have derivatives of all orders: see [18]. Given the rich variety of PDEs there is no general theory of solvability. Instead, researches focus on particular PDEs that are important for applications. It would be desirable when solving PDEs to prove the existence and uniqueness of a regular solution that depends on the initial data already given in the problem, but perhaps we are asking too much. A solution with enough smoothness is called a classical solution, but in most cases, as far as conservation laws are concerned, we cannot do so much, and only generalized or weak solutions are allowed. The point is this: looking for weak solutions allows us to investigate a larger class of candidates, so it is more reasonable to consider as separate the existence and the regularity problems. For various PDEs this is the best that can be done, and naturally nonlinear equations are more difficult than linear ones. In overall, we know too much about linear PDEs and in best cases we can express their solutions, but too little about nonlinear equations. For linear PDEs, various methods and techniques can use as separation of variables, method of characteristics, integral transform, change of variables, superposition principle or even finding a fundamental solution and taking a convolution product to get the solution. Moreover, the variational theories are the most accessible and useful methods for nonlinear PDEs. However, there are other non-variational techniques of use for nonlinear elliptic and parabolic PDEs such as monotonicity and fixed point methods, semigroup theory and sub-super solutions method that played an important role in the study of nonlinear boundary value problems for a long time. In addition, Scorza-Dragoni’s work in [13] was one of the earliest papers using a pair of ordered solutions of differential inequalities to establish the existence of solution to a given boundary value problem for a nonlinear second order ordinary differential equation. His work was followed later by Nagumo in [45] and [76] which inspired much work on both ordinary and PDEs during the decade of the sixties. Likewise, Knobloch in [19] introduced the sub-supersolution method to the study of periodic boundary value problems for nonlinear second order ordinary differential equations using Cesari’s method. Similar problems and techniques were studied in [27,34] and still assuming the sub- and supersolutions to be smooth solutions of differential inequalities. Thus, the SSM were also used to study Dirichlet and Neumann boundary value problems for semilinear elliptic problems in [10,20] and even for nonlinear boundary value problems in [28, 40] and also for systems of nonlinear ordinary differential equations in [21,22] and [29]. The concept of weak sub- and supersolutions was first formulated by Hess and Deuel in [33,51] to obtain existence results for weak solutions of semilinear elliptic Dirichlet problems, and was subsequently continued by several authors (see, e.g., [3,11,46,60,61,64,70,72,73] and [74]). The study of differential equations and variational problems with nonstandardp (x)- growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [7]). Many existence results have been obtained on this kind of problems, see for example [4,7,15,56–59,62,63] and [68]. In addition, in [4] a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term has been investigated. In the last few years, the regularity and existence of solutions for differential equations with nonstandardp (x)-growth conditions have been studied, andp-Laplacian elliptic systems withp (x) = q (x) = p (a constant) have been archived in [41,65]. This work aims to study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions by using sub-super solutions method. Motivated by the ideas introduced in ([59]) and the properties of Kirchhoff type operators in [59], it will be possible to study in this work the existence of positive solutions for hyperbolic system by using the sub- and super solutions techniques. So far, this is a new research topic for nonlocal problems. The outline of this paper is organized as follows. In Section 2, some preliminary results on the variable exponent Sobolev space W 0 1 , p ( x ) ( Ω ) are presented as well as the method of sub- and super solutions. Section 3 is devoted to state and prove the main results. Finally, a numerical example is presented to illustrate our stationary results.

2 Preliminaries, assumptions and statement of the problem

In order to discuss problem(1), we need some theories on W 0 1 , p ( x ) ( Ω ) which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces W 0 1 , p ( x ) ( Ω ) which will be used later (for details, see [68]). Let us define

L p ( x ) ( Ω ) = { u : u is a measurable real - valued function such that ∫ Ω | u ( x ) | p ( x ) dx < ∞ . }

We introduce the norm onL^p(x) ( Ω) by | u ( x ) | p ( x ) = inf { λ > 0 : ∫ Ω | u ( x ) λ | p ( x ) dx ≤ 1 } , and W 1 , p ( x ) ( Ω ) = { u ∈ L p ( x ) ( Ω ) ; | ∇ u | ∈ L p ( x ) ( Ω ) } with the norm: ∀u ∈ W^1,p(x) ( Ω) . ‖ u ‖ = | u | p ( x ) + | ∇ u | p ( x ) . We denote by W 0 1 , p ( x ) ( Ω ) the closure of C 0 ∞ ( Ω ) inW^1,p(x) ( Ω) . Now, using Euler time scheme of problem(1), we obtain the following problems { u k + τ ′ 2 M ( I 0 ( u k ) ) ▵ p ( x ) u k = u k + 1 + u k - 1 2 - τ ′ 2 λ p ( x ) [ λ 1 f ( v ) + μ 1 h ( u k ) ] in Ω , u k + τ ′ 2 M ( I 0 ( v ) ) ▵ p ( x ) v = u k + 1 + u k - 1 2 - τ ′ 2 λ p ( x ) [ λ 2 g ( u k ) + μ 2 τ ( v ) ] in Ω , u k = v = 0 on ∂ Ω , u 0 = ρ 1 , u . = ρ 2 ,(3)

whereNτ′ = T, 0 < τ′ < 1, and for 1 ≤ k ≤ N .

Proposition 1. (See [65]). The spacesL^p(x) ( Ω) ,W^1,p(x) ( Ω) and W 0 1 , p ( x ) ( Ω ) are separable and reflexive Banach spaces.

Throughout the paper, we will assume that:

(H1)M : [0, + ∞) → [m₀,m_∞] is a continuous and increasing function withm₀> 0 ;

(H2) p ∈ C 1 ( Ω ¯ ) and 1< p^- ≤ p⁺ ;

(H3)f,g,h,τ : [0, + ∞ [→ R areC¹, monotone functions such that lim u k → + ∞ f ( u k ) = + ∞ , lim u k → + ∞ g ( u k ) = + ∞ , lim u k → + ∞ h ( u k ) = + ∞ , lim u k → + ∞ τ ( u k ) = + ∞ ;

(H4) lim u k → + ∞ f ( L ( g ( u k ) ) 1 p - - 1 ) u k p - - 1 = 0 , for allL> 0 ;

(H5) lim u k → + ∞ h ( u ) u k p - - 1 = 0 , and lim u → + ∞ τ ( u k ) u k p - - 1 = 0 .

Definition 1. If u k , v ∈ W 0 1 , p ( x ) ( Ω ) ,We say that - M ( I 0 ( u k ) ) ▵ p ( x ) u k ≤ - M ( I 0 ( v ) ) ▵ p ( x ) v if for all φ ∈ W 0 1 , p ( x ) ( Ω ) withφ ≥ 0, M ( I 0 ( u ) ) ∫ Ω | ∇ u k | p ( x ) - 2 ∇ u k . ∇ φ dx ≤ M ( I 0 ( v ) ) ∫ Ω | ∇ v | p ( x ) - 2 ∇ v . ∇ φ dx ,(4) where I 0 ( u k ) = ∫ Ω 1 p ( x ) | ∇ u k | p ( x ) dx .

Definition 2.(1) Ifu_k, v ∈ W 0 1 , p x Ω , u k , v is called a weak solution of(3) if it satisfies M I 0 u k ∫ Ω ∇ u k p x − 2 ∇ u k . ∇ φ d x = ∫ Ω λ p x λ 1 f v + μ 1 h u k − u k + 1 − 2 u k + u k − 1 2 τ ′ 2 φ d x , in Ω , M I 0 v ∫ Ω ∇ v p x − 2 ∇ v . ∇ φ d x = ∫ Ω λ p x λ 2 g u k + μ 2 τ v − u k + 1 − 2 u k + u k − 1 2 τ ′ 2 φ d x , in Ω for all φ ∈ W 0 1 , p x Ω . withφ ≥ 0.

(2) We say that(u, v) is called a sub solution (respectively a super solution) of(3) if M I 0 u k ∫ Ω ∇ u k p x − 2 ∇ u k . ∇ φ d x ≤ respectively ≥ ∫ Ω λ p x λ 1 f v + μ 1 h u k − u k + 1 − 2 u k + u k − 1 2 τ ′ 2 φ d x in Ω and M I 0 v ∫ Ω ∇ v p x − 2 ∇ v . ∇ φ d x ≤ respectively ≥ ∫ Ω λ p x λ 2 g u k + μ 2 τ v − u k + 1 − 2 u k + u k − 1 2 τ ′ 2 φ d x , in Ω .

Lemma 1. (See [48] Comparison principle) Let u k , v ∈ W 1 , p x Ω and( H1) holds. If − M I 0 u k △ p x u ≤ − M I 0 v △ p x v and u k − v + ∈ W 0 1 , p x Ω . Then u k ≤ v i n Ω .

Lemma 2.(See [7]). Let (H1) hold. η > 0 and let u be the unique solution of the problem in Ω − ∫ Ω M I 0 u k ÷ ∇ u k p x − 2 ∇ u k = μ . Set h = m 0 p − 2 Ω 1 N C 0 . Then, whenμ ≥ h, we have u k ∞ ≤ C ∗ μ 1 p − − 1 and when μ < h, we have u k ∞ ≤ C ∗ μ 1 p + − 1 , where C^∗ and C^∗ are positive constants depending p + , p − , N , Ω , C 0 and m₀.

Here and hereafter, we will use the notationd(x, ∂Ω) to denote the distance ofx ∈ Ω to denote the distance of Ω. Denoted(x) =d(x, ∂ Ω) and ∂ Ω ε = x ∈ Ω : d x , ∂ Ω < ε .

Since∂ Ω isC² regularly, there exists a constantδ ∈ ( 0, 1) such that d x ∈ C 2 ∂ Ω 3 δ ¯ and|∇ d(x)| = 1.

Denote v 1 x = γ d x , d x < δ γ δ + ∫ δ d x γ 2 δ − t δ 2 p − − 1 λ + μ 1 2 p − − 1 d t , δ ≤ d x < 2 δ , γ δ + ∫ δ 2 δ γ 2 δ − t δ 2 p − − 1 λ 1 + μ 1 2 p − − 1 d t , 2 δ ≤ d x(5) and v 2 x ​ = γ d x , d x < δ γ δ + ∫ δ d x γ 2 δ − t δ 2 q − − 1 λ 2 + μ 2 2 q − − 1 d t , δ ≤ d x < 2 δ , γ δ + ∫ δ 2 δ γ 2 δ − t δ 2 q − − 1 λ 2 + μ 2 2 q − − 1 d t , 2 δ ≤ d x .(6) Obviously, 0 ≤ v 1 x , v 2 x ∈ C 1 Ω ¯ .

Considering − M ∫ Ω 1 p x ∇ u k p x d x △ p x ω x = η in Ω , u = 0 on ∂ Ω .(2.2) We have the following result

Lemma 3. (See [58]). If positive parameter η is large enough and ω is the unique solution of(2.2), then we have

For any θ ∈ (0, 1)there exists a positive constant C₁ such that C 1 η 1 p + − 1 + θ ≤ max x ∈ Ω ¯ ω x ;

In the following, when there is no misunderstanding, we always useC_i to denote positive constants.

Theorem 1. Assume that the conditions (H1)-(H6) are satisfied. Then problem(3) has a positive solution when λ is large enough.

We shall establish Theorem 1 by constructing a positive subsolution(ϕ_k,ϕ₁) and supersolution(z_k,z₁) of(1). such thatϕ_k ≤ z_k andϕ₁ ≤z₁. that is, (ϕ_k, ϕ₁) and(z_k,z₁) satisfies M I 0 ϕ k ∫ Ω ∇ ϕ k p x − 2 ∇ ϕ k . ∇ q d x ≤ ∫ Ω λ p x λ 1 f ϕ 1 + μ 1 h ϕ k − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 q d x , in Ω M I 0 ϕ 1 ∫ Ω ∇ Φ 1 p x − 2 ∇ ϕ 1 . ∇ q d x ≤ ∫ Ω λ p x λ 2 g ϕ k + μ 2 τ ϕ 1 − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 q d x in Ω and M I 0 z k ∫ Ω ∇ z k p x − 2 ∇ z k . ∇ q d x ≥ ∫ Ω λ p x λ 1 f z 1 + μ 1 h z k − z k + 1 − 2 z k + z k − 1 2 τ ′ 2 q d x , in Ω , M I 0 z 1 ∫ Ω ∇ z 1 p x − 2 ∇ z 1 . ∇ q d x ≥ ∫ Ω λ p x λ 2 g z k + μ 2 τ z 1 − z k + 1 − 2 z k + z k − 1 2 τ ′ 2 q d x , in Ω for all q ∈ W 0 1 , p x Ω withq ≥ 0. According to the sub-super solution method forp(x)-Kirchhoff type equations (see [48]), then(1) has a positive solution.

Step 1. We will construct a subsolution of(1). Let σ ∈ ( 0 , δ ) is small enough.

Denote ϕ k x = e k ′ d x − 1 , d x < σ e k ′ σ − 1 + ∫ σ d x k ′ e k ′ σ 2 δ − t 2 δ − σ 2 p − − 1 λ 1 + μ 1 2 p − − 1 d t , σ ≤ d x < 2 δ , e k ′ σ − 1 + ∫ σ 2 δ k ′ e k ′ σ 2 δ − t 2 δ − σ 2 p − − 1 λ 1 + μ 1 2 p − − 1 d t , 2 δ ≤ d x and ϕ 1 x ​ = ​ e k ′ d x − 1 , d x < σ e k ′ σ − 1 ​ + ∫ σ d x k ′ e k ′ σ ​ 2 δ − t 2 δ − σ 2 q − − 1 ​ λ 2 + μ 2 2 q − − 1 d t , σ ≤ d x < 2 δ , e k ′ σ − 1 ​ + ∫ σ 2 δ k ′ e k ′ σ 2 δ − t 2 δ − σ 2 q − − 1 λ 2 + μ 2 2 q − − 1 d t , 2 δ ≤ d x . It is easy to see that ϕ k , ϕ 1 ∈ C 1 Ω ¯ , Denote α = min inf p x − 1 4 sup ∇ p x + 1 , 1 , ζ ​ = ​ min { λ 1 f 0 ​ + ​ μ 1 h 0 , λ 2 g ( 0 ) ​ + ​ μ 2 τ ( 0 ) , − 1 } .

By some simple computations we can obtain − △ p x ϕ k = − k ′ e k ′ d x p x − 1 × p x − 1 + d x + ln k ′ k ′ ∇ p ∇ d + △ d k ′ , d x < σ 1 2 δ − σ 2 p x − 1 p − − 1 − 2 δ − d 2 δ − σ × ln k ′ e k ′ σ 2 δ − d 2 δ − σ 2 p − − 1 ∇ p ∇ d + △ d × K e k σ p x − 1 2 δ − d 2 δ − σ 2 p x − 1 p − − 1 − 1 × λ 1 + μ 1 , σ ≤ d x < 2 δ , 0 , 2 δ ≤ d x and − △ p x ϕ 1 = − k e k d x p x − 1 × p x − 1 + d x + ln k k ∇ p ∇ d + △ d k , d x < σ 1 2 δ − σ 2 p x − 1 p − − 1 − 2 δ − d 2 δ − σ × ln k e k σ 2 δ − d 2 δ − σ 2 p − − 1 ∇ p ∇ d + △ d × K e k σ p x − 1 2 δ − d 2 δ − σ 2 p x − 1 p − − 1 − 1 λ 2 + μ 2 , σ ≤ d x < 2 δ , 0 , 2 δ ≤ d x . From(H4) there exists a positive constantL > 1 such that f L − 1 ≥ 1 , g L − 1 ≥ 1 , h L − 1 ≥ 1 , τ L − 1 ≥ 1. Let σ = 1 k ′ ln L , then σ k ′ = ln L .(7) Ifk^′ is sufficiently large, from(7), we have for d x < σ − △ p x ϕ 1 ≤ − k ′ p x α .(8) Let λ ζ m ∞ = k ′ α , then − k ′ p x α ≥ − λ p x ζ m ∞ . From(8), we have − M I 0 ϕ k △ p x ϕ k ≤ M I 0 ϕ k λ p x ζ m ∞ ≤ λ p x ζ ≤ λ p x λ 1 f 0 + μ 1 h 0 ≤ λ p x λ 1 f ϕ 1 + μ 1 h ϕ k − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , d x < σ . Since d x ∈ C 2 ∂ Ω 3 δ ¯ , there exists a positive constantC₃ such that − M I 0 ϕ k △ p x ϕ k ≤ m ∞ K e k ′ σ p x − 1 × 2 δ − d 2 δ − σ 2 p x − 1 p − − 1 − 1 λ 1 + μ 1 × 1 2 δ − σ 2 p x − 1 p − − 1 − 2 δ − d 2 δ − σ × ln k ′ e k ′ σ 2 δ − d 2 δ − σ 2 p − − 1 ∇ p ∇ d + △ d ≤ C 3 m ∞ K e k σ p x − 1 λ 1 a 1 + μ 1 c 1 ln k ′ , σ ≤ d x < 2 δ . Ifk^′ is sufficiently large, let λ ζ m ∞ = k ′ α , then we have C 3 m ∞ K e k σ p x − 1 λ 1 a 1 + μ 1 c 1 ln k ′ = C 3 m ∞ K L p x − 1 λ 1 + μ 1 ln k ′ ≤ λ p x λ 1 + μ 1 − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 . Then − M I 0 ϕ k △ p x ϕ k ≤ λ p x λ 1 + μ 1 − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , σ ≤ d x < 2 δ .(9) Since ϕ 1 x , ϕ 2 x andf, h are monotone, whenλ is large enough we have − M ∫ Ω 1 p x ∇ ϕ k p x d x △ p x ϕ k ≤ λ p x λ 1 f ϕ 1 + μ 1 h ϕ k − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , σ ≤ d x < 2 δ , − M I 0 ϕ k △ p x ϕ k = 0 ≤ λ p x λ 1 + μ 1 ≤ λ p x λ 1 f ϕ 1 + μ 1 h ϕ k − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , 2 δ ≤ d x .(10) Combining(9) and(10), we can conclude that − M I 0 ϕ k △ p x ϕ k ≤ λ p x λ 1 f ϕ 1 + μ 1 h ϕ k − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , a . e . on Ω .(11) Similarly − M I 0 ϕ 1 △ p x ϕ 1 ≤ λ p x λ 2 g ϕ k + μ 2 τ ϕ 1 − ϕ k + 1 − 2 ϕ k + ϕ k − 1 2 τ ′ 2 , a . e . on Ω .(12) From(11) and(12), we can see that ϕ k , ϕ 1 is a subsolution of problem(3).

Step 2. We will construct a supersolution of problem(3). We consider - M ( I 0 ( z k ) ) ▵ p ( x ) z k = λ p + m 0 ( λ 1 + μ 1 ) μ - z k + 1 - 2 z k + z k - 1 2 τ ′ 2 in Ω , - M ( I 0 ( z 1 ) ) ▵ p ( x ) z 1 = λ p + m 0 ( λ 2 + μ 2 ) g ( β ( λ p + ( λ 1 + μ 1 ) μ ) ) - z k + 1 - 2 z k + z k - 1 2 τ ′ 2 in Ω , z k = z 1 = 0 on ∂ Ω , where β = β ( λ p + ( λ 1 + μ 1 ) μ ) = max x ∈ Ω ¯ z k ( x ) . We shall prove that (z _k ,z₁) is a supersolution of problem(3). For q ∈ W 0 1 , p ( x ) ( Ω ) withq ≥ 0, it is easy to see that M I 0 z 1 ∫ Ω ∇ z 1 p x − 2 ∇ z 1 . ∇ q d x = 1 m 0 M I 0 z 1 × ∫ Ω λ p + λ 2 + μ 2 g ( β ( λ p + λ 1 + μ 1 μ ) ) q d x ≥ ∫ Ω λ p + λ 2 b x g z k q d x + ∫ Ω λ p + μ 2 d x g ( β ( λ p + λ 1 + μ 1 μ ) ) q d x .(13) By (H6), forμ large enough, using Lemma 2.6, we have g ( β ( λ p + λ 1 + μ 1 μ ) ) ≥ τ C 2 λ p + λ 2 + μ 2 × g ( β ( λ p + λ 1 + μ 1 μ ) ) 1 p − − 1 ≥ τ z 1 .(14) Hence M ( I 0 ( z 1 ) ) ∫ Ω | ∇ z 1 | p ( x ) - 2 ∇ z 1 . ∇ qdx ≥ ∫ Ω λ p + λ 2 g ( z k ) qdx + ∫ Ω λ p + μ 2 τ ( z 1 ) qdx - ∫ Ω z k + 1 - 2 z k + z k - 1 2 τ ′ 2 qdx .(15) Also M I 0 z k ∫ Ω ∇ z k p x − 2 ∇ z k . ∇ q d x = 1 m 0 M I 0 z k ∫ Ω λ p + λ 1 + μ 1 μ q d x ≥ ∫ Ω λ p + λ 1 + μ 1 μ q d x . By (H4) , (H5) and Lemma 3, whenμ is sufficiently large, we have λ 1 + μ 1 μ ≥ 1 λ p + 1 C 2 β ( λ p + λ 1 + μ 1 μ ) p − − 1 ≥ μ 1 h ( β ( λ p + λ 1 + μ 1 μ ) ) + λ 1 f C 2 λ p + λ 2 + μ 2 × g ( β ( λ p + λ 1 + μ 1 μ ) ) 1 p − − 1 . Then M I 0 z k ∫ Ω ∇ z k p x − 2 ∇ z k . ∇ q d x ≥ ∫ Ω λ p + λ 1 f z 1 q d x + ∫ Ω λ p + μ 1 h z k q d x − ∫ Ω z k + 1 − 2 z k + z k − 1 2 τ ′ 2 q d x .(16) According to(15) and(16), we can conclude that (z _k ,z₁) is a supersolution of problem(3).It only remains to prove thatϕ _k  ≤ z _k andϕ₁ ≤ z₁ .

In the definition ofv₁ (x), let γ = 2 δ ( max Ω ¯ ϕ k ( x ) + max Ω ¯ | ∇ ϕ k | ( x ) ) . We claim that ϕ k ( x ) ≤ v 1 ( x ) , ∀ x ∈ Ω .(17)

From the definition ofv₁, it is easy to see that whend (x) = δ ϕ k ( x ) ≤ 2 max Ω ¯ ϕ k ( x ) ≤ v 1 ( x ) and whend (x) ≥ δ . ϕ k ( x ) ≤ 2 max Ω ¯ ϕ k ( x ) ≤ v 1 ( x ) , ϕ k ( x ) ≤ v 1 ( x ) , when d ( x ) < δ . Since v 1 - ϕ k ∈ C 1 ( ∂ Ω δ ¯ ) , there exists a point x 0 ∈ ∂ Ω δ ¯ such that v 1 ( x 0 ) - ϕ k ( x 0 ) = min x 0 ∈ ∂ Ω δ ¯ ( v 1 ( x 0 ) - ϕ k ( x 0 ) ) . Ifv₁ (x₀) - ϕ _k  (x₀) < 0, it is easy to see that 0<d (x) < δ and then ∇ v 1 ( x 0 ) - ∇ ϕ k ( x 0 ) = 0 . From the definition ofv₁, we have | ∇ v 1 ( x 0 ) | = γ = 2 δ ( max Ω ¯ ϕ k ( x 0 ) + max Ω ¯ | ∇ ϕ k | ( x 0 ) ) > | ∇ ϕ k | ( x 0 ) . It is a contradiction to ∇ v 1 ( x 0 ) - ∇ ϕ k ( x 0 ) = 0 . Thus(17) is valid. Obviously, there exists a positive constantC₃ such that γ ≤ C 3 λ . Since d ( x ) ∈ C 2 ( ∂ Ω 3 δ ¯ ) , according to the proof of Lemma 3, there exists a positive constantC₄ such that for allθ ∈ (0, 1) . M ( I 0 ( v 1 ) ) - ▵ p ( x ) v 1 ( x ) ≤ C ∗ γ p ( x ) - 1 + θ ≤ C 4 λ p ( x ) - 1 + θ . a . e in Ω . Whenη ≥ λ ^{p +} is large enough, we have - ▵ p ( x ) v 1 ( x ) ≤ η . According to the comparison principle, we have v 1 ( x ) ≤ ω ( x ) , ∀ x ∈ Ω .(18) From(17) and(18), whenη ≥ λ ^{p +} andλ ≥ 1 is sufficiently large, we have for allx ∈  Ω . ϕ k ( x ) ≤ v 1 ( x ) ≤ ω ( x ) .(19) According to the comparison principle, whenμ is large enough, we have for allx ∈  Ω . v 1 ( x ) ≤ ω ( x ) ≤ z k ( x ) .

Combining the definition ofv₁ (x) and(19), it is easy to see that fpr allx ∈  Ω . ϕ k ( x ) ≤ v 1 ( x ) ≤ ω ( x ) ≤ z k ( x ) . Whenμ ≥ 1 andλ is large enough, from Lemma 3, we can see that β ( λ p + λ 1 + μ 1 μ ) is large enough, then λ p + m 0 λ 2 + μ 2 × g ( β ( λ p + λ 1 + μ 1 μ ) ) is large enough. Similarly, we haveϕ₁ ≤ z₁. This completes the proof.