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Article

Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

1
Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Buraydah 51452, Saudi Arabia
2
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1 Ahmed Benbella, 31000 Oran, Algeria
3
Department of Mathematics, College of Sciences, Qassim University, Buraydah 51452, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(2), 253; https://doi.org/10.3390/sym11020253
Submission received: 17 January 2019 / Revised: 9 February 2019 / Accepted: 13 February 2019 / Published: 18 February 2019

Abstract

:
Motivated by the idea which has been introduced by Boulaaras and Guefaifia (Math. Meth. Appl. Sci. 41 (2018), 5203–5210 and, by Afrouzi and Shakeri (Afr. Mat. (2015) 26:159–168) combined with some properties of Kirchhoff type operators, we prove the existence of positive solutions for a class of nonlocal p x -Kirchhoff evolutionary systems by using the sub and super solutions concept.

1. Introduction

The study of differential equations and variational problems with nonstandard p ( x ) -growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1,2]). Many existence results have been obtained on this kind of problems—see, for example, [3,4,5,6]. In Refs. [7,8,9,10,11,12], Fan et al. studied the regularity of solutions for differential equations with nonstandard p ( x ) -growth conditions.
In this article, we are interested in the p ( x ) -Kirchhoff parabolic systems of the form
u t M I 0 u Δ p x u = λ p x λ 1 a x f v + μ 1 c x h u in Q T = Ω × 0 , T , u t M I 0 v Δ p x v = λ p x λ 2 b x g u + μ 2 d x τ v in Q T = Ω × 0 , T , u = v = 0 on Q T , u ( x , 0 ) = φ ( x ) ,
where Ω R N is a bounded smooth domain with C 2 boundary Ω , 1 < p x C 1 Ω ¯ is a functions with 1 < p : = inf Ω p x p + : = sup Ω p x < , Δ p x u = ÷ u p x 2 u is called p ( x ) -Laplacian, λ , λ 1 , λ 2 , μ 1 , and μ 2 are positive parameters, I 0 u = Ω 1 p x u p x d x and M ( t ) is a continuous function.
Problem (1) is a generalization of a model introduced by Kirchhoff [13]. More precisely, Kirchhoff proposed a model given by the equation
ρ 2 u t 2 P 0 h + E 2 L 0 L u x 2 d x 2 u x 2 = 0 ,
where ρ , P 0 , h , E , L are constants, which extends the classical D’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [14,15,16,17,18,19,20,21], in which the authors have used a variational method and topological method to get the existence of solutions.
In this paper, motivated by the ideas introduced in [22] and the properties of Kirchhoff type operators in [22], we study the existence of positive solutions for system (2) by using the sub- and super solutions techniques. To our best knowledge, this is a new research topic for nonlocal problems. The remainder of this paper is organized as follows. In Section 2, we present some preleminary results on the variable exponent Sobolev space W 0 1 , p x Ω and the method of sub- and super solutions. Section 3 is devoted to stating and proving the main result.

2. Preliminary Results

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 [3]).
Let us define
L p x Ω = u : u is a measurable real - valued function such that Ω u x p x d x < .
We introduce the norm on L p x Ω by
u x 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 p x + u p x , u W 1 , p x Ω .
We denote by W 0 1 , p x Ω the closure of C 0 Ω in W 1 , p x Ω .
Now, using Euler time scheme of problem (1), we obtain the following problems:
u k τ M I 0 u k Δ p x u k = λ p x τ λ 1 a x f v + μ 1 c x h u k + u k 1 in Ω , u k τ M I 0 v Δ p x v = λ p x τ λ 2 b x g u k + μ 2 d x τ v + u k 1 in Ω , u k = v = 0 on Ω , u 0 = φ 0 ,
where N τ = T , 0 < τ < 1 , and for 1 k N .
Proposition 1
(see [12]). The spaces L 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:
Hypothesis 1 (H1).
M : [ 0 , + ) m 0 , m is a continuous and increasing function with m 0 > 0 ;
Hypothesis 2 (H2).
p C 1 Ω ¯ and 1 < p p + ;
Hypothesis 3 (H3).
f , g , h , τ : 0 , + R are C 1 , 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 = + ,
Hypothesis 4 (H4).
lim u k + f L g u k 1 p 1 u k p 1 = 0 , for all L > 0 ,
Hypothesis 5 (H5).
lim u k + h u u k p 1 = 0 , and lim u + τ u k u k p 1 = 0 .
Hypothesis 6 (H6).
a , b , c , d : Ω ¯ 0 , + are continuous functions, such that
a 1 = min x Ω ¯ a x , b 1 = min x Ω ¯ b x , c 1 = min x Ω ¯ c x , d 1 = min x Ω ¯ d x , a 2 = max x Ω ¯ a x , b 2 = max x Ω ¯ b x , c 2 = max x Ω ¯ c x , d 2 = max x Ω ¯ d x .
Definition 1.
If u k , v W 0 1 , p x Ω , we say that
M I 0 u k Δ p x u k M I 0 v Δ p x v
if, for all φ W 0 1 , p x Ω with φ 0 ,
M I 0 u Ω u k p x 2 u k · φ d x M I 0 v Ω v p x 2 v · φ d x ,
where
I 0 u k = Ω 1 p x u k p x d x .
Definition 2.
1 If u k , v W 0 1 , p x Ω , u k , v is called a weak solution of the problem defined in (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 a x f v + μ 1 c x h u k u k u k 1 τ φ d x , M I 0 v Ω v p x 2 v · φ d x = Ω λ p x λ 2 b x g u k + μ 2 d x τ v u k u k 1 τ φ d x ,
for all φ W 0 1 , p x Ω with φ 0 .
2 We say that ( u , v ) is called a sub solution (respectively a super solution) of the problem defined in (3) if
M I 0 u k Ω u k p x 2 u k · φ d x r e s p e c t i v e l y Ω λ p x λ 1 a x f v + μ 1 c x h u k u k u k 1 τ φ d x , M I 0 v Ω v p x 2 v · φ d x r e s p e c t i v e l y Ω λ p x λ 2 b x g u k + μ 2 d x τ v u k u k 1 τ φ d x ,
Lemma 1
(see [22] comparison principle). Let u k , v W 1 , p x Ω and (H1) hold. If
M I 0 u k Δ p x u M I 0 v Δ p x v
and u k v + W 0 1 , p x Ω , then u k v in Ω .
Lemma 2
(see [22]). Let (H1) hold. η > 0 and let u be the unique solution of the problem
Ω M I 0 u k ÷ u k p x 2 u k = μ in Ω .
Set h = m 0 p 2 Ω 1 N C 0 . Then, when
μ h , u k C * μ 1 p 1 ,
and when
μ < h , u k C * μ 1 p + 1 ,
where C * and C * are positive constants depending p + , p , N , Ω , C 0 and m 0 .
Here and hereafter, we will use the notation d x , Ω to denote the distance of x Ω to denote the distance of Ω . Denote d x = d x , Ω and
Ω ε = x Ω : d x , Ω < ε .
Since Ω is C 2 regularly, there exists a constant δ 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 1 a 1 + μ 1 c 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 b 1 + μ 1 d 1 2 p 1 d t , 2 δ d x .
v 2 x = γ d x , d x < δ , γ δ + δ d x γ 2 δ t δ 2 q 1 λ 2 a 2 + μ 2 c 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 b 2 + μ 2 d 2 2 q 1 d t , 2 δ d x .
Obviously, 0 v 1 x , v 2 x C 1 Ω ¯ . Considering
M Ω 1 p x u k p x d x Δ p x ω x = η in Ω , u = 0 on Ω .
we have the following Lemma:
Lemma 3
(see [23]). If positive parameter η is large enough and ω is the unique solution of (5), then we have
(i) 
For any θ 0 , 1 , there exists a positive constant C 1 such that
C 1 η 1 p + 1 + θ max x Ω ¯ ω x .
(ii) 
There exists a positive constant C 2 such that
max x Ω ¯ ω x C 2 η 1 p 1 .

3. Main Result

In the following, when there is no misunderstanding, we always use C 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.
Proof. 
We shall establish Theorem 1 by constructing a positive subsolution ϕ k , ϕ 1 and supersolution z k , z 1 of the problem defined in (1) such that ϕ k z k and ϕ 1 z 1 . That is, ϕ k , ϕ 1 and z k , z 1 satisfies
M I 0 ϕ k Ω ϕ k p x 2 ϕ k · q d x Ω λ p x λ 1 a x f ϕ 1 + μ 1 c x h ϕ k ϕ k ϕ k 1 τ q d x , M I 0 ϕ 1 Ω Φ 1 p x 2 ϕ 1 · q d x Ω λ p x λ 2 b x g ϕ k + μ 2 d x τ ϕ 1 ϕ k ϕ k 1 τ q d x ,
and
M I 0 z k Ω z k p x 2 z k · q d x Ω λ p x λ 1 a x f z 1 + μ 1 c x h z k z k z k 1 τ q d x , M I 0 z 1 Ω z 1 p x 2 z 1 · q d x Ω λ p x λ 2 b x g z k + μ 2 d x τ z 1 z k z k 1 τ q d x ,
for all q W 0 1 , p x Ω with q 0 . According to the sub-super solution method for p ( x ) -Kirchhoff type equations (see [22]), then the problem defined in (1) has a positive solution.
Step 1. We will construct a subsolution of (1). Let σ ( 0 , δ ) be 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 a 1 + μ 1 c 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 a 1 + μ 1 c 1 2 p 1 d t , 2 δ d x .
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 b 1 + μ 2 d 1 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 b 1 + μ 2 d 1 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
Δ p x ϕ k = k e k d x p x 1 p x 1 + d x + ln k k p d + Δ 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 + Δ d × K e k σ p x 1 2 δ d 2 δ σ 2 p x 1 p 1 1 λ 1 a 1 + μ 1 c 1 , σ d x < 2 δ , 0 , 2 δ d x .
Δ p x ϕ 1 = k e k d x p x 1 p x 1 + d x + ln k k p d + Δ 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 + Δ d × K e k σ p x 1 2 δ d 2 δ σ 2 p x 1 p 1 1 λ 2 b 1 + μ 2 d 1 , σ d x < 2 δ , 0 , 2 δ d x .
From (H4), there exists a positive constant L > 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 .
If k is sufficiently large, from the problem defined in (6), we have
Δ p x ϕ 1 k p x α , d x < σ .
Let λ ζ m = k α . Then,
k p x α λ p x ζ m .
From the problem defined in (7), we have
M I 0 ϕ k Δ p x ϕ k M I 0 ϕ k λ p x ζ m , λ p x ζ , λ p x λ 1 a 1 f 0 + μ 1 c 1 h 0 , λ p x λ 1 a x f ϕ 1 + μ 1 c x h ϕ k ϕ k ϕ k 1 τ , d x < σ .
Since d ( x ) C 2 Ω 3 δ ¯ , there exists a positive constant C 3 such that
M I 0 ϕ k Δ 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 + Δ 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 .
If k 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 a 1 + μ 1 c 1 ln k λ p x λ 1 a 1 + μ 1 c 1 ϕ k ϕ k 1 τ .
Then,
M I 0 ϕ k Δ p x ϕ k λ p x λ 1 a 1 + μ 1 c 1 ϕ k ϕ k 1 τ , σ d x < 2 δ .
Since ϕ 1 x , ϕ 2 x and f , h are monotone, when λ is large enough, we have
M Ω 1 p x ϕ k p x d x Δ p x ϕ k λ p x λ 1 a x f ϕ 1 + μ 1 c x h ϕ k ϕ k ϕ k 1 τ , σ d x < 2 δ ,
M I 0 ϕ k Δ p x ϕ k = 0 λ p x λ 1 a 1 + μ 1 c 1 λ p x λ 1 a x f ϕ 1 + μ 1 c x h ϕ k ϕ k ϕ k 1 τ , 2 δ d x .
Combining two problems which defined in (8) and (9), we can conclude that
M I 0 ϕ k Δ p x ϕ k λ p x λ 1 a x f ϕ 1 + μ 1 c x h ϕ k ϕ k ϕ k 1 τ , a . e . on Ω .
Similarly,
M I 0 ϕ 1 Δ p x ϕ 1 λ p x λ 2 b x g ϕ k + μ 2 d x τ ϕ 1 ϕ k ϕ k 1 τ , a . e . on Ω .
From the problems defined in (10) and (11), 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 Δ p x z k = λ p + m 0 λ 1 a 2 + μ 1 c 2 μ z k z k 1 τ in Ω , M I 0 z 1 Δ p x z 1 = λ p + m 0 λ 2 b 2 + μ 2 d 2 g β λ p + λ 1 a 2 + μ 1 c 2 μ z k z k 1 τ in Ω , z k = z 1 = 0 on Ω ,
where β = β λ p + λ 1 a 2 + μ 1 c 2 μ = max x Ω ¯ z k x . We shall prove that z k , z 1 is a supersolution of problem (3).
For q W 0 1 , p x Ω with q 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 b 2 + μ 2 d 2 g β λ p + λ 1 a 2 + μ 1 c 2 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 .
By (H6), for large enough μ , using Lemma 2, we have
g β λ p + λ 1 a 2 + μ 1 c 2 μ τ C 2 λ p + λ 2 b 2 + μ 2 d 2 g β λ p + λ 1 a 2 + μ 1 c 2 μ 1 p 1 τ z 1 .
Hence,
M I 0 z 1 Ω z 1 p x 2 z 1 · q d x Ω λ p + λ 2 b x g z k q d x + Ω λ p + μ 2 d x τ z 1 q d x Ω z k z k 1 τ q d x .
In addition,
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 a 2 + μ 1 c 2 μ q d x Ω λ p + λ 1 a 2 + μ 1 c 2 μ q d x .
By (H4), (H5) and Lemma 2, when μ is sufficiently large, we have
λ 1 a 2 + μ 1 c 2 μ 1 λ p + 1 C 2 β λ p + λ 1 a 2 + μ 1 c 2 μ p 1 μ 1 h β λ p + λ 1 a 2 + μ 1 c 2 μ + λ 1 f C 2 λ p + λ 2 b 2 + μ 2 d 2 g β λ p + λ 1 a 2 + μ 1 c 2 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 a x f z 1 q d x + Ω λ p + μ 1 c x h z k q d x Ω z k z k 1 τ q d x .
According to the problems (14) and (15), we can conclude that z k , z 1 is a supersolution of problem (3). It only remains to prove that ϕ k z k and ϕ 1 z 1 .
In the definition of v 1 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 .
From the definition of v 1 , it is easy to see that
ϕ k x 2 max Ω ¯ ϕ k x v 1 x , when d x = δ
and
ϕ k x 2 max Ω ¯ ϕ k x v 1 x , when d x δ .
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 .
If v 1 x 0 ϕ k x 0 < 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 of v 1 , 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, problem (16) is valid.
Obviously, there exists a positive constant C 3 such that
γ C 3 λ .
Since d x C 2 Ω 3 δ ¯ , according to the proof of Lemma 2, there exists a positive constant C 4 such that
M I 0 v 1 Δ p x v 1 x C * γ p x 1 + θ C 4 λ p x 1 + θ . a . e in Ω , where θ 0 , 1 .
When η λ p + is large enough, we have
Δ p x v 1 x η .
According to the comparison principle, we have
v 1 x ω x , x Ω .
From problems (16) and (17), when η λ p + and λ 1 is sufficiently large, we have
ϕ k x v 1 x ω x , x Ω .
According to the comparison principle, when μ is large enough, we have
v 1 x ω x z k x , x Ω .
Combining the definition of v 1 x and the problem defined in (18), it is easy to see that
ϕ k x v 1 x ω x z k x , x Ω .
When μ 1 and λ is large enough, from Lemma 2.6 (see [22]), we can see that β λ p + λ 1 a 2 + μ 1 c 2 μ is large enough, and then
λ p + m 0 λ 2 b 2 + μ 2 d 2 g β λ p + λ 1 a 2 + μ 1 c 2 μ
is large enough. Similarly, we have ϕ 1 z 1 . This completes the proof. □

Author Contributions

All authors contributed equally.

Funding

The authors gratefully acknowledges Qassim University, represented by the Deanship of Scientific Research, for the material support of this research under Number 3733-alrasscac-2018-1-14-S during the academic year 1439AH /2018.

Acknowledgments

The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions which helped them to improve the paper. The authors gratefully acknowledges Qassim University, represented by the Deanship of Scientific Research, for the material support of this research under Number 3733-alrasscac-2018-1-14-S during the academic year 1439AH /2018.

Conflicts of Interest

The authors declare no conflict of interest.

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MDPI and ACS Style

Boulaaras, S.; Allahem, A. Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept. Symmetry 2019, 11, 253. https://doi.org/10.3390/sym11020253

AMA Style

Boulaaras S, Allahem A. Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept. Symmetry. 2019; 11(2):253. https://doi.org/10.3390/sym11020253

Chicago/Turabian Style

Boulaaras, Salah, and Ali Allahem. 2019. "Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept" Symmetry 11, no. 2: 253. https://doi.org/10.3390/sym11020253

APA Style

Boulaaras, S., & Allahem, A. (2019). Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept. Symmetry, 11(2), 253. https://doi.org/10.3390/sym11020253

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