# Existence of Positive Solutions for a Class ofpx,qx-Laplacian Elliptic Systems with Multiplication of Two Separate Functions

Authors: Youcef Bouizem; Salah Mahmoud Boulaaras; Ali Allahem
Journal: Complexity
 (2020)

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## Abstract

The paper deals with the study of the existence of weak positive solutions for a new class of the system of elliptic differential equations with respect to the symmetry conditions and the right hand side which has been defined as multiplication of two separate functions by using the sub-supersolutions method (1991 Mathematics Subject Classification: 35J60, 35B30, and 35B40).


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## Body

## 1. Introduction

Elliptic systems of differential equations 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 (see, for example, [1–15]).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 nonanalytic coefficients even if they have derivatives of all orders (see [16]). Given the rich variety of PDEs, there is no general theory of solvability. Instead, research focuses on particular PDEs that are important for applications. It would be desirable when solving a PDE to prove the existence and uniqueness of a regular solution that depends on the initial data 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 for conservation laws, we cannot achieve that much and allow generalized or weak solutions. 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. 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 be used for 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 obtain the solution. Variational theory is the most accessible and useful of the methods for nonlinear PDEs, but there are other nonvariational techniques of use for nonlinear elliptic and parabolic PDEs such as monotonicity and fixed point methods, semigroup theory, and sub-supersolutions method that played an important role in the study of nonlinear boundary value problems for a long time. Scorza-Dragoni’s work in [17] 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 [18, 19] which inspired much work on both ordinary and PDEs during the decade of the sixties. Knobloch in [20] 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 [21, 22] and still the sub-supersolutions and supersolutions are assumed to be smooth solutions of differential inequalities. Then, the SSM were also used to study Dirichlet and Neumann boundary value problems for semilinear elliptic problems in [23, 24], and even for nonlinear boundary value problems in [25–27] and also for systems of nonlinear ordinary differential equations in [28–30]. The concept of weak sub-supersolutions and supersolutions was first formulated by Hess and Deuel in [31, 32] to obtain existence results for weak solutions of semilinear elliptic Dirichlet problems and was subsequently continued by several authors (see, e.g., [33–43]).The study of differential equations and variational problems with nonstandardpx-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc (see [44]). Many existence results have been obtained on this kind of problems (see, for example, [44–57]) and in [45] a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term has been investigated.In the last few years in [51, 58–60], the regularity and existence of solutions for differential equations with nonstandard px-growth conditions have been studied and p-Laplacian elliptic systems with px=qx=p (a constant) have been archived. In this work, we 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-supersolution method.


## 2. Preliminaries, Assumptions, and Statement of the Problem

### 2.1. Plate Problems and Its History

In this paper, we consider the system of differential equations:(1)−Δpxu=λpxaxfuhvinΩ,−Δqxv=λqxbxguτvinΩ,u=v=0on∂Ω,where Ω⊂ℝN is a bounded smooth domain with C2 boundary ∂Ω and 1<px,qx∈C1Ω¯ are functions with 1<p−≔infΩpx≤p+≔supΩpx<∞, 1<q−≔infΩqx≤q+≔supΩqx, and Δpx is a px-Laplacian defined as(2)Δpxu=div∇upx−2∇u,and a,b:Ω¯⟶ℝ+ are continuous functions, while f,g,h,andτ are monotone functions in ℝ+ such that(3)limu⟶+∞fu=+∞,limu⟶+∞gu=+∞,limu⟶+∞hu=+∞,limu⟶+∞τu=+∞,satisfying some natural growth condition at u=∞.We point out that the extension fromp-Laplace operator to px-Laplace operator is not trivial, since the px-Laplacian has a more complicated structure then the p-Laplace operator, such as it is nonhomogeneous. Moreover, many results and methods for p-Laplacians are not valid for the px-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient(4)λpx=infu∈W01,pxΩ\0∫Ω1/px∇upxdx∫Ω1/pxupxdx,is zero in general, and only under some special conditions, λpx is positive (see [53]). Maybe the first eigenvalue and the first eigenfunction of the px-Laplacian do not exist, but the fact that the first eigenvalue λp is positive and the existence of the first eigenfunction are very important in the study of p-Laplacian problem. There are more difficulties in discussing the existence of solutions of variable exponent problems. In [59], the authors considered the existence of positive weak solutions for the following p-Laplacian problem:(5)−Δpu=λfvinΩ,−Δpu=λguinΩ,u=v=0on∂Ω,where the first eigenfunction has been used to construct the subsolution of p-Laplacian problem. Under the condition that for all M>0,(6)limu⟶+∞fMgu1/p−1up−1=0,the authors gave the existence of positive solutions for problem (5) provided that λ is large enough.In [48], the existence and nonexistence of positive weak solutions to the following quasilinear elliptic system:(7)−Δpu=λuαvγinΩ,−Δqu=λuδvβinΩ,u=v=0on∂Ω,has been considered where the first eigenfunction has been used to construct the subsolution of problem (7) and the following results were obtained:(i)
Ifα,β≥0,γ,δ>0,θ=p−1−αq−1−β−γδ>0, then problem (7) has a positive weak solution for each λ>0.(ii)
Ifθ=0 and pγ=qp−1−α, then there exists λ0>0 such that for 0<λ<λ0, problem (7) has no nontrivial nonnegative weak solution. For further generalizations of system (7), we refer to [49, 50].As already discussed before, on thepx-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of the px-Laplacian do not exist even if the first eigenfunction of the px-Laplacian exists. Because of the nonhomogeneous property of the px-Laplacian, the first eigenfunction cannot be used to construct the subsolutions of px-Laplacian problems. Moreover, in [47, 61], the authors studied the existence of solutions for problem (5), where some symmetry conditions are imposed. Then, in [46], the existence of positive solutions of the system was investigated:(8)−Δpxu=λpxfvinΩ,−Δpxu=λpxguinΩ,u=v=0onΩ,without any symmetry conditions. Motivated by the ideas introduced in [47], the authors proved the existence of a positive solution when λ is large enough and satisfies condition (6) and they did not assume any symmetric condition and did not assume any sign condition on f0 and g0. Also the authors proved the existence of positive solutions with multiparameters; in this paper, we extend this given system of differential equations, where we establish the existence of a positive solution for a new class of this system with respect to the symmetry conditions by constructing a positive subsolution and supersolution and p,q∈C1Ω¯ are functions, λ,λ1,λ2,μ1, and μ2 are positive parameters, and Ω⊂ℝN is a bounded domain and we did not assume any sign condition on f0, g0, h0, and τ0.


### 2.2. Preliminary Results

In order to discuss problem (1), we need some theories on W01,pxΩ which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces W01,pxΩ which will be used later (for details, see [54]).Let us define(9)LpxΩ=u:uis a measurable real−valued function such that∫Ωuxpxdx<∞.We introduce the norm onLpxΩ by(10)uxpx=infλ>0:∫Ωuxλpxdx≤1,W1,pxΩ=u∈LpxΩ;∇u∈LpxΩ,with the norm(11)u=upx+∇upx,∀u∈W1,pxΩ.We denote byW01,pxΩ the closure of C0∞Ω in W1,pxΩ.Proposition 1 (see [59]).
The spacesLpxΩ,W1,pxΩ, and W01,pxΩ are separable and reflexive Banach spaces.Throughout the paper, we will assume thatH1p,q∈C1Ω¯ and 1<p−≤p+,1<q−≤q+H2f,g,h,τ:ℝ+⟶ℝ are C1 monotone functions such that(12)limu⟶+∞fu=+∞,limu⟶+∞gu=+∞;limu⟶+∞hu=+∞,limu⟶+∞τu=+∞.H3∃r>0 such that(13)limu⟶+∞fuhcur/q−−1up−−1=0,for allc>0,(14)limu⟶+∞guτkur/q−−1ur=0,for allk>0.H4a,b:Ω¯⟶ℝ+ are continuous functions, such that(15)a1=minx∈Ω¯ax,b1=minx∈Ω¯bx,a2=maxx∈Ω¯ax,b2=maxx∈Ω¯bx.We define(16)Lu,v=∫Ω∇upx−2∇u∇vdx,∀u,v∈W01,pxΩ.Then,L:W01,pxΩ⟶W01,pxΩ∗ is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [61], Theorem 3.1).DefineA:W01,pxΩ⟶W01,pxΩ∗ as for all u,φ∈W01,pxΩ,(17)Au,φ=∫Ω∇upx−2∇u∇φ+hx,uφdx,where hx,u is continuous on Ω¯×ℝ and hx is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [44], we have the following lemma:Lemma 1 (see [45]) (comparison principle).
Letu,v∈W01,px(Ω) satisfy(18)Au−Av≥0inW01,pxΩ∗,φx=minux−vx,0.
If(19)φx∈W01,pxΩ,i.eu≥von∂Ω,
thenu≥va.e. in Ω.Definition 1.
Letu,v∈W01,pxΩ×W01,qxΩ; the couple u,v is said to be a weak solution of (1) if it satisfies(20)∫Ω∇upx−2∇u⋅∇φdx=∫Ωλpxaxfuhvφdx,∫Ω∇vqx−2∇v⋅∇φdx=∫Ωλqxbxguτvψdx,for allφ,ψ∈W01,pxΩ×W01,qxΩwithφ,ψ≥0.
Here, and hereafter, we will use the notationdx,∂Ω to denote the distance of x∈Ω to denote the distance of Ω. Denote dx=dx,∂Ω and ∂Ωε=x∈Ω:dx,∂Ω<ε.
Since∂Ω is C2 regularly, there exists a constant δ∈0,1 such that dx∈C2∂Ω3δ¯ and ∇dx=1.
Denote(21)v1x=γdx,dx<δ,γδ+∫δdxγ2δ−tδ2/p−−1a22/p−−1dt,δ≤dx<2δ,γδ+∫δ2δγ2δ−tδ2/p−−1a22/p−−1dt,2δ≤dx.v2x=γdx,dx<δ,γδ+∫δdxγ2δ−tδ2/p−−1b22/p−−1dt,δ≤dx<2δ,γδ+∫δ2δγ2δ−tδ2/p−−1b22/p−−1dt,2δ≤dx.
Obviously,0≤v1x,v2x∈C1Ω¯. Considering(22)−Δpxωx=ηinΩ,ω=on∂Ω,we have the following resultLemma 2 (Lemma 2.1 in [52]).
If positive parameter η is large enough and ω is the unique solution of (22), then we have(i)
For anyθ∈0,1, there exists a positive constant C1 such that(23)C1η1/p+−1+θ≤maxx∈Ω¯ωx,(ii)
and, there exists a positive constantC2 such that(24)maxx∈Ω¯ωx≤C2η1/p−−1.


## 2.1. Plate Problems and Its History

In this paper, we consider the system of differential equations:(1)−Δpxu=λpxaxfuhvinΩ,−Δqxv=λqxbxguτvinΩ,u=v=0on∂Ω,where Ω⊂ℝN is a bounded smooth domain with C2 boundary ∂Ω and 1<px,qx∈C1Ω¯ are functions with 1<p−≔infΩpx≤p+≔supΩpx<∞, 1<q−≔infΩqx≤q+≔supΩqx, and Δpx is a px-Laplacian defined as(2)Δpxu=div∇upx−2∇u,and a,b:Ω¯⟶ℝ+ are continuous functions, while f,g,h,andτ are monotone functions in ℝ+ such that(3)limu⟶+∞fu=+∞,limu⟶+∞gu=+∞,limu⟶+∞hu=+∞,limu⟶+∞τu=+∞,satisfying some natural growth condition at u=∞.We point out that the extension fromp-Laplace operator to px-Laplace operator is not trivial, since the px-Laplacian has a more complicated structure then the p-Laplace operator, such as it is nonhomogeneous. Moreover, many results and methods for p-Laplacians are not valid for the px-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient(4)λpx=infu∈W01,pxΩ\0∫Ω1/px∇upxdx∫Ω1/pxupxdx,is zero in general, and only under some special conditions, λpx is positive (see [53]). Maybe the first eigenvalue and the first eigenfunction of the px-Laplacian do not exist, but the fact that the first eigenvalue λp is positive and the existence of the first eigenfunction are very important in the study of p-Laplacian problem. There are more difficulties in discussing the existence of solutions of variable exponent problems. In [59], the authors considered the existence of positive weak solutions for the following p-Laplacian problem:(5)−Δpu=λfvinΩ,−Δpu=λguinΩ,u=v=0on∂Ω,where the first eigenfunction has been used to construct the subsolution of p-Laplacian problem. Under the condition that for all M>0,(6)limu⟶+∞fMgu1/p−1up−1=0,the authors gave the existence of positive solutions for problem (5) provided that λ is large enough.In [48], the existence and nonexistence of positive weak solutions to the following quasilinear elliptic system:(7)−Δpu=λuαvγinΩ,−Δqu=λuδvβinΩ,u=v=0on∂Ω,has been considered where the first eigenfunction has been used to construct the subsolution of problem (7) and the following results were obtained:(i)
Ifα,β≥0,γ,δ>0,θ=p−1−αq−1−β−γδ>0, then problem (7) has a positive weak solution for each λ>0.(ii)
Ifθ=0 and pγ=qp−1−α, then there exists λ0>0 such that for 0<λ<λ0, problem (7) has no nontrivial nonnegative weak solution. For further generalizations of system (7), we refer to [49, 50].As already discussed before, on thepx-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of the px-Laplacian do not exist even if the first eigenfunction of the px-Laplacian exists. Because of the nonhomogeneous property of the px-Laplacian, the first eigenfunction cannot be used to construct the subsolutions of px-Laplacian problems. Moreover, in [47, 61], the authors studied the existence of solutions for problem (5), where some symmetry conditions are imposed. Then, in [46], the existence of positive solutions of the system was investigated:(8)−Δpxu=λpxfvinΩ,−Δpxu=λpxguinΩ,u=v=0onΩ,without any symmetry conditions. Motivated by the ideas introduced in [47], the authors proved the existence of a positive solution when λ is large enough and satisfies condition (6) and they did not assume any symmetric condition and did not assume any sign condition on f0 and g0. Also the authors proved the existence of positive solutions with multiparameters; in this paper, we extend this given system of differential equations, where we establish the existence of a positive solution for a new class of this system with respect to the symmetry conditions by constructing a positive subsolution and supersolution and p,q∈C1Ω¯ are functions, λ,λ1,λ2,μ1, and μ2 are positive parameters, and Ω⊂ℝN is a bounded domain and we did not assume any sign condition on f0, g0, h0, and τ0.


## 2.2. Preliminary Results

In order to discuss problem (1), we need some theories on W01,pxΩ which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces W01,pxΩ which will be used later (for details, see [54]).Let us define(9)LpxΩ=u:uis a measurable real−valued function such that∫Ωuxpxdx<∞.We introduce the norm onLpxΩ by(10)uxpx=infλ>0:∫Ωuxλpxdx≤1,W1,pxΩ=u∈LpxΩ;∇u∈LpxΩ,with the norm(11)u=upx+∇upx,∀u∈W1,pxΩ.We denote byW01,pxΩ the closure of C0∞Ω in W1,pxΩ.Proposition 1 (see [59]).
The spacesLpxΩ,W1,pxΩ, and W01,pxΩ are separable and reflexive Banach spaces.Throughout the paper, we will assume thatH1p,q∈C1Ω¯ and 1<p−≤p+,1<q−≤q+H2f,g,h,τ:ℝ+⟶ℝ are C1 monotone functions such that(12)limu⟶+∞fu=+∞,limu⟶+∞gu=+∞;limu⟶+∞hu=+∞,limu⟶+∞τu=+∞.H3∃r>0 such that(13)limu⟶+∞fuhcur/q−−1up−−1=0,for allc>0,(14)limu⟶+∞guτkur/q−−1ur=0,for allk>0.H4a,b:Ω¯⟶ℝ+ are continuous functions, such that(15)a1=minx∈Ω¯ax,b1=minx∈Ω¯bx,a2=maxx∈Ω¯ax,b2=maxx∈Ω¯bx.We define(16)Lu,v=∫Ω∇upx−2∇u∇vdx,∀u,v∈W01,pxΩ.Then,L:W01,pxΩ⟶W01,pxΩ∗ is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [61], Theorem 3.1).DefineA:W01,pxΩ⟶W01,pxΩ∗ as for all u,φ∈W01,pxΩ,(17)Au,φ=∫Ω∇upx−2∇u∇φ+hx,uφdx,where hx,u is continuous on Ω¯×ℝ and hx is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [44], we have the following lemma:Lemma 1 (see [45]) (comparison principle).
Letu,v∈W01,px(Ω) satisfy(18)Au−Av≥0inW01,pxΩ∗,φx=minux−vx,0.
If(19)φx∈W01,pxΩ,i.eu≥von∂Ω,
thenu≥va.e. in Ω.Definition 1.
Letu,v∈W01,pxΩ×W01,qxΩ; the couple u,v is said to be a weak solution of (1) if it satisfies(20)∫Ω∇upx−2∇u⋅∇φdx=∫Ωλpxaxfuhvφdx,∫Ω∇vqx−2∇v⋅∇φdx=∫Ωλqxbxguτvψdx,for allφ,ψ∈W01,pxΩ×W01,qxΩwithφ,ψ≥0.
Here, and hereafter, we will use the notationdx,∂Ω to denote the distance of x∈Ω to denote the distance of Ω. Denote dx=dx,∂Ω and ∂Ωε=x∈Ω:dx,∂Ω<ε.
Since∂Ω is C2 regularly, there exists a constant δ∈0,1 such that dx∈C2∂Ω3δ¯ and ∇dx=1.
Denote(21)v1x=γdx,dx<δ,γδ+∫δdxγ2δ−tδ2/p−−1a22/p−−1dt,δ≤dx<2δ,γδ+∫δ2δγ2δ−tδ2/p−−1a22/p−−1dt,2δ≤dx.v2x=γdx,dx<δ,γδ+∫δdxγ2δ−tδ2/p−−1b22/p−−1dt,δ≤dx<2δ,γδ+∫δ2δγ2δ−tδ2/p−−1b22/p−−1dt,2δ≤dx.
Obviously,0≤v1x,v2x∈C1Ω¯. Considering(22)−Δpxωx=ηinΩ,ω=on∂Ω,we have the following resultLemma 2 (Lemma 2.1 in [52]).
If positive parameter η is large enough and ω is the unique solution of (22), then we have(i)
For anyθ∈0,1, there exists a positive constant C1 such that(23)C1η1/p+−1+θ≤maxx∈Ω¯ωx,(ii)
and, there exists a positive constantC2 such that(24)maxx∈Ω¯ωx≤C2η1/p−−1.


## 3. Main Result

In the following, when there is no misunderstanding, we always useCi to denote positive constants.Theorem 1.
Assume that the conditionsH1−H4 are satisfied. Then problem (1) has a positive solution when λ is large enough.Proof.
We shall establish Theorem1 by constructing a positive subsolution ϕ1,ϕ2 and supersolution z1,z2 of (1) such that ϕ1≤z1 and ϕ2≤z2, that is, ϕ1,ϕ2 and z1,z2 satisfy(25)∫Ω∇ϕ1px−2∇ϕ1⋅∇φdx≤∫Ωλpxaxfϕ1hϕ2φdx,∫Ω∇ϕ2qx−2∇ϕ2⋅∇ψdx≤∫Ωλqxbxgϕ1τϕ2ψdx,∫Ω∇z1px−2∇z1⋅∇φdx≥∫Ωλpxaxfz1hz2φdx,∫Ω∇z2qx−2∇z2⋅∇ψdx≥∫Ωλqxbxgz1τz2ψdx,for all φ,ψ∈W01,pxΩ×W01,qxΩ with φ,ψ≥0. According to the sub-supersolution method for px-Laplacian equations (see [52]), problem (1) has a positive solution.Step 1.
We will construct a subsolution of (1). Let σ∈0,δ be small enough. Denote(26)ϕ1x=ekdx−1,dx<σ,ekσ−1+∫σdxkekσ2δ−t2δ−σ2/p−−1a12/p−−1dt,σ≤dx<2δ,ekσ−1+∫σ2δkekσ2δ−t2δ−σ2/p−−1a12/p−−1dt,2δ≤dx,ϕ2x=ekdx−1,dx<σ,ekσ−1+∫σdxkekσ2δ−t2δ−σ2/p−−1b12/q−−1dt,σ≤dx<2δ,ekσ−1+∫σ2δkekσ2δ−t2δ−σ2/p−−1b12/q−−1dt,2δ≤dx.
It is easy to see thatϕ1,ϕ2∈C1Ω¯.
Denote(27)α=mininfpx−14sup∇px+1,infqx−14sup∇qx+1,1,ζ=min−αa1f0h01/p+;−αa1f0h01/p−;−αb1g0τ01/q+;−αb1g0τ01/q−,iff0h0<0,g0τ0<0,min−αa1f0h01/p+;−αa1f0h01/p−,iff0h0<0,g0τ0>0,min−αb1g0τ01/q+;−αb1g0τ01/q−,iff0h0>0,g0τ0<0,1,iff0h0>0,g0τ0>0.
By some simple computations, we can obtain(28)−Δpxϕ1=−kekdxpx−1×px−1+dx+lnkk∇p∇d+Δdk,dx<σ,12δ−σ2px−1p−−1−2δ−d2δ−σ×lnkekσ×2δ−d2δ−σ2/p−−1∇p∇d+Δd×Kekσpx−12δ−d2δ−σ2px−1/p−−1−1a1,σ≤dx<2δ,0,2δ≤dx,−Δqxϕ2=−kekdxqx−1×qx−1+dx+lnkk∇q∇d+Δdk,dx<σ,12δ−σ2qx−1q−−1−2δ−d2δ−σ×lnkekσ×2δ−d2δ−σ2/q−−1∇q∇d+Δd×Kekσqx−12δ−d2δ−σ2qx−1/q−−1−1b1,σ≤dx<2δ,0,2δ≤dx.
FromH2, there exists a positive constant M>1 such that(29)fM−1hM−1≥1,gM−1τM−1≥1.
Letσ=1/klnM, then(30)σk=lnM.
Ifk is sufficiently large, from (30), we have(31)−Δpxϕ1≤−kpxα,dx<σ.
Letλ=ζk. We claim that(32)−kpxα≤a1f0h0λpx,∀x∈Ω,Indeed, by definition of λ, the last inequality is obvious when f0h0>0.
Whenf0h0<0, we can notice that(33)λk≤−αa1f0h01/px,∀x∈Ω,Then, we have(34)−Δpxϕ1≤−kpxα≤λpxa1fϕ1hϕ2,dx<σ.
Sincedx∈C2∂Ω3δ¯, there exists a positive constant C3 such that(35)−Δpxϕ1≤Kekσpx−1×2δ−d2δ−σ2px−1/p−−1−1a1×12δ−σ2px−1p−−1−2δ−d2δ−σ×lnkekσ×2δ−d2δ−σ2/p−−1∇p∇d+Δd,≤C3Kekσpx−1a1lnk,σ≤dx<2δ.
Ifk is sufficiently large, we have(36)C3kekσpx−1a1lnk=C3kMpx−1a1lnk≤λpxa1.Then,(37)−Δpxϕ1≤λpxa1,σ≤dx<2δ.
Sinceϕ1x,ϕ2x, and f,h are monotone, when λ is large enough, we have(38)−Δpxϕ1≤λpxa1fϕ1hϕ2,σ≤dx<2δ,(39)−Δpxϕ1=0≤λpxa1≤λpxa1fϕ1hϕ2,2δ≤dx.
Combining (34), (38), and (39), we can deduce that(40)−Δpxϕ1≤λpxaxfϕ1hϕ2,a.e.onΩ.
Similarly,(41)−Δqxϕ2≤λqxbxgϕ1τϕ2,a.e.onΩ.
From (40) and (41), we can see that ϕ1,ϕ2 is a subsolution of problem (1).Step 2.
We will construct a supersolution of problem (1); we consider(42)−Δpxz1=λp+a2μinΩ,−Δqxz2=λq+b2βrinΩ,z1=z2=0on∂Ω,where r>0 is the positive number that verifies H3 and β=maxx∈Ω¯z1x. We shall prove that z1,z2 is a supersolution of problem (1).
Forψ∈W01,qxΩ with ψ≥0, it is easy to see that(43)∫Ω∇z2qx−2∇z2⋅∇ψdx=∫Ωλq+b2βrψdx.
ByH4, for a μ large enough, using Lemma 2, we have(44)βr≥gβτC2λq+b2βr1/q−−1.
Hence,(45)∫Ω∇z2qx−2∇z2⋅∇ψdx≥∫Ωλq+b2gmaxz1τmaxz2ψdx,≥∫Ωλqxbxgz1τz2ψdx.
Also, forφ∈W01,pxΩ with φ≥0, it is easy to see that(46)∫Ω∇z1px−2∇z1⋅∇φdx=∫Ωλp+a2μφdx.
ByH3 and Lemma 2, when μ is sufficiently large, we have(47)a2λp+μ≥1C2βp−−1≥a2λp+fβhC2λq+b2βr1/q−−1.
Then,(48)∫Ω∇z1px−2∇z1⋅∇φdx≥∫Ωλpxaxfz1hz2φdx.
According to (45) and (48), we can conclude that z1,z2 is a supersolution of problem (1). It only remains to prove that ϕ1≤z1 and ϕ2≤z2.
In the definition ofv1x, let(49)γ=2δmaxΩ¯ϕ1x+maxΩ¯∇ϕ1x.
We claim that(50)ϕ1x≤v1x,∀x∈Ω.
From the definition ofv1, it is easy to see that(51)ϕ1x≤2maxΩ¯ϕ1x≤v1x,whendx=δ,ϕ1x≤2maxΩ¯ϕ1x≤v1x,whendx≥δ,ϕ1x≤v1x,whendx<δ.
Sincev1−ϕ1∈C1∂Ωδ¯, there exists a point x0∈∂Ωδ¯ such that(52)v1x0−ϕ1x0=minx0∈∂Ωδ¯v1x0−ϕ1x0.
Ifv1x0−ϕ1x0<0, it is easy to see that 0<dx<δ and then(53)∇v1x0−∇ϕ1x0=0.
From the definition ofv1, we have(54)∇v1x0=γ=2δmaxΩ¯ϕ1x0+maxΩ¯∇ϕ1x0>∇ϕ1x0.
It is a contradiction to(55)∇v1x0−∇ϕ1x0=0.
Thus, (50) is valid.
Obviously, there exists a positive constantC3 such that γ≤C3λ.
Sincedx∈C2∂Ω3δ¯, according to the proof of Lemma 2, there exists a positive constant C4 such that(56)−Δpxv1x≤C∗γpx−1+θ≤C4λpx−1+θ,a.e,inΩ,whereθ∈0,1.
Whenη≥λp+ is large enough, we have −Δpxv1x≤η.
According to the comparison principle, we have(57)v1x≤ωx.
From (50) and (57), when η≥λp+ and λ≥1 are sufficiently large, we have, for all x∈Ω,(58)ϕ1x≤v1x≤ωx.
According to the comparison principle, whenμ is large enough, we have, for all x∈Ω,(59)v1x≤ωx≤z1x.
Combining the definition ofv1x and (58), it is easy to see that, for all x∈Ω,(60)ϕ1x≤v1x≤ωx≤z1x.
Whenμ≥1 and λ is large enough, from Lemma 2, we can see that β is large enough, and then λq+b2βr is a large enough. Similarly, we have ϕ2≤z2. This completes the proof.


## 4. Conclusion

Validity of the comparison principle and of the SSM for local and nonlocal problems as the stationary and evolutionary Kirchhoff Equation was an important subject in the last few years (see, for example, [44, 53, 58, 62–66]), where the authors showed by giving different counterexamples that the simple assumption M increasing somewhere is enough to make the comparison principle and SSM hold false contradiction and clear up some results in the literature. Moreover, the two conditions that M is nonincreasing and H is increasing turn out to be necessary and sufficient, at least for the validity of the comparison principle. It is worth to note that in [45, 67], C. O. Alves and F. J. S. A. Correa developed a new SSM for problem (1) to deal with the increasing M case. The result is obtained by using a kind of Minty–Browder theorem for a suitable pseudomonotone operator, but instead of constructing a subsolution, the authors assumed the existence of a whole family of functions which satisfy a stronger condition than just being subsolutions; the inconvenience is that these stronger conditions restrict the possible right hand sides in (1). Another SSM for nonlocal problems is obtained in [45] for a problem involving a nonlocal term with a Lebesgue norm, instead of the Sobolev norm appearing in (1).



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*Source: 3756406-2020-01-20.xml*