# A Sub-Supersolution Method forp-Laplacian Equation with Non-Local Term

Authors: Mei Rong; Qing Miao
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper is concerned with the existence of solutions forp-Laplace problems with non-local term. We prove the sub-supersolution theorem using the pseudomonotone operator theorem and Minty–Browder theorem with appropriate assumptions on M,gii=1,2. Then, using the sub-supersolution approach, we determine the existence of positive solutions to specific problems.


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## Body

## 1. Introduction

We study a problem for non-locality elliptic equations modelled after thep-Laplacian(1)−Δpu=M∫Ωg1x,udxg1x,u+g2x,u,inΩ,u=0,on∂Ω,where Ω is a bounded domain in ℝNN>1, −Δpu=−div∇up−2∇u, p>1, is the p-Laplacian operator, 1/p+1/p′=1,f,gii=1,2:Ω¯×ℝ⟶ℝ are continuous and satisfy certain conditions.p-Laplacian equation is a very important part of solving certain issues. For instance, it can be used in the famous non-Newtonian fluid theory (for more information, see [1, 2]). In addition to this, it can also be used to explain other physical phenomena: reaction-diffusion problem, flow through porous media, non-linear elasticity, etc. Certainly, it is also widely used in practical problems such as oil exploitation.The interest in these problems comes from [3–5]. The authors studied the categories of non-local problems, which appear in some fields of applied mathematics. More precisely, in [3], the authors mentioned that the importance of this model is that the measurements used to determine the physical constants are not carried out at a certain point, but represent the average value near a certain point, so these physical constants depend on the local average value.This paper mainly uses the sub-supersolution method, which is of great significance and application value to study the existence of solutions of equation. In particular, scholars have used this method to study thep=2 case, that is, Δpu=Δu, and have got many results. More exactly, in [6], the authors discussed the semilinear problem:(2)−Δu=fλx,u,inΩ,u=0,on∂Ω.Using the sub-supersolution and variational methods, whereλ is the real parameter, fλx,u=λu+up−1u1<p<N+2/N−2 which verified the existence and multiplicity of solutions of the equation. In addition, many scholars have made many conclusions on the study of Laplacian problems by using the sub-supersolution method [4, 5, 7, 8].Ifp≠2, the sub-supersolution method can also be used for research. For example, in [9], the authors studied quasilinear elliptic equations with homogeneous Dirichlet boundary conditions:(3)−Δpu=fx,u,inΩ,u=0,on∂Ω.On the one hand, the authors obtained the minimum positive solution and the maximum negative solution of the problem by using the sub-supersolution method. On the other hand, combined with the mountain pass lemma and the second deformation theorem, it was proved that the equation had five solutions: two positive solutions, two negative solutions, and one sign changing solution. In addition, in [10], the existence of maximum and minimum solutions of quasilinear problem(4)−Δpu=fx,u,∇u,inΩ,u=0,on∂Ω,was explored by using the sub-supersolution method. Firstly, when f=fx,s, the authors explained the sub-supersolution theorem of the equation by monotone iterative method. Secondly, when f=fx,s,t, the authors obtained the sub-supersolution theorem by using pseudomonotone theorem. Such problems have produced many research results (see [11–17] for details).In this paper, the sub-supersolution method is used mainly to prove the existence of the solution top-Laplacian problem. Even though there are many research theories, there are still research difficulties. For one thing, there are many ways to prove the sub-supersolution theorem. For example, Schaefer’s fixed point theorem is used in [7, 8, 11, 18, 19], pseudomonotone operator theorem is used in [4, 5, 10], and the idea of monotone iteration is used in [20–22] to explore the sub-supersolution theorem of the equation. We must consider the applicability and feasibility of the above methods. For another, because the equation has non-local term, the processing process is more complex.Theorem 1.
There is a pair of sub-supersolutionu¯,u¯ for (1), and if equation (1) satisfy the following conditions:
M0Mt:ℝ+⟶ℝ is continuous, there are constants m1>1, such that(5)Mt≤m1.
inΩ¯;
G0gix,sx is a continuous function with respect to sx and for allx∈Ω,sx∈u¯,u¯ and has growth condition:(6)gix,sx≤k0+θ0sxr,where k0,θ0 are positive constants, and r<p−1.
Then, (1) has at least one solution u∈W01,pΩ,u¯≤u≤u¯.Theorem 2.
Problem (1) has a set of sub-supersolutions u¯,u¯, and if problem (1) satisfies conditions G0,r<p−1/2, and M1Mt:ℝ+⟶ℝ and is continuous function, there are positive constants a,b such that(7)Mt≤a+bt,in Ω¯.
Then, (1) has at least one solution u∈W01,pΩ,u¯≤u≤u¯.
In [5], the authors studied the quasilinear Kirchhoff type equation with p=2; in contrast with [5], we use the s+ condition of p-Laplacian operator to prove the strong convergence of sequences in space W01,pΩ. Moreover, compared with [4, 5], this paper studies the p-Laplace problem, which further expands the scope of applying pseudomonotone operator theory to this kind of problem.


## 2. Preliminaries

The following notations will be used later. In this paper, it is discussed in spaceW01,pΩ,p>1, and its norm is defined as u=∫Ω∇updx1/p. Denote LpΩ norm by up=∫Ωupdx1/p.Proposition 1.
(Sobolev embedding theorem). LetΩ be a bounded region in ℝN,1≤p<∞,∀u∈W01,pΩ and W01,pΩ be separable reflexive Banach spaces; then, W01,pΩ embedded into LpΩ is compact and continuous, and up≤Cu, where C is a constant.
Later, we will considerS+ condition of the operator Δp:(8)∀un⇀uinW01,pΩwithlimsupn⟶+∞−Δpun,un−u≤0,and then ||un−u||⟶0.
If necessary, we will give the definition of the first eigenvalueλ1Ω of operator −Δp, i.e., λ1Ω=min∫Ω∇updx:u∈W01,pΩand∫Ωupdx=1. The eigenvalue function φx∈W01,pΩ has the following properties: φΩ>0,forx∈Ω, and ∫Ωφxpdx=1.Definition 1.
(1)
We say thatu∈W01,pΩ is a weak solution of (1) if(9)∫Ω∇up−2∇u∇φdx=∫ΩM∫Ωg1x,udxg1x,u+g2x,uφdx,for allφ∈W01,pΩ.(2)
We say thatu∈W01,pΩ is a weak subsolution (or supersolution) of (1) if u≤0or≥0 on ∂Ω and(10)∫Ω∇up−2∇u∇φdx≤or≥∫ΩM∫Ωg1x,udxg1x,u+g2x,uφdx,for allφ∈W01,pΩ satisfying φ≥0.Proposition 2.
Consideru,v∈W1,pΩ. If −Δpu≤−Δpv and u≤v on ∂Ω, then u≤v in Ω.Definition 2.
LetX be Banach space, X∗ be its dual space, and B:X⟶X∗ be an operator. If un⇀u and(11)limsupn⟶+∞Bun,un−u≤0,liminfn⟶+∞Bun,un−v≥Bu,u−v,∀v∈Xcan be obtained, then B is called a pseudomonotone operator.Proposition 3 (minty–browder).
LetX be a reflexive and separable Banach space and B:X⟶X∗ be an operator satisfying the following:(i)
B is coercive, i.e.,Bu,u/u⟶+∞ as u⟶+∞.(ii)
B is bounded and continuous.(iii)
B is pseudomonotone.
Then,B is surjective, that is, BX=X∗.


### 2.1. Proofs of Main Results

Proof of Theorem 1.
DefinitionB is an operator function from W01,pΩto its dual space W−1,p′Ω, and is given by(12)Bu,v=∫Ω∇up−2∇u∇vdx−∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uvdx,where u¯,u¯ is the supersolution and subsolution in Theorem 1, respectively, and(13)gi˜x,u=gix,u¯u≤u¯,gix,uu¯≤u≤u¯,gix,u¯u≥u¯,where i=1,2. Define function(14)γx,u=−u¯x−ux+l+ux−u¯x+l,for all l∈0,1,x∈Ω. The following will prove Bu=0 by using the Minty–Browder theorem.
Firstly, we need to prove thatB is coercive. On the one hand, according to the condition that gix,u satisfies G0, using Proposition 2, we can get(15)∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,uudx≤m1∫Ωg1˜x,u+g2˜x,uudx≤m1∫Ωc1u+c2ur+1dx≤m1c1u+c2ur+1.
On the other hand,(16)γx,u≤u¯x−ul+u−u¯xl≤c3+c4ul,and therefore −c3−c4ul≤γx,u≤c3+c4ul. Then,(17)∫Ωγx,uudx≥−∫Ωc3u+c4ul+1dx≥−c5u−c6ul+1.
According to (15) and (17), it is concluded that(18)Bu,u≥up−m1c1u+c2ur+1−c5u−c6ul+1,≥up−m1c1+c5u−m1c2ur+1−c6ul+1,since l∈0,1 and r+1<p; then, there are Bu,u/u⟶+∞ if u⟶+∞. It shows that operator B is continuous and bounded.
Next, we will prove thatB is pseudomonotone. Because(19)limsupn⟶+∞Bun,un−u=limsupn⟶+∞∫Ω∇unp−2∇un∇un−udx−∫ΩM∫Ωg1˜x,undxg1˜x,un+g2˜x,un−γx,unun−udx≤0,when n⟶+∞, it can be concluded that(20)∫Ω∇unp−2∇un∇un−udx≤∫ΩM∫Ωg1˜(x,undxg1˜x,un+g2˜x,un−γx,unun−udx≤∫ΩM∫Ωg1˜(x,undxg1˜x,un+g2˜x,un−γx,unp′dx1/p′∫Ωun−updx1/p.
Sinceun converges weakly in space W01,pΩ,un converges strongly in space LpΩ, that is, ∫Ωun−updx⟶0, so −Δpun,un−u⟶0; by condition S+, we know that un⟶u in W01,pΩ. Then,(21)liminfn⟶∞Bun,un−v=Bu,u−v.
Consequently, there isu∈W01,pΩ such that Bu=0.
Finally, we will prove thatu¯≤u≤u¯. Because Bu=0, u is the solution of equation(22)−Δpu=M∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uinΩ,u=0on∂Ω.
Take test functionv=u−u¯+=maxu−u¯,0, and thus(23)−Δpu,u−u¯+=∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uu−u¯+dx≤∫ΩM∫Ωg1˜x,u¯dxg1˜x,u¯+g2˜x,u¯−u−u¯+lu−u¯+dx≤−Δpu¯,u−u¯+−∫Ωu−u¯+l+1dx.
Then,0≤−Δpu+Δpu¯,u−u¯+≤−∫Ωu−u¯+l+1dx≤0, which implies u−u¯+=0, and hence u<u¯. By the same method, we can get u>u¯.Proof of Theorem 2.
On the basis of Theorem1, we only need to explain that B is coercive. Other processes are similar to Theorem 1, and we will not repeat it here.
According toM1 and G0, we obtain the following estimate:(24)M∫Ωg1˜x,udxg1˜x,u≤a+b∫Ωg1˜x,udxg1˜x,u≤ak0+θ0ur+b∫Ωk0+θ0urdxk0+θ0ur≤ak0+aθ0ur+bk02Ω+θ0k0ur+k0θ0Ω+θ02urur.
Then,(25)∫ΩM∫Ωg1˜x,udxg1˜x,uudx≤∫Ωak0+aθ0ur+bk02Ω+θ0k0ur+k0θ0Ω+θ02ur|u|rudx≤bk02Ω+ak0u+aθ0+bθ0k0+k0θ0Ωur+1+θ02u2r+1=Pu+Qur+1+θ02u2r+1,where P=Pa,b,k0,Ω,Q=a,b,k0,Ω,θ0. Also, ∫Ωg2˜x,uudx≤k0u+θ0ur+1, and r<p−1/2; therefore,(26)Bu,u≥up−c1′u−c2′ur+1−c3′u2r+1−c6ul+1.
That is,B is coercive.


## 2.1. Proofs of Main Results

Proof of Theorem 1.
DefinitionB is an operator function from W01,pΩto its dual space W−1,p′Ω, and is given by(12)Bu,v=∫Ω∇up−2∇u∇vdx−∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uvdx,where u¯,u¯ is the supersolution and subsolution in Theorem 1, respectively, and(13)gi˜x,u=gix,u¯u≤u¯,gix,uu¯≤u≤u¯,gix,u¯u≥u¯,where i=1,2. Define function(14)γx,u=−u¯x−ux+l+ux−u¯x+l,for all l∈0,1,x∈Ω. The following will prove Bu=0 by using the Minty–Browder theorem.
Firstly, we need to prove thatB is coercive. On the one hand, according to the condition that gix,u satisfies G0, using Proposition 2, we can get(15)∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,uudx≤m1∫Ωg1˜x,u+g2˜x,uudx≤m1∫Ωc1u+c2ur+1dx≤m1c1u+c2ur+1.
On the other hand,(16)γx,u≤u¯x−ul+u−u¯xl≤c3+c4ul,and therefore −c3−c4ul≤γx,u≤c3+c4ul. Then,(17)∫Ωγx,uudx≥−∫Ωc3u+c4ul+1dx≥−c5u−c6ul+1.
According to (15) and (17), it is concluded that(18)Bu,u≥up−m1c1u+c2ur+1−c5u−c6ul+1,≥up−m1c1+c5u−m1c2ur+1−c6ul+1,since l∈0,1 and r+1<p; then, there are Bu,u/u⟶+∞ if u⟶+∞. It shows that operator B is continuous and bounded.
Next, we will prove thatB is pseudomonotone. Because(19)limsupn⟶+∞Bun,un−u=limsupn⟶+∞∫Ω∇unp−2∇un∇un−udx−∫ΩM∫Ωg1˜x,undxg1˜x,un+g2˜x,un−γx,unun−udx≤0,when n⟶+∞, it can be concluded that(20)∫Ω∇unp−2∇un∇un−udx≤∫ΩM∫Ωg1˜(x,undxg1˜x,un+g2˜x,un−γx,unun−udx≤∫ΩM∫Ωg1˜(x,undxg1˜x,un+g2˜x,un−γx,unp′dx1/p′∫Ωun−updx1/p.
Sinceun converges weakly in space W01,pΩ,un converges strongly in space LpΩ, that is, ∫Ωun−updx⟶0, so −Δpun,un−u⟶0; by condition S+, we know that un⟶u in W01,pΩ. Then,(21)liminfn⟶∞Bun,un−v=Bu,u−v.
Consequently, there isu∈W01,pΩ such that Bu=0.
Finally, we will prove thatu¯≤u≤u¯. Because Bu=0, u is the solution of equation(22)−Δpu=M∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uinΩ,u=0on∂Ω.
Take test functionv=u−u¯+=maxu−u¯,0, and thus(23)−Δpu,u−u¯+=∫ΩM∫Ωg1˜x,udxg1˜x,u+g2˜x,u−γx,uu−u¯+dx≤∫ΩM∫Ωg1˜x,u¯dxg1˜x,u¯+g2˜x,u¯−u−u¯+lu−u¯+dx≤−Δpu¯,u−u¯+−∫Ωu−u¯+l+1dx.
Then,0≤−Δpu+Δpu¯,u−u¯+≤−∫Ωu−u¯+l+1dx≤0, which implies u−u¯+=0, and hence u<u¯. By the same method, we can get u>u¯.Proof of Theorem 2.
On the basis of Theorem1, we only need to explain that B is coercive. Other processes are similar to Theorem 1, and we will not repeat it here.
According toM1 and G0, we obtain the following estimate:(24)M∫Ωg1˜x,udxg1˜x,u≤a+b∫Ωg1˜x,udxg1˜x,u≤ak0+θ0ur+b∫Ωk0+θ0urdxk0+θ0ur≤ak0+aθ0ur+bk02Ω+θ0k0ur+k0θ0Ω+θ02urur.
Then,(25)∫ΩM∫Ωg1˜x,udxg1˜x,uudx≤∫Ωak0+aθ0ur+bk02Ω+θ0k0ur+k0θ0Ω+θ02ur|u|rudx≤bk02Ω+ak0u+aθ0+bθ0k0+k0θ0Ωur+1+θ02u2r+1=Pu+Qur+1+θ02u2r+1,where P=Pa,b,k0,Ω,Q=a,b,k0,Ω,θ0. Also, ∫Ωg2˜x,uudx≤k0u+θ0ur+1, and r<p−1/2; therefore,(26)Bu,u≥up−c1′u−c2′ur+1−c3′u2r+1−c6ul+1.
That is,B is coercive.


## 3. Applications

This section will introduce the application of Theorem1, and we will consider the positive solution of equation(27)−Δpu=M∫Ωurdxur+λuqinΩ,u=0on∂Ω.M2Mt:0,+∞⟶ℝ,t=∫Ωurdx, and there is a constant m∞>0 such that limt⟶+∞Mt=m∞ is uniform in Ω¯.Theorem 3.
If the functionMt satisfies conditions M0 and M2, q≤minr,p−1<+∞,λ>0, then (27) has a positive solution.Proof of Theorem 3.
First construct the supersolutionu¯. From(28)M∫Ωurdxur+λuq≤m1ur+λuq.
The equation:(29)−Δpu=m1ur+λuqinΩ,u=0on∂Ω,has a positive solution. Note that operator Km1,λ=−Δpu−m1ur−λuq because −Δpu is a continuous and bounded operator, W1,pΩ embedded intoLpΩ is compact, and Km1,λ is bounded and continuous.
On the other hand, iflimsupn⟶+∞−Δpun−m1unr−λunq,un−u≤0, because −Δpu,m1ur+λuq is bounded in W01,pΩ, there is a subsequence, which can be recorded as un, so that(30)limsupn⟶+∞−Δpun,un−u≤limsupn⟶+∞m1unr+λunq,un−u.
Sinceun⇀u in W1,p, un⟶u in LpΩ, so the right side of (30) tends to 0; then, limsupn⟶+∞−Δpun,un−u≤0; according to s+ condition of operator −Δp, we can get un⟶u in W01,pΩ. Therefore,(31)liminfn⟶+∞Km1,λun,un−v=Km1,λu,u−v,and then Km1,λ is pseudomonotone.
The last verification operatorKm1,λ will be coercive.(32)1p∫Ω∇updx−∫Ωm11r+1ur+1+λ1q+1uq+1dx≥1pup−cm1r+1ur+1−cλq+1uq+1.
Sincer+1,q+1<p and λ>0, Km1,λ is coercive. Then, there is u¯∈W01,p, s.t.(33)−Δpu¯=m1u¯r+λu¯qinΩ,u¯=0on∂Ω.
Take test function−u¯−=−max0,−u¯:(34)∫Ω∇u¯−pdx=−∫Ω∇u¯p−2∇u¯∇u¯−dx=−∫Ωm1u¯r+λu¯qu¯−dx≤0,and u≠0; therefore, u¯>0.
Now construct the subsolutionu¯. According to the definition of subsolution, we need to prove that(35)−Δpu¯≤M∫Ωurdxu¯r+λu¯q.
Becauselimt⟶+∞Mt=m∞uniforly in Ω¯, there is n0>0, when t>n0, Mt>m∞/2, and take u¯=εφ1, ε≥0 is enough small, and φ1∈W01,pΩ∩L∞Ω is the first eigenfunction of the operator −Δp, then the left side of the above inequality can be sorted as −Δpεφ1=λ1εφ1p−1. The right side is(36)M∫Ωurdxu¯r+λu¯q≥m0εφ1r+λεφ1q=m0εrφ1r+λεqφ1q,where m0=minm∞/2,a0,a0=minMt:t∈0,n0. The following will explain that λ1εp−1φ1p−1≤m0εrφ1r+λεqφ1q, and it may be proved that(37)λ1εp−1φ1∞p−1≤m0εrφ1r+λεqφ1q,that is,(38)λ1φ1∞p−1≤m0εr−p+1φ1r+λεq−p+1φ1q.
Therefore, whenε is sufficiently small, the above formula holds, so u¯=εφ1 is the subsolution of (27), and −Δpu¯≤−Δpu¯; then, u¯≥u¯.



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