# Strictly or Semitrivial Principal Eigensurface forp,q-Biharmonic Systems

Authors: Têlé Jonas Doumate; Liamidi Leadi; Lawouè Robert Toyou
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This paper extends the eigensurface ofp-bilaplacian operator to examine existence and simplicity of the first eigensurface for the third-order spectrum of p,q-biharmonic systems subject to boundary conditions.


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## Body

## 1. Introduction and Preliminary Results

We wish to investigate existence and simplicity of the first eigensurface for the following system:(1)Σ:Findu,v,β,λ∈YpqΩ\0,0×ℝN×ℝsuchthatΔp2u+H1β,u−λm1xup−2u=mxvα2+1uα1−1uinΩ,Δq2v+H2β,v−λm2xvq−2v=mxuα1+1vα2−1vinΩ,u=Δu=v=Δv=0on∂Ω,where
(i)
Ω⊂ℝN (with N≥1) is a bounded domain with smooth boundary ∂Ω(ii)
α1, α2, p, and q are constants such that α1≥0, α2≥0, p>1, q>1, and α1+1/p+α2+1/q=1(iii)
H1β,u=2β·∇Δup−2Δu+β2Δup−2Δu(iv)
H2β,v=2β·∇Δvq−2Δv+β2Δvq−2Δv(v)
YpqΩ=W2,pΩ∩W01,pΩ×W2,qΩ∩W01,qΩ(vi)
the weightsm, m1, m2∈L∞Ω are assumed to be nonnegatives in Ω(vii)
Δr2w=ΔΔwr−2Δw with r,w∈p,u,q,v denoting the r-biharmonic operatorNote that this operator describes a huge class of physical phenomena and its linear partΔ2≔Δ∘Δ is often seen as a prototypical example of biharmonic operator well known in the theory of elasticity. The spectrum of p-biharmonic operator has drawn much attention in recent works, and problems like Σ type appear in several branches of pure and applied mathematics, such as surface diffusion on solids, interface dynamics, thin plate theory, electrorheological fluids, thermorheological fluids, image restoration, and other phenomena related to electrical resistivity and polycrystal plasticity (see, for example, [1–10]).In [11], Ben Haddouch et al. have investigated the scalar version of problem Σ with m≡0, which reads
(2)Pa,p:Findu,β,Γ∈Xp\0×ℝN×ℝsuchthatΔΔup−2Δu+2β·∇Δup−2Δu+β2Δup−2Δu=Γaxup−2uinΩ,u=Δu=0on∂Ω,where a∈L∞Ω and Xp=W2,pΩ∩W01,pΩ. They proved that Pa,p possesses at least one sequence of positive eigensurfaces Γnp.,an defined by
(3)∀β∈ℝN,Γnpβ,a=infK⊂Bnsupu∈K∫Ωeβ.xΔupdx,where(4)Bn=K⊂Nβ:Kiscompact,SymmetricandγK≥n,Nβ=u∈Xp:∫Ωaeβ.xupdx=1,and Γnpβ,a⟶∞asn⟶∞.The authors in [12] gave the first eigensurface Γ1p.,a and showed that if a≥0 a.e. in Ω, then Γ1p.,a is simple (i.e., the associated eigenfunctions are a constant multiple of one another) and principal, i.e., the associated eigenfunction, denoted by φp,a is positive or negative on Ω with
(5)∀β∈ℝN,Γ1pβ,a=infu∈Nβ∫Ωeβ·xΔupdx.It is of interest to know thatu,β,λ is a solution of problem Pm1,p if and only if u,0;β,λ is a solution of Σ. This kind of solution is called “semitrivial solution” of Σ. Furthermore, if u,0;β,λβ is a solution of Σ with u of one sign on Ω, then λ. is called “semitrivial principal eigensurface” of Σ. Consequently, there are two forms of semitrivial solutions for problem Σ: one of the type u,0;β,λβ with u≡0 and u,β,λβ solution of the problem Pm1,p and the second of the type 0,v;β,λβ with v≡0 and v,β,λβ solution of the problem Pm2,q. In particular, Γ1p.,m1 and Γ1q.,m2 are semitrivial principal eigensurfaces of Σ.Motivated by the recent work in [13] where Σ was considered in the case β=0, our main goal in this paper is to show, in the presence of β, the simplicity and the existence of the first strictly principal eigensurface or semitrivial principal eigensurface of the system Σ.Throughout this work, the Lebesgue norm inLrΩ will be denoted by .r, ∀r∈1,∞ and the norm in a normed space X by .X. We denote by YpqΩ=W2,pΩ∩W01,pΩ×W2,qΩ∩W01,qΩ which is a reflexive Banach space endowed with the norm [7, 14]
(6)u,v=Δup+Δvq.The weak convergence inYpqΩ is denoted by ⇀. The positive and negative parts of a function w are denoted by w+=maxw,0 and w−=max−w,0. Equalities (and inequalities) between two functions must be understood a.e., and for all f∈LrΩ, the Poisson equation associated with the Dirichlet problem
(7)−Δu=fx,inΩ,u=0,on∂Ω,is uniquely solvable in Xr=W2,rΩ∩W01,rΩ (see [15]). The following lemma gives us some properties of the inverse operator of −Δ:Xr↦LrΩ denoted by Λ.Lemma 1 [16, 17].
(1)
Continuity: there exists a constantcr>0 such that(8)ΛfW2,r≤crfrholds for all r∈1,∞ and f∈LrΩ.
(2)
Continuity: givenk∈ℕ∗, there exists a constant cr,k>0 such that(9)ΛfWk+2,r≤cr,kfWk,rholds for all r∈1,∞ and f∈Wk,rΩ.
(3)
Symmetry: the identity(10)∫ΩΛu·vdx=∫Ωu.Λvdxholds for u∈LrΩ and v∈Lr′Ω with r′=r/r−1 and r∈1,∞.
(4)
Regularity: givenf∈L∞Ω, we have Λf∈C1,νΩ¯ for all ν∈0,1. Moreover, there exists cν>0 such that(11)ΛfC1,νΩ≤cνf∞.(5)
Regularity and Hopf-type maximum principle: letf∈CΩ¯ and f≥0 then
w=Λf∈C1,νΩ¯, for all ν∈0,1 and w satisfies: w>0 in Ω, ∂w/∂n<0 on ∂Ω.
(6)
Order-preserving property: givenf, g∈LrΩ if f≤g in Ω, then Λf<Λg in Ω.The rest of the paper is organized as follows. We construct an eigensurface curve associated to systemΣ in Section 2, and Section 3 is devoted to the study of strictly principal eigensurface or semitrivial principal eigensurface of Σ followed by simplicity results.


## 2. An Eigensurface Curve Associated to SystemΣ

To address the question of existence of solutions ofΣ, our analysis is partly based upon the “λ-dependent approach” used in several recent works, for instance, [18–22] and references therein. To this end, let us fix λ∈ℝ and embed the system Σ into a λ-dependent system Σλ by introducing a new real parameter μ as follows:
(12)Σλ:Findu,v,β,μ∈YpqΩ\0,0×ℝN×ℝsuchthatΔp2u+H1β,u−mxvα2+1uα1−1u−λm1xup−2u=μup−2uinΩ,Δq2v+H2β,v−mxuα1+1vα2−1v−λm2xvq−2v=μvq−2vinΩ,u=Δu=v=Δv=0on∂Ω.We now give equivalent versions of both systemsΣ and Σλ in the following lemma which is straightforward from [11].Lemma 2.
Forβ∈ℝN, the system Σλ) (resp. Σ) is equivalent to the system
(13)Sλ:Findu,v,μ∈YpqΩ\0,0×ℝsuchthatΔp2,βu−eβ·xmxvα2+1uα1−1u−λeβ·xm1xup−2u=μeβ·xup−2uinΩ,Δq2,βv−eβ·xmxuα1+1vα2−1v−λeβ·xm2xvq−2v=μeβ·xuvq−2vinΩ,u=Δu=0on∂Ω,(14)resp.S:Findu,v,λ∈YpqΩ\0,0×ℝsuchthatΔp2,βu−eβ·xmxvα2+1uα1−1u=λeβ·xm1xup−2uinΩΔq2,βv−eβ·xmxuα1+1vα2−1v=λeβ·xm2xvq−2v=inΩu=Δu=0on∂Ω,where Δp2,βu=Δeβ·xΔup−2Δu and Δq2,βv=Δeβ·xΔvq−2Δv.For the sake of clarity, we give a series of definitions that is used throughout this work.Definition 1.
(1)
The set of couplesβ,λ∈ℝN×ℝresp.β,μ∈ℝN×ℝ such that there exists a solution u,v,β,λ∈YpqΩ\0,0×ℝN×ℝ (resp. u,v,β,μ∈YpqΩ\0,0×ℝN×ℝ) of Σ (resp. Σλ) is called the third-order spectrum of the p,q-biharmonic operator plus potential. The couple β,λ (resp. β,μ) is then called a third-order eigenvalue and u,v is said to be an associated eigenfunction of Σ (resp. Σλ). Moreover, a set of third-order eigenvalues of the form β,fβ, for β∈ℝN and some function f:ℝN⟶ℝ, is called an eigensurface.(2)
Forβ∈ℝN, u,v;μ∈Yp,qΩ×ℝ is a (weak) solution to problem Sλ if(15)∫Ωeβ·xΔup−2ΔuΔφ1dx−∫Ωmeβ·xvβ+1uα−1uφ1dx−λ∫Ωm1eβ·xup−2uφ1dx=μ∫Ωeβ·xup−2uφ1dx,(16)∫Ωeβ·xΔvq−2ΔvΔφ2dx−∫Ωmeβ·xuα+1vβ−1vφ2dx−λ∫Ωm2eβ·xvq−2vφ2dx=μ∫Ωeβ·xvq−2vφ1dx,
for allφ1,φ2∈YpqΩ.
(3)
Forβ∈ℝN, u,v;λ∈Yp,qΩ×ℝ is a (weak) solution to problem Σ if(17)∫Ωeβ·xΔup−2ΔuΔφ1dx−∫Ωmeβ·xvβ+1uα−1uφ1dx=λ∫Ωm1eβ·xup−2uφ1dx,∫Ωeβ·xΔvq−2ΔvΔφ2dx−∫Ωmeβ·xuα+1vβ−1vφ2dx=λ∫Ωm2eβ·xvq−2vφ2dx,for all φ1,φ2∈YpqΩ.
(4)
Ifu,v;λ∈Yp,qΩ×ℝ (resp. u,v;μ∈Yp,qΩ×ℝ) is a (weak) solution to problem S (resp. Sλ), u,v shall be called an eigenfunction of the problem S (resp. Sλ) associated to the eigenvalue λ (resp. μλ). Let us agree to say that an eigenvalue of S or Sλ is strictly principal (resp., semitrivial principal) if it is associated to an eigenfunction u,v such that u>0 or u<0 and v>0 or v<0 (resp. [u>0 and v≡0 or u<0 and v≡0] or [u≡0 and v>0 or u≡0 and v<0]).(5)
Ifu,v,μβ,λ∈YpqΩ×ℝresp.u,v,λm,β,m1,m2∈YpqΩ×ℝ is a weak solution to problem Sλ (resp. S), u,v shall be called an eigenfunction of the problem Sλ (resp. S) associated to the eigenvalue μβ,λ (resp. λm,β,m1,m2). So u,v shall be called an eigenfunction of the problem Σ (resp. Σλ) associated to the eigensurface λm,.,m1,m2 (resp., μ.,λ). We can say that an eigensurface of Σ or Σλ is strictly principal (resp., semitrivial principal) if it is associated to an eigenfunction u,v such that u>0 or u<0 and v>0 or v<0 (resp. [u>0 and v≡0 or u<0 and v≡0] or [u≡0 and v>0 or u≡0 and v<0]).In a context of fixingλ and finding μ in terms of λ, we focus on the smallest eigenvalue μ. of system Σλ. For this, we define the energy functional
(18)Jλ,β:Yp,qΩ⟶ℝ,u,v↦Jλ,βu,v=Eβu,v−Vβu,v−λMβu,v,where
(19)Eβu,v=α1+1p∫Ωeβ·xΔupdx+α2+1q∫Ωeβ·xΔvqdx,Vβu,v=∫Ωmeβ·xuα1+1vα2+1dx,Mβu,v=α1+1pM1,βu+α2+1qM2,βv,with
(20)M1,βu=∫Ωm1eβ·xupdx,M2,βv=∫Ωm2eβ·xvqdx,∀u,v∈YpqΩ.Equalities (13) and (15) are equivalent to
(21)∇Jλ,βu,v=μ∇Iβu,v,where
(22)Iβu,v=α1+1p∫Ωeβ·xupdx+α2+1q∫Ωeβ·xvqdx,∀u,v∈YpqΩ.Lemma 3.
Letω1,ω2∈L∞Ω×L∞Ω. If ω1, ω2>0 on Ω, then there exist three positive constants c1,β, c2,β, and c3,β such that
(23)Δupp+Δvqq≤c1,βJλ,βu,v+c2,β∫Ωω1eβ·xupdx+c3,β∫Ωω2eβ·xvqdx,for every u,v∈YpqΩ and β∈ℝN.Proof.
We know, following ideas of [13], that there exist three positive constants c1, c2, and c3 such that
(24)Δupp+Δvqq≤c1Jλ,0u,v+c2∫Ωω1updx+c3∫Ωω2vqdx.
Considering the following function:(25)fβ:Ω¯⟶ℝ,x↦fβx=eβ·x.We can deduce the existence of two positive constants k1,β=minx∈Ω¯fβx and k2,β=maxx∈Ω¯fβx satisfying
(26)k1,β∫Ωw1updx≤∫Ωw1eβ·xupdx≤k2,β∫Ωw1updx,k1,β∫Ωw2vqdx≤∫Ωw2eβ·xvqdx≤k2,β∫Ωw2vqdx,k1,βJλ,0u,v≤Jλ,βu,v≤k2,βJλ,0u,v.
This gives(27)k1,βΔupp+Δvqq≤c1Jλ,βu,v+c2∫Ωω1eβ·xupdx+c3∫Ωω2eβ·xvqdx,leading to c1,β=c1/k1,β, c2,β=c2/k1,β, and c3,β=c3/k1,β and the result follows.Let us now consider the manifold(28)Mβ=u,v∈YpqΩ:Iβu,v=1.In the following, we construct for eachλ:β∈ℝN↦λβ∈ℝ the smallest eigensurface β∈ℝN↦μ1β,λβ from which we get the eigensurface curve λβ∈ℝ↦μ1β,λβ. This is the main result of this section.Proposition 1.
μ1.,λ. is the smallest eigensurface of Σλ with
(29)μ1β,λβ≔infJλ,βu,v:u,v∈Mβ,for β∈ℝN.Proof.
In the first step, Lemma3 ensures the existence of μ1β,λβ for all β∈ℝN. Furthermore, any sequence un,vn that minimizes Jλ,β on Mβ is bounded in YpqΩ. Thus, there exists u0,v0∈YpqΩ such that, up to a subsequence, the sequence un,vn converges weakly to u0,v0 in YpqΩ and strongly in LpΩ×LqΩ. Therefore,
(30)Jλ,βu0,v0≤limn⟶∞Jλ,βun,vn=μ1β,λβ,u0,v0∈M,and consequently, Jλ,βu0,v0=μ1β,λβ. By the Lagrange multipliers rule, μ1β,λβ is an eigenvalue for Sλ and u0,v0 is an associated eigenfunction. As a result, the value μ1.,λ. is an eigensurface for Σλ and u0,v0 is an associated eigenfunction.
In the second step, for any eigenvalueμβ,λβ associated to uλ,vλ∈YpqΩ\0,0, one has Jλ,βuλ,vλ=μβ,λβIβuλ,vλ with Iβuλ,vλ>0. Then,
(31)μ1β,λβ≤Jλ,βuλIβuλ,vλ1/p,vλIβuλ,vλ1/q=Jλ,βuλ,vλIβuλ,vλ=μβ,λβ,and we can draw the conclusion that μ1.,λ. is the smallest eigensurface of Σλ putting an end to the proof.Form=m1=m2≡0, we denote by
(32)μ0β=μ1β,λβ=infα1+1p∫Ωeβ·xΔupdx+α2+1q∫Ωeβ·xΔvqdx:u,v∈Mβ.Since the spaceW2,rΩ∩W01,rΩ with r∈p,q does not contain any constant nontrivial function, one has μ0β>0. The following properties are well known (see details of their proof in [13]) and are useful in proving the main result of the next section.Proposition 2.
(1)
μ1β,.:ℝ⟶ℝ is concave differentiable with(33)μ1′β,λ=∂μ1β,λ∂λ=−Mβu0,v0,where u0,v0 is some eigenfunction of Sλ associated to μ1β,λ for all λ∈ℝ.
(34)limλ⟶∞μ1β,λ=−∞.(2)
μ1β,. is strictly decreasing.


## 3. Strictly or Semitrivial Principal Eigensurface for SystemΣ

In the following lemma, we build a sufficient condition forμ1.,λ. to vanish as it is known that its zeros solve Σ.Lemma 4.
Ifm∞<μ0., then μ1.,0>0 and μ1.,λ.≡0 has a unique positive solution λ. (eigensurface of Σ).Proof.
We haveVβu,v≤m∞Iβu,v, ∀u,v∈YpqΩ so that
(35)Eβu,v−m∞Iβu,v≤Eβu,v−Vβu,v,∀u,v∈YpqΩ.
Then,(36)μ0β≤Eβu,v−Vβu,v+m∞,∀u,v∈Mβ,μ0β−m∞≤infEβu,v−Vβu,v,u,v∈Mβ≤μ1β,0.
Thus,μ1.,0>0, and from Proposition 1, μ1β,. is strictly decreasing. Consequently μ1.,λ.≡0 has a unique positive solution λ. which is an eigensurface of Σ.We will denote by(37)LΩ≔LpΩ×LqΩ\0,0×ℝ,L0Ω≔LpΩ×LqΩ\0,0×0.We apply some results proved by Drábek and Ôtani [16] and Talbi and Tsouli [17] and some ideas used by Leadi and Toyou [13] to make the following remarks.Remark 1.
(1)
∀u∈Xr, ∀v∈LrΩ with r∈1,∞: v=−Δu⇔u=Λv(2)
LetNr be the Nemytskii operator with r∈1,∞, defined by(38)Nrux=uxr−2ux,ifux≠0,0,ifux=0.
We have(39)∀v∈LrΩ,∀w∈Lr′Ω:Nrv=w⇔v=Nr′w,with r′=r/r−1.
(3)
Ifu,v is an eigenfunction of Σλ associated with μ., then for β∈ℝN, u,v is an eigenfunction of Sλ associated with μβ and φ=−Δu, w=−Δv satisfy(40)eβ·xNpφ=Λμλ+λm1eβ·xNpΛφ+meβ·xΛwα2+1Λφα1−1Λφ,eβ·xNqw=Λμλ+λm2eβ·xNqΛw+meβ·xΛφα1+1Λwα2−1Λw.
Hence,(a)
u0,v0;β,μλ is a solution of Σλ if and only if φ0,w0;μλ is a solution of problem(41)Sλ′:Findφ,w;μλ∈LΩsuchthateβ·xNpφ=Λμλ+λm1eβ·xNpΛφ+meβ·xΛwα2+1Λφα1−1Λφ,eβ·xNqw=Λμλ+λm2eβ·xNqΛw+meβ·xΛφα1+1Λwα2−1Λw,with φ0=−Δu0 and w0=−Δv0.
(b)
u0,v0;β,μλ is a solution of Σ if and only if φ0,w0;λ is a solution of problem(42)S′:Findφ,w;λ∈LΩsuchthateβ·xNpφ=Λλm1eβ·xNpΛφ+meβ·xΛwα2+1Λφα1−1Λφ,eβ·xNqw=Λλm2eβ·xNqΛw+meβ·xΛφα1+1Λwα2−1Λw,with φ0=−Δu0 and w0=−Δv0.
(c)
φ0,w0;μλ∈L0Ω is a solution of Sλ′ if and only if φ0,w0;λ∈LΩ is a solution of problem S′(43)μ1β,λβ≔infFβ,λφ,w:φ,w∈LpΩ×LqΩ,Rβφ,w=1,where
(44)Fβ,λφ,w=α1+1p∫Ωeβ·xφpdx−λ∫Ωm1eβ·xΛφpdx+α2+1q∫Ωeβ·xwqdx−λ∫Ωm2eβ·xΛwqdx−∫Ωmeβ·xΛφα1+1Λwα2+1dx,Rβφ,w=α1+1p∫Ωeβ·xΛφpdx+α2+1q∫Ωeβ·xΛwqdx.In a sequel, we name previous sufficient condition of solvability ofΣ(45)Hm:m∞<μ0..Lemma 5.
Ifu,v;β,μλ is a solution of Σλ, then −Δu, −Δv∈CΩ¯, and u, v∈C1,νΩ¯, for all ν∈0,1.Proof.
An easy adaptation of ideas of the proof of Lemma4 in [13] and we just omit it.Lemma 6.
φ1,w1;μ1β,λ∈LΩ is a solution of problem Sλ′, if and only if
(46)Gβ,λφ1,w1=0=minφ,w∈L∗ΩGβ,λφ,w,where
(47)Gβ,λφ,w=Fβ,λφ,w−μ1β,λRβφ,w,L∗Ω=LpΩ×LqΩ\0,0.Proof.
AssumingFβ,λφ1,w1=μ1β,λRβφ1,w1, then
(48)Gβ,λφ1,w1=Fβ,λφ1,w1−μ1β,λRβφ1,w1=0,and one can set φ¯=φ/Rβφ,w1/p and w¯=w/Rβφ,w1/q for every φ,w∈L∗Ω. We readily check that Rβφ¯,w¯=1 and deduce
(49)μ1β,λ≤Fβ,λφ¯,w¯=Fβ,λφ,wRβφ,w,Gβ,λφ,w=Fβ,λφ,w−μ1β,λRβφ,w≥0,
for allφ,w∈L∗Ω. This proves (45).
Conversely, let us suppose that (45) holds. We deduce that ∇Gβ,λφ1,w1=0,0 and then
(50)∂Gβ,λ∂φφ1,w1,Ψ=∂Gβ,λ∂wφ1,w1,θ=0,for all Ψ,θ∈LpΩ×LqΩ which proves that φ1,w1;μ1β,λ∈LΩ is a solution of Sλ′.Lemma 7.
IfHm is satisfied and φ1,w1;μ1β,λ∈L0Ω is a solution of problem Sλ′, then φ1,w1;μ1β,λ∈L0Ω is a solution of problem Sλ′.Proof.
From the assumptionsGβ,λφ1,w1=0, μ1β,λ=0, λ>0, and φ1,w1∈LpΩ×LqΩ\0,0; we have Gβ,λφ1,w1≥0. In addition, Λφ1r≥Λφ1r and Λw1r≥Λw1r, for all r∈1;∞ and imply
(51)−λ∫Ωm1eβ·xΛφ1pdx≤−λ∫Ωm1eβ·xΛφ1pdx,−λ∫Ωm2eβ·xΛw1qdx≤−λ∫Ωm2eβ·xΛw1qdx,−∫Ωmeβ·xΛφ1α+1Λw1β+1dx≤−∫Ωmeβ·xΛφ1α+1Λw1β+1dx.
Therefore,Fβ,λφ1,w1≤Fβ,λφ1,w1 and Gβ,λφ1,w1≤Gβ,λφ1,w1=0. It then reads Gβ,λφ1,w1=0 and φ1,w1;μ1β,λ is a solution of Sλ′.Proposition 3.
Assume thatHm holds and μ1.,λ.≡0. Then, λ. is a semitrivial principal eigensurface or strictly principal eigensurface of system Σ.Proof.
The valueλ. is an eigensurface of problem Σ associated with u,v∈YpqΩ\0,0. Let β∈ℝN with μ1β,λβ=0.
First, ifu≡0 and v≡0, then φ,w;μ1β,λβ and φ,w;μ1β,λβ belong to L0Ω and are solutions of problem Sλ′ with φ=−Δu≡0 and w=−Δv≡0. Since φ≥0 and w≥0, and then, Λφ>0 and Λw>0. Therefore, NpΛφ>0, NqΛw>0, Λwα2+1Λφα1>0, Λφα1+1Λwα2>0, and
(52)φ=Np′eβ·xΛλβm1eβ·xNpΛφ+meβ·xΛwα2+1Λφα1−1Λφ>0,w=Nq′eβ·xΛλβm2eβ·xNqΛw+meβ·xΛφα1+1Λwα2−1Λw>0.
We then conclude thatφ,w;μ1β,λβ is a solution of problem Sλ′ with both φ and w positive in Ω or negative in Ω. On the other hand, Lemma 5 expresses that φ, w∈CΩ¯ yielding to u=Λφ positive in Ω or negative in Ω and v=Λw positive in Ω or negative in Ω (see Lemma 1). It follows immediately that λ. is a strictly principal eigensurface of Σ.
Second, if [u≡0 and v≡0] or [u≡0 and v≡0], then we also prove that [u≡0 and v>0 in Ω or v<0 in Ω] or [u>0 in Ω or u<0 in Ω and v≡0]. We conclude that λ. is a semitrivial principal eigensurface of Σ.Lemma 8 [13].
LetA, B, C, and r be real numbers satisfying A≥0, B≥0, C≥maxB−A,0, and r∈1,+∞. Then,
(53)A+Cr+B−Cr≥Ar+Br.Lemma 9.
Suppose thatHm holds. If φ1,w1 and φ2,w2 are positive eigenfunctions of problem Sλ′ associated with μ1β,λ=0, then φ12,w12, φ12,w21, φ21,w12, and φ21,w21 with
(54)φ12x≔maxφ1x,φ2x=φ1x+φ2−φ1+x,w12x≔maxw1x,w2x=w1x+w2−w1+x,φ21x≔minφ1x,φ2x=φ2x−φ2−φ1+x,w21x≔minw1x,w2x=w2x−w2−w1+x,for all x∈Ω, are eigenfunctions of Sλ′ associated with μ1β,λ=0.Proof.
From Lemma8, we have
(55)Λφ12p+Λφ21p≥Λφ1p+Λφ2p,Λw12q+Λw21q≥Λw1q+Λw2q,Λφ12α1+1+Λφ21α1+1≥Λφ1α1+1+Λφ2α1+1,Λw12α2+1+Λw21α2+1≥Λw1α2+1+Λw2α2+1.
Then,(56)−λ∫Ωm1eβ·xΛφ12pdx−λ∫Ωm1eβ·xΛφ21pdx≤−λ∫Ωm1eβ·xΛφ1∣pdx−λ∫Ωm1eβ·xΛφ2pdx,(57)−λ∫Ωm2eβ·xΛw12qdx−λ∫Ωm2eβ·xΛw21qdx≤−λ∫Ωm2eβ·xΛw1qdx−λ∫Ωm2eβ·xΛw2qdx.
Similarly, we have(58)Z1φ,w≤Z2φ,w≤−∫Ωmeβ·xΛφ1α1+1Λw1α2+1dx−∫Ωmeβ·xΛφ2α1+1Λw2α2+1dx,with
(59)Z1φ,w=−∫Ωmeβ·xΛφ12α1+1Λw12α2+1dx−∫Ωmeβ·xΛφ12α1+1Λw21α2+1dx,(60)−∫Ωmeβ·xΛφ21α1+1Λw12α2+1dx−∫Ωmeβ·xΛφ21α1+1Λw21α2+1dx,(61)Z2φ,w=−∫Ωmeβ·xΛφ1α1+1Λw1α2+1dx−∫Ωmeβ·xΛφ1α1+1Λw2α2+1dx,(62)−∫Ωmeβ·xΛφ2α1+1Λw1α2+1dx−∫Ωmeβ·xΛφ2α1+1Λw2α2+1dx.
Moreover,(63)∫Ωeβ·xφ12pdx+∫Ωeβ·xφ21pdx=∫Ωeβ·xφ1∣pdx+∫Ωeβ·xφ2pdx,(64)∫Ωeβ·xw12qdx+∫Ωeβ·xw21qdx=∫Ωeβ·xw1qdx+∫Ωeβ·xw2qdx.
Combining (54), (56), (57), (59), and (63), we establish
(65)Fβ,λφ12,w12+Fβ,λφ12,w21+Fβ,λφ21,w12+Fβ,λφ21,w21≤Fβ,λφ1,w1+Fβ,λφ2,w2,Gβ,λφ12,w12+Gβ,λφ12,w21+Gβ,λφ21,w12+Gβ,λφ21,w21≤Gβ,λφ1,w1+Gβ,λφ2,w2=0,which imply
(66)Gβ,λφ12,w12=Gβ,λφ12,w21=Gβ,λφ21,w12=Gβ,λφ21,w21=0,and ensure that φ12,w12, φ12,w21, φ21,w12, and φ21,w21 are eigenfunctions of Sλ′ associated with μ1β,λ=0.Proposition 4.
Assume thatHm holds and μ1.,λ.≡0. Then λ. is a semitrivial principal eigensurface or strictly principal eigensurface of problem Σ and simple.Proof.
From Proposition3, the value λ. is a semitrivial principal eigensurface or strictly principal eigensurface of problem Σ. The rest of the proof falls naturally into two cases as follows.Case 1:λ. is a strictly principal eigensurface of Σ.Letu11,u12 and u21,u22 be two positive eigenfunctions of Σ associated with λ.. Then, v,w;0, φ,ψ;0, v,w;0, φ,ψ;0∈L0Ω, are solutions of Sλ′ with v=−Δu11>0, w=−Δu12>0, φ=−Δu21>0, and ψ=−Δu22>0. Let us fix x0∈Ω and set
(67)k=φx0vx0,ω1x=maxφx,kvxandω2x=maxψx,kp/qwx,for all x∈Ω.Using Lemma9, ω1,ω2;0 is a solution of problem Sλ′ as kv,kp/qw;0 and φ,ψ;0 are solutions of Σλ. We infer that Npv, Nqw, Npφ, Nqψ, Npω1, Nqω2∈C1,νΩ¯, and Npφ/Npv, Nqψ/Nqw∈C1Ω, and for any unit vector e=0,⋯,ei,⋯,0 with i∈1,⋯,N and t∈ℝ, we obtain
(68)Npφx0+te−Npφx0≤Npω1x0+te−Npω1x0,Npkvx0+te−Npkvx0≤Npω1x0+te−Npω1x0.Dividing these inequalities byt>0 and t<0 and letting t tend to 0±, we get
(69)∂∂xiNpφx0≤∂∂xiNpω1x0,∂∂xiNpkvx0≤∂∂xiNpω1x0,∂∂xiNpφx0≥∂∂xiNpω1x0,∂∂xiNpkvx0≥∂∂xiNpω1x0,for all i∈1,⋯,N. That is
(70)∂∂xiNpφx0=∂∂xiNpω1x0,∂∂xiNpkvx0=∂∂xiNpω1x0,for all i∈1,⋯,N. In other words,
(71)∇Npφx0=∇Npω1x0=∇Npkvx0=kp−1∇Npvx0.Furthermore,(72)∇NpφNpvx0=∇Npφx0Npvx0−Npφx0∇Npvx0Npvx02=Npvx0−k1−pNpφx0∇Npφx0Npvx02=0,for all x0∈Ω. Then, Npφ/v=Npφ/Npv=cst=kp−1 in Ω, i.e., φ=kv in Ω. In the same manner, we can see that ψ=hw if for x0∈Ω. Setting
(73)h=ψx0wx0,ω¯1x=maxψx,hwxandω¯2x=maxφx,kq/pvx,for all x∈Ω, we can write φ,ψ=kv,hw with k=hq/p. We deduce that u21,u22=ku11,hu12 with k=hq/p. Further, let u11,u12 and u21,u22 be two eigenfunctions of Σ associated with λ.. If there exist i, j∈1,2 such that uij<0, then we can set u¯ij=−uij and the result follows.Case 2:λ. is a semitrivial principal eigensurface of Σ.Let [u11,0 and u21,0] or [0,u12 and 0,u22] be two eigenfunctions of Σ associated with λ.. It is obvious to see that there exist [k≠0 real number] or [h≠0 real number] such that [u11=ku21] or [u12=hu22]. The proof is complete.We are now ready to state the main result of this section concerningΣ.Theorem 1.
Assume thatHm holds. The lowest positive eigensurface of problem Σ is λ1m,.,m1,m2 defined by
(74)λ1m,β,m1,m2=minu,v∈SβEβ,mu,v,for all β∈ℝN and
(75)Sβ=u,v∈YpqΩ:Mβu,v=1.
Moreover,(1)
λ1m,.,m1,m2≤minΓ1p.,m1,Γ1q.,m2(2)
λ1m,.,,m1,m2 is semitrivial principal eigensurface or strictly principal eigensurface(3)
λ1m,.,m1,m2 is simpleProof.
Combining Proposition2 and Lemma 4, there exists a unique λ1m,.,m1,m2 solution of equation μ1.,λ≡0, that is, λ1m,.,m1,m2 is an eigensurface of Σ and
(76)μ1′β,λ1m,β,m1,m2=−Mβu0,v0<0=μ1β,λ1m,β,m1,m2=Eβ,mu0,v0−λ1m,β,m1,m2Mβu0,v0,for all β∈ℝN, with u0,v0∈Mβ. Then, Eβ,mu0,v0=λ1m,β,m1,m2Mβu0,v0>0 and we can set
(77)u¯0=u0Mβu0,v01/p,v¯0=v0Mβu0,v01/q.
We easily prove thatu¯0,v¯0∈Sβ and infer Eβ,mu¯0,v¯0=λ1m,β,m1,m2. On the other hand, for each u,v∈Sβ, we have
(78)Eβ,muIβu,v1/p,vIβu,v1/q≥λ1m,β,m1,m2MβuIβu,v1/p,vIβu,v1/q,
i.e.,Eβ,mu,v≥λ1m,β,m1,m2. Therefore, (73) holds, and applying Proposition 4, we get that λ1m,.,m1,m2 is a strictly principal eigenvalue or semitrivial principal eigenvalue and simple.
Finally, what is left is to show thatλ1m,.,m1,m2≤minΓ1p.,m1,Γ1q.,m2. To do this, consider φp=p/α1+11/pφp,m1 and φq=q/α2+11/qφq,m2. Then, for all β∈ℝN, we have
(79)α1+1pM1,βφp+α2+1qM2,β0=1,α1+1pM1,β0+α2+1qM2,βφq=1.
Consequently,(80)λ1m,β,m1,m2≤Eβ,mφp,0=α1+1pEβφp,0=Γ1pβ,m1,λ1m,β,m1,m2≤Eβ,m0,φq=β+1qEβ0,φq=Γ1qβ,m2,and the result follows.



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