# Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions,  Solutions with Positive Components

Authors: Luisa Toscano; Speranza Toscano
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.


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## Body

## 1. Introduction

In this paper, we present some significant applications of the results got in [1] to Dirichlet problems (Section 2) of the type:
(1.1)-div⁡(Ai(x,u1,…,un,∇u1,…,∇un))=λibi|ui|p-2ui+di(x,u1,…,un,∇u1,…,∇un)+fiinΩ,ui=0on⁡∂Ωasi=1,…,n,
and to Neumann problems (Section 3) of the type:
(1.2)-div⁡(Ai(x,u1,…,un,∇u1,…,∇un))=λibi|ui|p-2ui+di(x,u1,…,un,∇u1,…,∇un)+fiinΩ,Ai(x,u1,…,un,∇u1,…,∇un)⋅ν=μib̂i|ui|p-2ui+d̂i(x,u1,…,un,∇u1,…,∇un)+f̂ion⁡∂Ωasi=1,…,n,
where n≥1,λi,μi are real parameters, Ω is a bounded connected open set of RN with regular boundary ∂Ω, and ν is the outward orthogonal unitary vector to ∂Ω.The study deals with the solvability of the problems, the existence of multiple solutions with all the components not identically equal to zero and, in the homogeneous case, the existence of solutions with positive components, bounded and locally Hölderian with their first derivatives. It is suitable to recall the problem studied in [1] with some notations and hypotheses.LetW1,…,Wn real reflexive Banach spaces (n≥1). Let W be the product space Xℓ=1nWℓ. Let ∥·∥ be the norm on W, ∥·∥* the norm on W* (dual space of W), and 〈·,·〉ℓ  (resp.〈〈·,·〉〉) the duality between  Wℓ* (dual space of Wℓ) and Wℓ   (resp. W* and W). Let us denote by “∂” Fréchet differential operator and by “∂uℓ” Fréchet differential operator with respect to uℓ. Let A≢0 and Dj≢0(j=1,…,m;m≥1) be real functionals defined in W,Bℓ and B̂ℓ(ℓ=1,…,n) real functionals defined in Wℓ satisfying the conditions:(i11)
A is lower weakly semicontinuous in W and C1(W∖{0}),BℓandB̂ℓ are weakly continuous in Wℓ andC1(Wℓ),∃p>1:A(tv)=tpA(v)forallt≥0and forallv∈W,Bℓ(tvℓ)=tpBℓ(vℓ) and B̂ℓ(tvℓ)=tpB̂ℓ(vℓ)forallt≥0 andforallvℓ∈Wℓ;(i12)
Dj is weakly continuous in W and C1(W∖{0}),∃qj>1:Dj(tv)=tqjDj(v)forallt≥0 and forallv∈W,1<q1<⋯<qmifm>1.LetF=(F1,…,Fn)with Fℓ∈Wℓ*,λℓandμℓ∈R; let us consider the following problem.Problem (P).
Findu=(u1,…,un)∈W∖{0}such that
(1.3)〈∂uiA(u),vi〉i=λi〈∂Bi(ui),vi〉i+μi〈∂B̂i(ui),vi〉i+∑j=1m〈∂uiDj(u),vi〉i+〈Fi,vi〉i∀i∈{1,…,n},∀vi∈Wi.
Obviously Problem (P) means to find the critical points u∈W∖{0} of the Euler functional:
(1.4)E(v)=A(v)-∑l=1n[λlBl(vl)+μlB̂l(vl)]-∑j=1mDj(v)-〈〈F,v〉〉∀v=(v1,…,vn)∈W,
where 〈〈F,v〉〉=∑ℓ=1n〈Fℓ,vℓ〉ℓ.Let us set(1.5)Hλμ(v)=A(v)-∑l=1n[λlBl(vl)+μlB̂l(vl)]∀v=(v1,…,vn)∈W,∀λ=(λ1,…,λn),μ=(μ1,…,μn)∈Rn,Sλμ={v∈W:Hλμ(v)=1},Vλμ-={v∈W:Hλμ(v)<0},asm1=1,…,mV+(Dm1,…,Dm)={v∈W:∑j=m1mDj(v)>0},S+(D1,…,Dm)={v∈W:∑j=1mDj(v)=1},S(Dj)={v∈W:Dj(v)=-1},V+(F)={v∈W:〈〈F,v〉〉>0}.
About Problem (P), using Lagrange multipliers and the “fibering method,” different existence theorems have been proved in [1]. They base on one of the following hypotheses: (i13)
∃c(λ,μ)>0:∥v∥p≤c(λ,μ)Hλμ(v)forallv∈W;(i14)
∃c(λ,μ)>0:∥v∥p≤c(λ,μ)Hλμ(v)forallv∈V+(Dm)(ifV+(Dm)≠∅);(i15)
∃m1∈{1,…,m}:Vλμ-∩S(Dm1) is not empty and bounded in W.Remark 1.1.
In this paper, we use some existence theorems ([1], Theorems 2.1, 2.2, 3.1, and 3.2), in which as n>1, in relation to a set 𝔉⊆Sλμ, we suppose(i16h)
for eachv=(v1,…,vn)∈𝔉 with vh=0, there exist  v̅h∈Wh∖{0} and the real functions ϕ1,…,ϕn  such that  ϕh∈C0([0,1])∩C1([0,1[)  and ϕh(1)=0,  ϕℓ∈C1([0,1])  and ϕℓ(1)=1  as  ℓ≠h,   v(s)=(ϕ1(s)v1,…,ϕh(s)v̅h,…,ϕn(s)vn)∈𝔉    for all   s∈[s0,1](0≤s0<1), lims→1-′′(d/ds)Dj(v(s))<+∞  for all  j∈{1,…,m},  lims→1-(d/ds)Dj(v(s))=-∞   for some  j∈{1,…,m}.
The condition (i16h) assures that for the solutions u=(u1,…,un) of Problem (P), found with the method used in the recalled theorems, we have uh≠0 if Fh≡0.Before showing Dirichlet problems (including the problem studied in [2] by Drábek and Pohozaev when n=1 and m=1) we give Propositions 2.2–2.6 which show some cases in which hypotheses (i13)−(i15) hold. These propositions are based on the comparison between the parameters λi with suitable eigenvalues connected to p-Laplacian. About Neumann problems (including the one studied in [3] by Pohozaev and Véron when n=1) the same question is solved by Propositions 3.1–3.5 in which the parameters λi and μi have compared with zero. Finally, the results in Appendix are very useful: Propositions A.1 and A.2 in order to get condition (i16h), Propositions A.3 and A.4 to get qualitative properties of the solutions and the positive sign of the components of the found solutions.


## 2. Dirichlet Problems

LetΩ⊆RN be an open, bounded, connected and C2,β set with 0<β≤1. Let|·|N the Lebesgue measure on RN,1<p<∞,p̃=Np/(N-p)ifN>p,p̃=∞otherwise.Let us assume(2.1)W=(W01,p(Ω))n(n≥1)with‖v‖=(∑l=1n∫Ω|∇vl|pdx)1/p∀v=(v1,…,vn)∈W,Bl(vl)=p-1∫Ωbl|vl|pdx∀vl∈W01,p(Ω)wherebl∈L∞(Ω)∖{0},bl≥0,B̂l≡0.
Moreover we consider the functionals A (as in (i11)) such that(2.2)∃c̃>0:A(v)≥p-1c̃‖v‖p∀v∈W.
Let us use the notation Hλ(SλandVλ-,resp.) instead of Hλμ(SλμandVλμ-,resp.).Asℓ=1,…,n let λℓ*anduℓ*, respectively, the first eigenvalue and the first eigenfunction of the problem:
(2.3)ul∈W01,p(Ω):-c̃div⁡(|∇ul|p-2∇ul)=θbl|ul|p-2ulinΩ.
Let us remember that [4]uℓ*∈C1,αℓ(Ω̅) with 0<αℓ<1,uℓ*>0 in Ω;λℓ*=c̃∫Ω|∇uℓ*|pdx/∫Ωbℓ|uℓ*|pdx=min{c̃∫Ω|∇vℓ|pdx/∫Ωbℓ|vℓ|pdx:∫Ωbℓ|vℓ|pdx>0};λℓ* is simple, that is, each eigenfunction of (2.3) related to λℓ* is of the type cℓuℓ* with cℓ∈R∖{0};λℓ* is isolate, that is, there exists a>0 such that λℓ* is the only eigenvalue of (2.3) belonging to ]0,a[.Remark 2.1.
About the results related to problem (2.3), it is sufficient to suppose bℓ∈L∞(Ω) and bℓ+=max{bℓ,0}≢0 as ℓ=1,…,n. This holds also for the results of this section if we limit to consider only the parameters λ1,…,λn nonnegative.Let us start by presenting some sufficient conditions such that(i13),(i14), and (i15) hold.Using the variational characterization ofλℓ* it is easy to verify the following proposition.Proposition 2.2.
Ifλℓ<λℓ*forallℓ∈{1,…,n}, then (i13) holds. Consequently, (i14) holds when V+(Dm)≠∅.Whenλℓ≥λℓ*  for some ℓ∈{1,…,n}, it is possible to fulfil (i14) with an additional condition on Dm. Let I={1,…,n}. For any I*⊆I let
(2.4)V*={v=(v1,…,vn)∈W:vl≡0ifl∈I∖I*,vl=clul*ifl∈I*withcl∈Randcl≠0forsomel},
and let us suppose(i21)
There existsI*⊆I:Dm(v)<0forallv∈V*.Proposition 2.3.
Let (i21) holds with I*≠I. Let V+(Dm)≠∅.  If we fix the parameters set (λℓ)ℓ∈I∖I* with λℓ<λℓ*, then there exists δ*>0 such that (i14) also holds for any (λℓ)ℓ∈I*∈Xℓ∈I*[λℓ*,λℓ*+δ*[.Proof.
Arguing by contradiction, for anyk∈ℕ there exist (λℓk)ℓ∈I*∈Xℓ∈I*[λℓ*,λℓ*+k-1[ and vk=(v1k,…,vnk)∈V+(Dm) such that
(2.5)A(vk)-p-1∑l∈I∖I*λl∫Ωbl|vl|pdx-p-1∑l∈I*λlk∫Ωbl|vlk|pdx<k-1‖vk‖p.
Set wk=∥vk∥-1vk, we have
(2.6)Dm(wk)>0,c̃∑l∈I∖I*∫Ω|∇wlk|pdx-∑l∈I∖I*λl∫Ωbl|wlk|pdx+c̃∑l∈I*∫Ω|∇wlk|pdx-∑l∈I*λlk∫Ωbl|wlk|pdx<pk-1,
moreover, since ∥wk∥=1, there exists w∈W such that (within a subsequence)
(2.7)wk⟶wweaklyinW,wk⟶wstronglyin(Lp(Ω))n.
Taking into account that Dm is weakly continuous in W, from (2.6) as k→+∞ we get
(2.8)Dm(w)≥0,(2.9)∑l∈I∖I*[c̃∫Ω|∇wl|pdx-λl∫Ωbl|wl|pdx]+∑l∈I*[c̃∫Ω|∇wl|pdx-λl*∫Ωbl|wl|pdx]≤0.
Since
(2.10)wl≢0⟹c̃∫Ω|∇wl|pdx-λl∫Ωbl|wl|pdx>0,c̃∫Ω|∇wl|pdx-λl*∫Ωbl|wl|pdx≥0,
from (2.9), we deduce that
(2.11)wl≡0∀l∈I∖I*,∀l∈I*∃cl∈R:wl=clul*.
Let us add thatcℓ≠0 for some ℓ∈I*, since if cℓ=0forallℓ∈I* we have the contradiction c̃=c̃limk→+∞∥wk∥p=0. Then w∈V*, and consequently Dm(w)<0 from (i21). This last inequality contradicts (2.8).In the same way the following propositions can be proved.Proposition 2.4.
Let (i21) holds with I*=I. Let V+(Dm)≠∅. Then, there exists δ*>0 such that (i14) also holds for any (λℓ)ℓ∈I∈Xℓ∈I[λℓ*,λℓ*+δ*[.Let us pass to (i15) and suppose(i22)
there existI*⊆I and m1∈{1,…,m}such thatDm1(v)<0andA(v)=c̃p-1∑ℓ∈I*∫Ω|∇vℓ|pdx for any v∈V*.Proposition 2.5.
If (i22) holds withI*≠I, then
(2.12)Vλ-∩S(Dm1)≠∅∀(λl)l∈Iwith(λl)l∈I*∈Xl∈I*[λl*,+∞[∖{(λl*)l∈I*}.
Moreover, if we fix the parameters set (λℓ)ℓ∈I∖I* with λℓ<λℓ*, then there exists δ*>0 such that
(2.13)Vλ-∩S(Dm1)isboundedinW∀(λl)l∈I*∈Xl∈I*[λl*,λl*+δ*[∖{(λl*)l∈I*}.Proof.
Let us prove (2.12). Let v∈V*with vℓ=uℓ*if ℓ∈I*, then Dm1(v)<0. Let w=|Dm1(v)|-1∖qm1v, we have
(2.14)Dm1(w)=|Dm1(v)|-1Dm1(v)=-1,Hλ(w)=p-1∑l∈I*[c̃∫Ω|∇wl|pdx-λl∫Ωbl|wl|pdx]<0.
Let us prove (2.13). Arguing by contradiction, for any k∈ℕ there exist (λℓk)ℓ∈I*∈Xℓ∈I*[λℓ*,λℓ*+k-1[ with (λℓk)ℓ∈I*≠(λℓ*)ℓ∈I* and (vk,h)h∈ℕ⊆Vλk-∩S(Dm1), where λℓk=λℓ if ℓ∈I∖I*, such that
(2.15)sup⁡h∈N‖vk,h‖=+∞.
Relation (2.15) implies that there exists (hk)k∈ℕ⊆ℕ strictly increasing such that
(2.16)δk=‖vk,hk‖⟶+∞ask⟶+∞.
Let wk=δk-1vk,hk, we have
(2.17)∑l∈I∖I*[c̃∫Ω|∇wlk|pdx-λl∫Ωbl|wlk|pdx]+∑l∈I*[c̃∫Ω|∇wlk|pdx-λlk∫Ωbl|wk|pdx]<0,Dm1(wk)=-δk-qm1,∃w∈W:(withinasubsequence)wk⟶wweaklyinW,wk⟶wstronglyin(Lp(Ω))n.
Then, as k→+∞ we get
(2.18)∑l∈I∖I*[c̃∫Ω|∇wl|pdx-λl∫Ωbl|wl|pdx]+∑l∈I*[c̃∫Ω|∇wl|pdx-λl*∫Ωbl|wl|pdx]≤0,(2.19)Dm1(w)=0.
From (2.18), we get that  w∈V*. Then since (i22) inequality Dm1(w)<0 holds, which contradicts (2.19).Proposition 2.6.
If (i22) holds with I*=I, then
(2.20)Vλ-∩S(Dm1)≠∅∀λ=(λl)l∈I∈Xl∈I[λl*,+∞[∖{(λl*)l∈I},∃δ*>0:Vλ-∩S(Dm1)isboundedinW∀λ=(λl)l∈I∈Xl∈I[λl*,λl*+δ*[∖{(λl*)l∈I}.
The proof as in Proposition2.5.Remark 2.7.
The applications we now show, except the first one, deal with systems withn>1 equations. We consider the functionals A with c̃=1, and we suppose bℓ∈L∞(Ω)∖{0},bℓ≥0.Application 2.8.
Letn=1. Let us consider the problem
(2.21)-div⁡(|∇u|p-2∇u)=λ1b1|u|p-2u+∑j=1mdj|u|qj-2uinΩ,u=0on⁡∂Ω,
where
(2.22)p<q1<p̃,d1∈L∞(Ω)∖{0}ifm=1,p<q1<⋯<qm<p̃,dj∈L∞(Ω)∖{0}asj=1,…,m,dj≤0asj=1,…,m-1ifm>1.
Evidently
(2.23)A(v)=p-1∫Ω|∇v|pdx,Dj(v)=qj-1∫Ωdj|v|qjdx∀v∈W.
Let us advance the conditions:
(2.24)dm+≢0(⟹V+(Dm)≠∅),(2.25)∫Ωdm(u1*)qmdx<0(⟹Dm(c1u1*)<0∀c1∈R∖{0}).
Let us note that (Propositions 2.2, 2.4, and 2.6)
(2.26)(2.24)⟹((i14)holdsifλ1<λ1*),(2.24)and(2.25)⟹(∃δ1*>0:(i14)holdsifλ1<λ1*+δ1*),(2.25)⟹(∃δ2*>0:(i15)holdsifλ1∈]λ1*,λ1*+δ2*[).Proposition 2.9 (see [1], Theorems 2.1, 2.2, 4.1, and 4.2; Remarks 2.1, 2.3, 4.1, and 4.4; Proposition A.3; [5, 6]).
Under assumptions (2.22) we have:(i)
When (2.24) holds, with λ1<λ1* [resp. (2.24) and (2.25) hold, with λ1<λ1*+δ1*] problem (2.21) has at least two weak solutions u0 and-u0(u0=τ0v0,τ0=const.>0,v0∈Sλ1∩V+(Dm)), and it results in u0∈L∞(Ω)∩Cℓoc1,α0(Ω),u0>0;(ii)
When (2.25) holds, with λ1∈]λ1*,λ1*+δ2*[ problem (2.21) has at least two weak solutions u̅and-u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλ1-∩S(Dm)), and it results in u̅∈L∞(Ω)∩Cℓoc1,α̅(Ω),u̅>0.
Consequently, when (2.24) and (2.25) hold, with λ1∈]λ1*,λ1*+min{δ1*,δ2*}[ problem (2.21) has at least four different weak solutions.Remark 2.10.
Our results include the ones of Drábek and Pohozaev [2] when m=1.Application 2.11.
Let us consider the system:(2.27)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+|∑l=1ndlul|q1-2(∑l=1ndlul)di-d̃i|ui|q1-2uiinΩ,ui=0on⁡∂Ωasi=1,…,n,
where
(2.28)1<q1<p̃,q1≠p,dl,d̃l∈L∞(Ω),dl,d̃l>0.
System (2.27) is included among Problem (P) with:
(2.29)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,D1(v)=q1-1[∫Ω|∑l=1ndlvl|q1dx-∑l=1n∫Ωd̃l|vl|q1dx]∀v=(v1,…,vn)∈W.
Let us advance the conditions (compatible):
(2.30)dlq1<d̃l∀l∈{1,…,n}(⟹D1(0,…,ciui*,…,0)<0asi=1,…,n,ci∈R∖{0}),
there exist Ω+⊆Ω and a constant c̃j>0 such that |Ω+|N>0 and
(2.31)(∑l≠jdl+c̃jdj)q1>∑l≠jd̃l+c̃jq1d̃jinΩ+(⟹V+(D1)≠∅(PropositionA.1)).
Then (Propositions 2.2, 2.3, and 2.5)
(2.32)(2.31)⟹((i14)holdsifλl<λl*∀l∈{1,…,n}),
and set i∈{1,…,n}(2.33)(2.30)and(2.31)⟹(withλl<λl*∀l≠i∃δ1*>0:(i14)holdsifλi<λi*+δ1*),(2.34)(2.30)⟹(withλl<λl*∀l≠i∃δ2*>0:(i15)holdsifλi∈]λi*,λi*+δ2*[).Taking into account thatD1(v1,…,vn)≤D1(|v1|,…,|vn|) and D1(-v)=D1(v), from ([1], Theorem 2.1, Remark 2.1, and Theorem 4.1) we get the following proposition.Proposition 2.12.
Under assumptions (2.28) we have:(i)
When (2.31) holds, ((2.30) and (2.31) hold resp.), choosing λ1,…,λn as in (2.32) (resp. (2.33)) system (2.27) has at least two weak solutions u0 and -u0 with uℓ0≥0 as ℓ=1,…,n(u0=τ0v0,τ0=const.>0,v0∈Sλ∩V+(D1));(ii)
When (2.30) holds, choosingλ1,…,λn as in (2.34) system (2.27) has at least two weak solutions u̅ and -u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλ-∩S(D1)).
Consequently, when (2.30) and (2.31) hold, with λℓ<λℓ*forallℓ≠i and λi∈]λi*,λi*+min{δ1*,δ2*}[ system (2.27) has at least four different weak solutions.The following proposition is obvious.Proposition 2.13.
The following relations hold:(2.35)ui0≢0asi=1,…,n,∃h,k∈{1,…,n}:u̅h≢0,u̅k≢0.Proposition 2.14.
Ifp<q1, then as i=1,…,n:
(2.36)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),ui0>0.Proof.
It is easy to prove that(2.37)∑i=1n∫Ω|∇ui0|p-2∇ui0⋅∇vidx≤∫Ωg(∑i=1nui0)p-1(∑i=1nvi)dx∀v=(v1,…,vn)∈(W01,p(Ω)∩L∞(Ω))nwithvi≥0,
where g∈Lq1/(q1-p)(Ω). Then (Proposition A.3) ui0∈L∞(Ω) and consequently [5] ui0∈Cℓoc1,αi0(Ω).
Let us note thatui0 is a weak supersolution to the equation:
(2.38)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui-d̃i|ui|q1-2uiinΩ.
Then, since (2.35), it must be [6] ui0>0.Let us continue the analysis of system (2.27) under the condition:
(2.39)(∑l≠idl)q1<min⁡⁡{d̃1,…,d̃n}∀i∈{1,…,n},
then
(2.40)D1(c1u1*,…,cnun*)<0∀(c1,…,cn)∈Rn∖{0}withci=0foratleastonei∈{1,…,n}.
Hence (Proposition 2.5) if I*⊆I and I*≠I:
(2.41)(2.39)⟹(asλl<λl*∀l∈I∖I*∃δ*>0:(i15)holdsif(λl)l∈I*∈Xl∈I*[λl*,λl*+δ*[∖(λl*)l∈I*).Proposition 2.15.
Under assumptions (2.28) and (2.39), choosing λ1,…,λn as in (2.41) system (2.27) has at least two weak solutions u̅and-u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλ-∩S(D1)) with u̅i≢0 as i=1,…,n.Proof.
Thanks to ([1], Theorem 4.1), there exists v̅∈Vλ-∩S(D1) such that
(2.42)Hλ(v̅)=inf⁡⁡{Hλ(v):v∈Vλ-∩S(D1)}=e̲,u̅=τ̅v̅isaweaksolutionofsystem(2.27),
where τ̅=(-pq1-1e̲)1/(q1-p).
Reasoning by contradiction, let, for example,u̅1≡0. Since -1=D1(v̅)≤D1(0,|v̅2|,…,|v̅n|) and from (2.39) D1(0,|v̅2|,…,|v̅n|)<0, setting δ=|D1(0,|v̅2|,…,|v̅n|)|-1/q1 we have
(2.43)D1(0,δ|v̅2|,…,δ|v̅n|)=-1,Hλ(0,δ|v̅2|,…,δ|v̅n|)=δpHλ(v̅)≤Hλ(v̅),
then Hλ(0,δ|v̅2|,…,δ|v̅n|)=Hλ(v̅). This implies that ([1], see the proof of Theorem 4.1) (0,τ̅δ|v̅2|,…,τ̅δ|v̅n|) is a weak solution of system (2.27). Then (∑ℓ=2ndℓ|v̅ℓ|)q1-1≡0from which u̅ℓ≡0too as ℓ=2,…,n.
Condition (2.39) holds in particular when
(2.44)(∑l=1ndl)q1<min⁡⁡{d̃1,…,d̃n}.Proposition 2.16.
Replacing in Proposition2.15 (2.39) with (2.44), it is right to say that u̅i≥0 and u̅i≢0 as i=1,…,n. Consequently, if  p<q1(2.45)u̅i∈L∞(Ω)∩Cloc1,a̅i(Ω),u̅i>0asi=1,…,n.Proof.
Setδ=|D1(|v̅1|,…,|v̅n|)|-1/q1, as in Proposition 2.15(τ̅δ|v̅1|,…,τ̅δ|v̅n|) is a weak solution to system (2.27).
Let us add that since (2.44)⇒D1(c1u1*,…,cnun*)<0forall(c1,…,cn)∈Rn∖{0}, there exists (Proposition 2.6) δ**>0 such that
(2.46)(i15)holdsif(λl)l∈I∈Xl=1n[λl*,λl*+δ**[∖{(λl*)l∈I}.
Then the existence of u̅ is assured also choosing λ1,…,λn as in (2.46), and the conclusions of Proposition 2.16 hold.Application 2.17.
Let us set(2.47)λ1=⋯=λn=λ̅,b1=⋯=bn=b(thenλ1*=⋯=λn*=λ*,u1*=⋯=un*=u*),A(v)=p-1∑l=1n∫Ω|∇vl|pdx,D1(v)=q1-1∫Ωd1(∑l=1n|vl|γ)q1/γdx,∀v=(v1,…,vn)∈W,
where
(2.48)1<γ<q1<p̃,q1≠p,d1∈L∞(Ω).
Let us consider the system:
(2.49)-div⁡(|∇ui|p-2∇ui)=λ̅b|ui|p-2ui+d1(∑l=1n|ul|γ)(q1/γ)-1|ui|γ-2uiinΩ,ui=0on⁡∂Ωasi=1,…,n.
We advance the conditions
(2.50)d1+≢0(⟹V+(D1)≠∅),(2.51)∫Ωd1(u*)q1dx<0(⟹D1(c1u*,…,cnu*)<0∀(c1,…,cn)∈Rn∖{0}).
Therefore,
(2.52)(2.50)⟹((i14)holdsifλ̅<λ*)(Proposition2.2),(2.50)and(2.51)⟹(∃δ1*>0:(i14)holdsifλ̅<λ*+δ1*)(Proposition2.4),(2.51)⟹(∃δ2*>0:(i15)holdsifλ̅∈]λ*,λ*+δ2*[)(Proposition2.6).
Then ([1], Theorems 2.1 and 4.1, and Remarks 2.1 and 4.1).Proposition 2.18.
Under assumption (2.48), we have:(i)
When (2.50) holds, ((2.50) and (2.51) hold resp.), if λ̅<λ*(resp.λ̅<λ*+δ1*) system (2.49) has at least two weak solutions u0and -u0 with uℓ0≥0 as ℓ=1,…,n(u0=τ0v0,τ0=const.>0,v0∈Sλ∩V+(D1));(ii)
When (2.51) holds,ifλ̅∈]λ*,λ*+δ2*[system (2.49) has at least two weak solutions u̅ and -u̅ with u̅ℓ≥0 as ℓ=1,…,n(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλ-∩S(D1)).
Consequently, when (2.50) and (2.51) hold, withλ̅∈]λ*,λ*+min{δ1*,δ2*}[ system (2.49) has at least four different weak solutions.In order to establish some properties ofu0 and u̅ it is useful to recall that ([1], Theorems 2.1 and 4.1)(2.53)D1(v0)=sup⁡{D1(v):v∈Sλ∩V+(D1)}=e̅,τ0=(q1p-1e̅)1/(p-q1),(2.54)Hλ(v̅)=inf⁡⁡{Hλ(v):v∈Vλ-∩S(D1)}=e̲,τ̅=(-pq1-1e̲)1/(q1-p).Proposition 2.19.
Whenp<q1, we have
(2.55)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),
besides
(2.56)ui0≢0∀i∈{1,…,n}ifγ<p.Proof.
The relationui0∈L∞(Ω) comes from Proposition A.3. Then [5] ui0∈Cℓoc1,αi0(Ω).
About (2.56), it is sufficiently (Remark 1.1) to prove that
(2.57)(i16h)holds∀h∈{1,…,n}withF=Sλ∩V+(D1).
Letv=(v1,…,vn)∈Sλ∩V+(D1) with vh≡0. Since
(2.58)v∈V+(D1)⟹(∃acompactsetK⊆Ω:|K|N>0,d1>0andψ=∑l≠h|vl|γ>0inK),
let (Proposition A.1) (φε)0<ε<ε0⊆C0∞(Ω) with 0≤φε≤1 such that
(2.59)φε⟶χstronglyinLs(Ω),∫Ω|∇φε|sdx⟶+∞asε⟶0+∀s∈[1,+∞[,
where χ is the characteristic function of 𝕂. Set ε such that
(2.60)∫Ωd1ψ(q1/γ)-1φεγdx>0,δ=p-1[∫Ω|∇φε|pdx-λ̅∫Ωbφεpdx]>0,
with v(s)=(s1/pv1,…,(1-s)1/pδ-1/pφε,…,s1/pvn) it results in
(2.61)Hλ(v(s))=δ-1(1-s)p-1[∫Ω|∇φε|pdx-λ̅∫Ωbφεpdx]+sHλ(v)=1∀s∈[0,1],∃s0∈[0,1[:D1(v(s))>0∀s∈[s0,1],lim⁡s→1-ddsD1(v(s))=-∞.Proposition 2.20.
Whenp<q1, we have
(2.62)u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω),(2.63)u̅i>0∀i∈{1,…,n}ifp<γ.Proof.
We can get (2.62) from Proposition A.3 and [5].
About (2.63), it is sufficiently [6] to prove that u̅i≢0 as i=1,…,n. Reasoning by contradiction, let, for example, v̅1≡0. We note that
(2.64)v̅∈Vλ-⟹(∃l∈{2,…,n}:∫Ω|∇v̅l|pdx-λ̅∫Ωbv̅lpdx<0).
Let us suppose ℓ=2 and set v(s)=((1-s)1/γv̅2,s1/γv̅2,v̅3,…,v̅n).  Then
(2.65)D1(v(s))=-1∀s∈[0,1],∃s0∈[0,1[:Hλ(v(s))<0∀s∈[s0,1],lim⁡s→1-ddsHλ(v(s))=+∞.
Set s1∈[s0,1[ such that (d/ds)Hλ(v(s))>0foralls∈[s1,1[ and taking into account (2.54), we get the contradiction:
(2.66)Hλ(v̅)≤Hλ(v(s))<Hλ(v̅)∀s∈[s1,1[.Proposition 2.21.
Whenγ=p<q1, we allow that as i=1,…,n:
(2.67)ui0>0,u̅i>0.Proof.
The assumptionγ=p implies that
(2.68)∀v=(v1,…,vn)∈W∖{0}withvh≡0forsomeh∈{1,…,n},∃ṽ=(ṽ1,…,ṽn)∈W:ṽl≢0asl=1,…,n,Hλ(ṽ)=Hλ(v),D1(ṽ)=D1(v).
Let, for example, v1≡0 and v2≢0. Set s∈]0,1[ and v11=(1-s)1/pv2,v21=s1/pv2,vℓ1=vℓ asℓ>2, with v1=(v11,…,vn1), we have
(2.69)Hλ(v1)=Hλ(v),D1(v1)=D1(v).
If v3≡0, set v12=(1-s)1/pv11,v32=s1/pv11,vℓ2=vℓ1 as ℓ∈{1,…,n}∖{1,3}, withv2=(v12,…,vn2), it results in
(2.70)Hλ(v2)=Hλ(v),D1(v2)=D1(v).
This method let us to find ṽ.
Then, ifvh0≡0(resp.v̅h≡0) for some h∈{1,…,n}, with ṽ0(resp.v̅̃) as in (2.68) we have from (2.53) (resp. (2.54)) D1(ṽ0)=e̅(resp.Hλ(v̅̃)=e̲). Consequently ([1], see the proof of Theorem 2.1 (resp. Theorem 4.1)) ũ0=τ0ṽ0(resp.u̅̃=τ̅v̅̃) is a weak solution of system (2.49). Therefore [6] ũi0>0(resp.u̅̃i>0) as i=1,…,n.Application 2.22.
Let us assumeλℓ,bℓ, and A as in Application 2.17,
(2.71)Dj(v)=qj-1∫Ωdj(∑l=1n|vl|γj)qj/γjdx∀v=(v1,…,vn)∈Wasj=1,…,m,
where
(2.72)p<q1<⋯<qm<p̃,1<γj<qj,dm∈L∞(Ω),dj∈L∞(Ω)∖{0},dj≤0ifj=1,…,m-1.
Let us consider the system:
(2.73)-div⁡(|∇ui|p-2∇ui)=λ̅b|ui|p-2ui+∑j=1mdj(∑l=1n|ul|γj)(qj/γj)-1|ui|γj-2uiinΩ,ui=0on⁡∂Ωasi=1,…,n,
under almost one of the conditions:
(2.74)dm+≢0,∫Ωdm(u*)qmdx<0.
By using some results ([1], Theorems 2.2 and 4.2, and Remarks 2.3 and 4.4), we can advance a proposition similar to Proposition 2.18 replacing in particular V+(D1) with V+(Dm) and S(D1) withS(Dm).
Thanks to PropositionA.3 and a result of [5], for the solutions u0 and u̅ to system (2.73), we have
(2.75)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω).
We continue to analyze the properties of u0 and u̅. To this aim we recall that ([1], Theorems 2.2 and 4.2), set for each v∈V+(Dm)(resp.v∈Vλ-∩S(Dm))ψ(t,v)=ptp-1Hλ(v)-∑j=1mqjtqj-1Dj(v), we have:
(2.76)∃∣t(v)>0:ψ(t(v),v)=0,∂ψ∂t(t(v),v)≠0.
Besides with Ẽ̃(v)=(t(v))pHλ(v)-∑j=1m(t(v))qjDj(v), it results in
(2.77)Ẽ̃(v0)=inf⁡{Ẽ̃(v):v∈Sλ∩V+(Dm)},τ0=t(v0),(2.78)Ẽ̃(v̅)=inf⁡{Ẽ̃(v):v∈Vλ-∩S(Dm)},τ̅=t(v̅).Proposition 2.23.
Whenγm<p≤γj as j=1,…,m-1, then
(2.79)ui0≢0∀i∈{1,…,n}.Proof.
It is sufficiently (Remark1.1) to prove that
(2.80)(i16h)holds∀h∈{1,…,n}withF=Sλ∩V+(Dm).
Let v=(v1,…,vn)∈Sλ∩V+(Dm) with vh≡0. As in Proposition 2.19, it is possible to find v̅h∈C0∞(Ω)∖{0} such that with v(s)=(s1/pv1,…,(1-s)1/pv̅h,…,s1/pvn), it results in
(2.81)Hλ(v(s))=1∀s∈[0,1],Dm(v(s))>0∀s∈[s0,1](0≤s0<1),lim⁡s→1-ddsDj(v(s))∈Rasj=1,…,m-1,lim⁡s→1-ddsDm(v(s))=-∞.Proposition 2.24.
Whenp<γm≤γj as j=1,…,m-1, then
(2.82)u̅i>0∀i∈{1,…,n}.Proof.
It is sufficiently [6] to prove that u̅i≢0foralli∈{1,…,n}. Reasoning by contradiction, let, for example, v̅1≡0andv̅2≢0 such that
(2.83)∫Ω|∇v̅2|pdx-λ̅∫Ωbv̅2pdx<0.
Since
(2.84)t(v̅)>0,ψ(t(v̅),v̅)=0,∂ψ∂t(t(v̅),v̅)≠0,
there exist an open ball B̃ofW with centre v̅ included in Vλ- and a unique functional t*(v) belongs to C1(B̃) such that
(2.85)t*(v)>0,ψ(t*(v),v)=0∀v∈B̃.
Then, the functional
(2.86)E*(v)=(t*(v))pHλ(v)-∑j=1m(t*(v))qjDj(v)∀v∈B̃
belongs to C1(B̃), and we have
(2.87)t(v)=t*(v)∀v∈B̃∩S(Dm).
Then, for (2.78)
(2.88)E*(v̅)=inf⁡{E*(v):v∈B̃∩S(Dm)}.
Now, let us remark that with v(s)=((1-s)1/γmv̅2,s1/γmv̅2,v̅3,…,v̅n), it results in
(2.89)Dm(v(s))=-1∀s∈[0,1],∃s0∈[0,1[:v(s)∈B̃∀s∈[s0,1],lim⁡s→1-ddsHλ(v(s))=+∞,lim⁡s→1-ddsDj(v(s))∈Rasj=1,…,m-1.
Then, since
(2.90)ddsE*(v(s))=(t*(v(s)))pddsHλ(v(s))-∑j=1m(t*(v(s)))qjddsDj(v(s))∀s∈[s0,1[,
we havelims→1-(d/ds)E*(v(s))=+∞. Consequently,
(2.91)∃s1∈[s0,1[:ddsE*(v(s))>0∀s∈[s1,1[,
from which we get the contradiction:
(2.92)E*(v̅)≤E*(v(s))<E*(v̅)∀s∈[s1,1[.Proposition 2.25.
Whenp=γ1=⋯=γm, we allow that
(2.93)ui0>0,u̅i>0∀i∈{1,…,n}.Proof.
We reason as in Proposition2.21, taking into account (2.77) and (2.78) ([1], see proofs of Theorems 2.2 and 4.2).Application 2.26.
Let for eachv=(v1,…,vn)∈W:
(2.94)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,Dj(v)=-∏l=1n∫Ω|vl|qjldxasj=1,…,m-1(m≥2),Dm(v)=qm-1[∫Ω(∑l=1ndl|vl|γ)qm/γdx-∑l=1n∫Ωd̃l|vl|qmdx],
where
(2.95)1<γ<p≤qjl,∑l=1nqjl=qj<qm<p̃,q1<⋯<qm-1,dl,d̃l∈L∞(Ω),dl,d̃l>0.
Let us consider the system:
(2.96)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui-∑j=1m-1(qji∏l≠i∫Ω|ul|qjldx)|ui|qji-2ui+(∑l=1ndl|ul|γ)(qm/γ)-1di|ui|γ-2ui-d̃i|ui|qm-2uiinΩ,ui=0on⁡∂Ωasi=1,…,n.
Let us introduce the conditions:
(2.97)∃Ω+⊆Ω:|Ω+|N>0,dl̅qm/γ>d̃l̅inΩ+forsomel̅∈{1,…,n-1}(⟹V+(Dm)≠∅),(2.98)dnqm/γ<d̃n(⟹Dm(0,…,0,cnun*)<0∀cn∈R∖{0}).
Then (Propositions 2.2, 2.3 and 2.5)
(2.99)(2.97)⟹(withλl<λl*∀l∈{1,…,n}(i14)holds),(2.100)(2.97)and(2.98)⟹(withλl<λl*∀l∈{1,…,n-1}∃δ1*>0:(i14)holdsifλn<λn*+δ1*),(2.101)(2.98)⟹(withλl<λl*∀l∈{1,…,n-1}∃δ2*>0:(i15)holdsifλn∈]λn*,λn*+δ2*[).
Since ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4), we get the following proposition.Proposition 2.27.
Under assumption (2.95) we have:(i)
When (2.97) holds ((2.97) and (2.98) hold, resp.), set λ1,…,λnas in (2.99) (resp. (2.100)) system (2.96) has at least two weak solutions u0 and -u0 with uℓ0≥0 as ℓ=1,…,n(u0=τ0v0,τ0=const.>0,v0∈Sλ∩V+(Dm));(ii)
When (2.98) holds, set λ1,…,λn as in (2.101) system (2.96) has at least two weak solutions u̅and -u̅ with u̅ℓ≥0 as ℓ=1,…,n(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλ-∩S(Dm)).
Consequently, when (2.97) and (2.98) hold, with λℓ<λℓ*forallℓ∈{1,…,n-1}and λn∈]λn*,λn*+min{δ1*,δ2*}[ system (2.96) has at least four different weak solutions.We remark that (PropositionA.3, [5]) as i=1,…,n:(2.102)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω).
Moreover, since ui0(resp.u̅i) is a weak supersolution of the equation:(2.103)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui-∑j=1m-1ajqji|ui|qji-2ui-d̃i|ui|qm-2uiinΩ,
where aj=∏ℓ≠i∫Ω(uℓ0)qjℓdx(resp.aj=∏ℓ≠i∫Ω(u̅ℓ)qjℓdx), we have [6]
(2.104)ui0>0ifui0≢0[resp.u̅i>0ifu̅i≢0].Proposition 2.28.
It results in(2.105)ui0>0asi=1,…,n,u̅l̅>0.Proof.
Since (2.104), we must show that
(2.106)ui0≢0asi=1,…,n,(2.107)u̅l̅≢0.
About (2.106), it is sufficient (Remark 1.1) to prove that
(2.108)(i16h)holds∀h∈{1,…,n}withF=Sλ∩V+(Dm).
Let v=(v1,…,vn)∈Sλ∩V+(Dm) with vh≡0. Let 𝕂⊆Ω be a compact set such that
(2.109)|K|N>0,ψ=∑l≠hdl|vl|γ>0inK.
From Proposition A.1, there exists φ∈C0∞(Ω), with 0≤φ≤1, such that
(2.110)δ=p-1[∫Ω|∇φ|pdx-λh∫Ωbhφpdx]>0,∫Ωψqm/γdhφpdx>0.
Then, with v(s)=(s1/pv1,…,(1-s)1/pδ-1/pφ,…,s1/pvn), we have
(2.111)Hλ(v(s))=1∀s∈[0,1],∃s0∈[0,1[:Dm(v(s))>0∀s∈[s0,1],lim⁡s→1-ddsDj(v(s))∈]-∞,0]asj=1,…,m-1,lim⁡s→1-ddsDm(v(s))=-∞.
Let us prove (2.107). We recall that ([1], Theorem 4.2):
(2.112)Ẽ̃(v̅)=inf⁡⁡{Ẽ̃(v):v∈Vλ-∩S(Dm)},
where Ẽ̃ as in Application 2.22. Reasoning by contradiction, let v̅ℓ̅≡0. Then, v̅ℓ≢0 for some ℓ≠ℓ̅ and consequently from (2.104) ∑ℓ≠ℓ̅dℓ(v̅ℓ)γ>0.
Letφ∈C0∞(Ω), with 0≤φ≤1, such that ∫Ωdℓ̅qm/γφqmdx>∫Ωd̃ℓ̅φqmdx. Let us consider the function:
(2.113)g(s,τ)=Dm(τv̅1,…,sφ,…,τv̅n)=qm-1[∫Ω(sγdl̅φγ+τγ∑l≠l̅dl(v̅l)γ)qm/γdx-sqm∫Ωd̃l̅φqmdx-τqm∑l≠l̅∫Ωd̃l(v̅l)qmdx]∀s≥0,∀τ≥1.
Since
(2.114)g(0,1)=-1,∂g∂s(s,τ)>0∀s>0,∀τ≥1,g(0,τ)=-τqm<-1∀τ>1,lim⁡s→+∞g(s,τ)=+∞∀τ≥1,
we have
(2.115)∀τ≥1∃∣s(τ)≥0(s(1)=0,s(τ)>0forτ>1):g(s(τ),τ)=-1.
We note that limτ→1+s(τ)=0. In fact, if {τn}⊆]1,+∞[andlimτn=1, being g(s(τn),τn)=-1, {s(τn)} is bounded (else (within a subsequence) limg(s(τn),τn)=+∞). Then (within a subsequence) lims(τn)=ω with g(ω,1)=0, from which ω=0.
We add thats(τ) belongs to C1(]1,+∞[), and its derivative has the form:
(2.116)s′(τ)=-1(s(τ))γ-1g̃(s(τ),τ)∀τ>1withlim⁡τ→1+g̃(s(τ),τ)∈]-∞,0[.
Hence, set v(τ)=(τv̅1,…,s(τ)φ,…,τv̅n), it results in
(2.117)Dm(v(τ))=-1∀τ≥1,lim⁡τ→1+ddτHλ(v(τ))=pHλ(v̅)<0,lim⁡τ→1+ddτDj(v(τ))=0asj=1,…,m-1.
As in Proposition2.24, we introduce the open ball B̃ with centre v̅ included in Vλ- and the functionals t*(v) and E*(v) belonging to C1(B̃). Chosen τ0>1 such that v(τ)∈B̃forallτ∈[1,τ0], we have
(2.118)ddτE*(v(τ))=(t*(v(τ)))pddτHλ(v(τ))-∑j=1m-1(t*(v(τ)))qjddτDj(v(τ))∀τ∈[1,τ0],
and consequently limτ→1+(d/dτ)E*(v(τ))<0. Then, taking into account (2.112), with τ1∈]1,τ0] such that (d/dτ)E*(v(τ))<0forallτ∈]1,τ1], we get the contradiction:
(2.119)E*(v̅)≤E*(v(τ))<E*(v̅)∀τ∈]1,τ1].Proposition 2.29.
Ifdℓqm/γ>d̃ℓasℓ=1,…,n-1, then
(2.120)u̅l>0asl=1,…,n.Proof.
In fact,(2.121)u̅l>0asl=1,…,n-1(Proposition2.23),u̅n≡0⟹Dm(u̅)>0.Application 2.30.
Let for eachv=(v1,…,vn)∈W:
(2.122)A(v)=p-1∫Ω(∑l=1n|∇vl|γ)p/γdx+∏l=1n∫Ω|∇vl|pldx,Dj(v)=∫Ωρj(∏l=1n|vl|qjl)dxasj=1,…,m-1,Dm(v)=qm-1[∫Ω|∑l=1ndlvl|qm-1(∑l=1ndlvl)dx-∫Ωd|vn|qmdx],
where
(2.123)1<γ<p,pl>1,∑l=1npl=p,qjl>1,∑l=1nqjl=qj,p<qm,q1<⋯<qm<p̃,ρj∈L∞(Ω)∖{0},ρj≤0,dl,d∈L∞(Ω),dl(x)≠0a.e.inΩ,d>0.
Let as ℓ=1,…,nFℓ∈W-1,p′(Ω)(p′=p/(p-1)). Let 〈〈F,v〉〉=∑ℓ=1n〈Fℓ,vℓ〉forallv∈W. Set ηi=0 as i=1,…,n-1 and ηn=1, let us consider the system:
(2.124)-div⁡([(∑l=1n|∇ul|γ)(p/γ)-1|∇ui|γ-2+pi(∏l≠i∫Ω|∇ul|pl)|∇ui|pi-2]∇ui)=λibi|ui|p-2ui+∑j=1m-1qjiρj(∏l≠i|ul|qjl)|ui|qji-2ui+|∑l=1ndlul|qm-1di-ηid|un|qm-2un+FiinΩ,ui=0on⁡∂Ωasi=1,…,n,
under at least one of the following conditions
(2.125)∃Ω+⊆Ω:|Ω+|N>0,dl>0inΩ+forsomel∈{1,…,n-1}(⟹V+(Dm)≠∅),(2.126)|dn|qm<d(⟹Dm(0,…,0,cnun*)<0∀cn∈R∖{0}).
Evidently, about the validity of (i14) we choose λ1,…,λn as in Application 2.26.Proposition 2.31 (see [1], Theorem 3.2).
Under assumptions (2.123), (2.125) ((2.125) and (2.126), resp.), if F≢0 and ∥F∥* is sufficiently small, for λ1,…,λn as in (2.99) (resp. (2.100)) system (2.124) has at least one weak solution ũ(ũ=τ̃ṽ, τ̃=const.>0,ṽ∈Sλ∩V+(Dm)).Let us note that(2.127)ũh≢0evenifFh≡0since(Fh≡0,ũh≡0)⟹∑l=1ndlũl≡0⟹Dm(ũ)≤0.Application 2.32.
Letλ1=⋯=λn=0, and for each v=(v1,…,vn)∈W:
(2.128)A(v)=p-1∫Ω[∑l=1n(|∇vl|γ+a|vl|γ)]p/γdx,Dj(v)=qj-1∫Ωdj(∑l=1n|vl|γj)qj/γjdxasj=1,…,m,withm>2,
under one of the following assumptions:
(2.129)a∈L∞(Ω),a≥0,dj∈L∞(Ω)∖{0}withd1≤0,dj≥0asj≥2,1<γj<γ<p<q2<⋯<qm<p̃asj≥2,γ≤γ1<q1<q2;(2.130)a∈L∞(Ω),a≥0,asj=1,…,mdj∈L∞(Ω)∖{0},dj≥0,1<γj<γ<p<q1<⋯<qm<p̃.
Set F as in Application 2.30. Let us consider the system:
(2.131)-div⁡([∑l=1n(|∇ul|γ+a|ul|γ)](p/γ)-1|∇ui|γ-2∇ui)=-[∑l=1n(|∇ul|γ+a|ul|γ)](p/γ)-1a|ui|γ-2ui+∑j=1mdj(∑l=1n|ul|γj)(qj/γj)-1|ui|γj-2ui+FiinΩ,ui=0on⁡∂Ωasi=1,…,n.
Let us verify that
(2.132)(2.129)[resp.(2.130)]⟹((i16h)holds∀h∈{1,…,n}withF=Sλ∩V+(D2,…,Dm)[resp.F=Sλ∩V+(D1,…,Dm)]).
Let v=(v1,…,vn)∈𝔉 with, for example, v1≡0. Let j0∈{2,…,m}(resp.j0∈{1,…,m}) and ℓ0∈{2,…,m} such that dj0vℓ0≢0. Let us suppose ℓ0=2 and set v(s)=((1-s)1/γv2,s1/γv2,v3,…,vn). Then,
(2.133)A(v(s))=1∀s∈[0,1],∃s0∈[0,1[:Dj0(v(s))>0∀s∈[s0,1],lim⁡s→1-ddsDj0(v(s))=-∞,lim⁡′′s→1-ddsDj(v(s))<+∞asj≠j0.Proposition 2.33.
Under assumption (2.129) (resp. (2.130)), system (2.131) with F≡0 has at least two weak solutions u0 and -u0, and we have as i=1,…,n:
(2.134)ui0∈L∞(Ω),ui0≥0,ui0≢0.
Consequently,
(2.135)a≡0⟹ui0∈Cloc1,αi0(Ω),a≡0and(2.129)holdswithp≤γ1[resp.(2.130)holds]⟹ui0>0.Proof.
The statement is due to ([1], Theorem 2.2, Remark 2.3), [5], Proposition A.3, [6].Proposition 2.34 (see [1], Theorems 3.1, 3.2).
Under assumption (2.129) (resp. (2.130)), system (2.131) with F≢0 and ∥F∥* sufficiently small has at least two different weak solutions u1andu2(ui=τivi,τi=const.>0,v1∈V+(F)∩Sλ,v2∈Sλ∩V+(D2,…,Dm)[resp.v2∈Sλ∩V+(D1,…,Dm)]), and we have uh2≢0 even if Fh≡0.Remark 2.35.
If⋃j=2m{x∈Ω:dj(x)>0}[resp.⋃j=1m{x∈Ω:dj(x)>0}]=Ω (within a set with measure equal to zero), with the same reasoning used about (2.132), we get that
(2.136)(i16h)holds∀h∈{1,…,n}withF=V+(F)∩Sλ,
hence, uh1≢0 even if Fh≡0.


## 3. Neumann Problems

LetΩ⊆RN be an open, bounded, and connected C0,1 set. Let |·|N,pandp̃ as in Section 2, σ the measure on ∂Ω,ν the outward unit normal to ∂Ω,p̂=(N-1)p/(N-p) if p<N,p̂=∞ if p≥N. Let us assume(3.1)W=(W1,p(Ω))n(n≥1)with‖v‖=(∑l=1n∫Ω[|∇vl|p+|vl|p]dx)1/p∀v=(v1,…,vn)∈W,Bl(vl)=p-1∫Ωbl|vl|pdx∀vl∈W1,p(Ω),wherebl∈L∞(Ω)∖{0},B̂l(vl)=p-1∫∂Ωb̂l|vl|pdσ∀vl∈W1,p(Ω),whereb̂l∈L∞(∂Ω)∖{0}.
We note that  for each vℓ∈W1,p(Ω) we set γ0(vℓ)=vℓ where γ0 is the trace operator from W1,p(Ω) into W1-(1/p),p(∂Ω). Morever we consider the functionals A (as in (i11)) such that(3.2)∃c̃>0:A(v)≥p-1c̃∑l=1n∫Ω|∇vl|pdx∀v∈W.
It is easy to verify the following.Proposition 3.1.
Letbℓ,b̂ℓ≥0 as ℓ=1,…,n. Then,
(3.3)(i13)holdsifλl,μl≤0,λl+μl<0asl=1,…,n.Let us setI={1,…,n} and for each I*⊆I(3.4)C*={c=(c1,…,cn)∈Rn:cl=0ifl∈I∖I*,cl≠0forsomel∈I*}.
Let us introduce the conditions:(i31)
there existsI*⊆I:Dm(c)<0forallc∈C*;(i32)
there existI*⊆I and m1∈{1,…,m}:Dm1(c)<0 and A(c)=0forallc∈C*.Proposition 3.2.
Let (i31) holds with I*≠I. Let V+(Dm)≠∅. Let bℓ,b̂ℓ≥0 as ℓ∈I∖I*. Then with λℓ,μℓ≤0 and λℓ+μℓ<0 as ℓ∈I∖I*∃δ*>0: (i14) holds if|λℓ|,|μℓ|≤δ* as ℓ∈I*.Proof.
Reasoning by contradiction, for eachk∈ℕ there exist λℓk,μℓk∈[-k-1,k-1], with ℓ∈I*, and vk=(v1k,…,vnk)∈V+(Dm) such that
(3.5)‖vk‖p>k{A(vk)-∑l∈I∖I*p-1[λl∫Ωbl|vlk|pdx+μl∫∂Ωb̂l|vlk|pdσ]-∑l∈I*p-1[λlk∫Ωbl|vlk|pdx+μlk∫∂Ωb̂l|vlk|pdσ]},
then, set wk=∥vk∥-1vk, we have
(3.6)Dm(wk)>0,p-1{c̃∑l=1n∫Ω|∇wlk|pdx-∑l∈I∖I*[λl∫Ωbl|wlk|pdx+μl∫∂Ωb̂l|wlk|pdσ]}<k-1+∑l∈I*p-1[λlk∫Ωbl|wlk|pdx+μlk∫∂Ωb̂l|wlk|pdσ].
Since ∥wk∥=1, there exists w∈W such that (within a subsequence)
(3.7)wk⟶wweaklyinW,wk⟶wstronglyin(Lp(Ω))n,wk⟶wstronglyin(Lp(∂Ω))n.
Consequently, from (3.6), passing to limit as k→+∞, we get
(3.8)Dm(w)≥0,∑l=1n∫Ω|∇wl|pdx=0,∑l∈I∖I*[λl∫Ωbl|wl|pdx+μl∫∂Ωb̂l|wl|pdσ]=0,
from which w=0, and then the contradiction 0=limk→+∞∥wk∥=1.Proposition 3.3.
Let (i31) holds with I*=I. Let V+(Dm)≠∅. Then,
(3.9)∃δ*>0:(i14)holdsif|λl|,|μl|≤δ*asl=1,…,n.The proof as in Proposition3.2.Proposition 3.4.
Let (i32) holds with I*≠I. Let ∫Ωbℓdx,∫∂Ωb̂ℓdσ>0 as ℓ∈I*. Then,
(3.10)Vλμ-∩S(Dm1)≠∅∀(λl,μl)l∈Iwithλl,μl≥0∀l∈I*,λl+μl>0forsomel∈I*.
Moreover, if bℓ,b̂ℓ≥0 as ℓ∈I∖I*, we havewithλℓ,μℓ≤0 and λℓ+μℓ<0 as ℓ∈I∖I*∃δ*>0:(i15) holdsifλℓ,μℓ∈[0,δ*]∀ℓ∈I* and λℓ+μℓ>0 for some ℓ∈I*.Proof.
The first statement is evident. Let us prove the second one. Reasoning by contradiction, for eachk∈ℕ there exist λℓk,μℓk∈[0,k-1], with ℓ∈I* and λℓk+μℓk>0 for some ℓ∈I*, and a sequence (vk,h)h∈ℕ such that
(3.11)(vk,h)h∈N⊆Vλkμk-∩S(Dm1)(λlk=λl,μlk=μlasl∈I∖I*),sup⁡h‖vk,h‖=+∞.
Let{hk}⊆ℕbe a strictly increasing sequence such that ∥vk,hk∥→+∞ as k→+∞. Let wk=∥vk,hk∥-1vk,hk. Then, Dm1(wk)=-∥vk,hk∥-qm1 and
(3.12)p-1{c̃∑l=1n∫Ω|∇wlk|pdx-∑l∈I∖I*[λl∫Ωbl|wlk|pdx+μl∫∂Ωb̂l|wlk|pdσ]}<p-1∑l∈I*[λlk∫Ωbl|wlk|pdx+μlk∫∂Ωb̂l|wlk|pdσ],
moreover, there exists w∈W such that (within a subsequence)
(3.13)wk⟶wweaklyinW,wk⟶wstronglyin(Lp(Ω))n,wk⟶wstronglyin(Lp(∂Ω))n.
Consequently,
(3.14)Dm1(w)=0,∑l=1n∫Ω|∇wl|pdx=0,∑l∈I∖I*[λl∫Ωbl|wl|pdx+μl∫∂Ωb̂l|wl|pdσ]=0,
then w=0, and the contradiction 0=limk→+∞∥wk∥=1.Proposition 3.5.
Let (i32) holds with I*=I. Let ∫Ωbℓdx,∫∂Ωb̂ℓdσ>0 as ℓ=1,…,n. Then,
(3.15)Vλμ-∩S(Dm1)≠∅ifλl,μl≥0∀l∈Iandλl+μl>0forsomel∈I,∃δ*>0:(i15)holdsifλl,μl∈[0,δ*]∀l∈Iandλl+μl>0forsomel∈I.The proof as in Proposition3.4.Remark 3.6.
It is suitable to make some clarifications.(i)
The assumption “bℓ,b̂ℓ≥0” (see Propositions 3.1, 3.2, and 3.4) can be replaced by “bℓ,b̂ℓ do not change sign.” In this case we can choose λℓ and μℓ such that λℓbℓ≤0,μℓb̂ℓ≤0 and |λℓ|+|μℓ|>0.(ii)
The assumption “∫Ωbℓdx,∫∂Ωb̂ℓdσ>0” (see Propositions 3.4 and 3.5) can be replaced by “∫Ωbℓdx,∫∂Ωb̂ℓdσ≠0”. In this case, we can choose λℓ and μℓ such that λℓ∫Ωbℓdx,μℓ∫∂Ωb̂ℓdσ≥0 and |λℓ|+|μℓ|>0 for some ℓ, with |λℓ|,|μℓ|≤δ* instead of λℓ,μℓ∈[0,δ*].(iii)
When for eachℓ∈{1,…,n}bℓ,b̂ℓ do not change sign, then the conclusion of the Proposition 3.2 [resp. Proposition 3.3] holds even if λℓbℓ,μℓb̂ℓ≤0 and |λℓ|+|μℓ|>δ* as ℓ∈I*(resp. as ℓ=1,…,n).
In order to simplify the presentation of the applications, we suppose in the nextbℓ∈L∞(Ω)∖{0} andb̂ℓ∈L∞(∂Ω)∖{0}, while the additional assumptions on bℓ,b̂ℓ and the assumptions on ∫Ωbℓdx,∫∂Ωb̂ℓdσ (the same of Propositions 3.1, 3.2, 3.4, and 3.5) will be pointed out case by case.Passing to the applications (withn>1), we recall that in [3] Pohozaev and Véron in the case n=1 have studied the Neumann problem:(3.16)-div⁡(|∇u|p-2∇u)=λb(x)|u|p-2u+c(x)|u|s-2u+a(x)|u|q-2uinΩ,|∇u|p-2∂u∂ν=k(x)|u|r-2uon⁡∂Ω.
The existence theorems proved by these authors can be got by using some results of ([1], Theorems 2.1, 2.2, 4.1, and 4.2; Remarks 2.1, 2.3, 4.1, and 4.4), Propositions 3.3 and 3.5.Application 3.7.
Let for eachv=(v1,…,vn)∈W:
(3.17)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,D1(v)=q1-1[∫∂Ω(∑l=1ndl|vl|γ)q1/γdσ-∑l=1n∫∂Ωd̂l|vl|q1dσ],
where
(3.18)1<γ<q1<p̂,q1≠p,dl,d̂l∈L∞(∂Ω),dl,d̂l>0.
Let us consider the system:
(3.19)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2uiinΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui+(∑l=1ndl|ul|γ)(q1/γ)-1di|ui|γ-2ui-d̂i|ui|q1-2uion⁡∂Ωasi=1,…,n.
Let us introduce the conditions:
(3.20)∫∂Ω(∑l=1ndl)q1/γdσ<∫∂Ωd̂dσ(d̂=min⁡⁡{d̂1,…,d̂n}),(3.21)∃Γ⊆∂Ω:σ(Γ)>0,(∑l=1ndl)q1/γ>∑l=1nd̂lon⁡Γ,(3.22)∫Ωbldx>0,∫∂Ωb̂ldσ>0asl=1,…,n.
Evidently (3.20)  ⇒D1(c)<0forallc∈Rn∖{0}. Moreover (3.21)  ⇒V+(D1)≠∅ (Proposition A.2). Hence (Propositions 3.3 and 3.5)
(3.23)(3.20)and(3.21)⟹(∃δ1*>0:(i14)holdsif|λl|,|μl|≤δ1*∀l∈{1,…,n}),(3.24)(3.20)and(3.22)⟹(∃δ2*>0(i15)holdsifλl,μl∈[0,δ2*]∀l∈{1,…,n},λl+μl>0forsomel).Proposition 3.8 (see ([1], Theorems 2.1 and 4.1; Remarks 2.1 and 4.1); Proposition A.4; [5, 6]).
Under assumption (3.18), we have:(i)
When (3.20) and (3.21) hold, with λℓ,μℓ as in (3.23) system (3.19) has at least two weak solutions u0 and -u0(u0=τ0v0,τ0=const.>0,v0∈Sλμ∩V+(D1)), and it results in
(3.25)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),ui0≥0asi=1,…,n,ui0>0ifui0≢0;(ii)
When (3.20) and (3.22) hold, with λℓ,μℓ as in (3.24) system (3.19) has at least two weak solutions u̅ and -u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλμ-∩S(D1)), and it results in(3.26)u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω),u̅i≥0asi=1,…,n,u̅i>0ifu̅i≢0.
Consequently, when (3.20)–(3.22) hold, with λℓ,μℓ as in (3.24) and min{δ1*,δ2*} instead of δ2* system (3.19) has at least four different weak solutions.Proposition 3.9.
Ifγ<p<q1, then ui0>0asi=1,…n.Proof.
It is sufficient (Remark1.1) to verify that
(3.27)(i16h)holdsash=1,…,nwithF=Sλμ∩V+(D1).
Letv=(v1,…,vn)∈V+(D1)∩Sλμ. Let, for example, v1≡0. Since ∫∂Ω(∑ℓ≠1dℓ|vℓ|γ)q1/γdσ>0, there exists Γ+⊆∂Ω such that
(3.28)σ(Γ+)>0,∑l≠1dl|vl|γ>0on⁡Γ+.
Let 𝕂⊆Ω a compact set and Ω′ an open set such that
(3.29)|K|N>0,K⊆Ω′,Ω′̅⊆Ω.
Since Propositions A.1 and A.2, there exist a compact set Γ̂+⊆Γ+, with σ(Γ̂+)>0, and (φ1ε)0<ε<ε0,(φ2ε)0<ε<ε0⊆C0∞(RN) such that
(3.30)0≤φ1ε≤1,supp⁡φ1ε⊆Ω′,φ1ε⟶χstronglyinLs(Ω)∫Ω|∇φ1ε|sdx⟶+∞asε⟶0+∀s∈[1,+∞[,0≤φ2ε≤1,supp⁡φ2ε⊆RN∖Ω′̅,φ2ε⟶χ̂stronglyinLs(∂Ω),∫Ωφ2εsdx⟶0asε⟶0+∀s∈[1,+∞[,
where χ(resp.χ̂) is the characteristic function of 𝕂(resp.Γ̂+). Let us choose ε such that
(3.31)δ=p-1[∫Ω|∇φε|pdx-λ1∫Ωb1φεpdx-μ1∫∂Ωb̂1φεpdσ]>0,∫∂Ω(∑l≠1dl|vl|γ)(q1/γ)-1d1φεγdσ>0(φε=φ1ε+φ2ε),
and we set v(s)=((1-s)1/pδ-1/pφε,s1/pv2,…,s1/pvn). Then,
(3.32)Hλμ(v(s))=1∀s∈[0,1],∃s0∈[0,1[:D1(v(s))>0∀s∈[s0,1],lim⁡s→1-ddsD1(v(s))=-∞.Proposition 3.10.
If(3.33)dlq1/γ<d̂lasl=1,…,n,(3.34)λl+μl>0asl=1,…,n,
then u̅i>0asi=1,…,n.Proof.
We recall that ([1], Theorem 4.1)
(3.35)Hλμ(v̅)=inf⁡⁡{Hλμ(v):v∈Vλμ-∩S(D1)}.
Reasoning by contradiction let, for example, v̅1≡0. As c1=const.>0 and g(s,τ)=D1(sc1,τv̅2,…,τv̅n)=q1-1[∫∂Ω(d1sγc1γ+τγ∑ℓ≠1dℓ(v̅ℓ)γ)q1∖γdσ-sq1c1q1∫∂Ωd̂1dσ-τq1∑ℓ≠1∫∂Ωd̂ℓ(v̅ℓ)q1dσ]   for all  s,τ≥0, we have g(0,τ)=-τq1>-1forallτ∈]0,1[ and since (3.33) lims→+∞g(s,τ)=-∞forallτ≥0. Thenforallτ∈]0,1[, it is possible to choose s(τ)>0 such that g(s(τ),τ)=-1. Let us add that there exist s0>0 and τ0∈]0,1[ such that (∂g/∂s)(s,τ)>0forall(s,τ)∈]0,s0[x]τ0,1[.
Let now{τn}⊆]τ0,1[ and limτn=1. Since g(s(τn),τn)=-1,{s(τn)} is necessarily bounded. Then (within a subsequence) lims(τn)=ω≥s0. Consequently, from the inequality:
(3.36)Hλμ(v̅)≤Hλμ(v(τn)),wherev(τn)=(s(τn)c1,τnv̅2,…,τnv̅n)∈Vλμ-∩S(D1),
as n→+∞ and from (3.34), we get the contradiction:
(3.37)Hλμ(v̅)≤-p-1ωpc1p(λ1∫Ωb1dx+μ1∫∂Ωb̂1dσ)+Hλμ(v̅)<Hλμ(v̅).Remark 3.11.
Let us note that the conditions (3.20), (3.21), and (3.33) are compatible.Application 3.12.
Let for eachv=(v1,…,vn)∈W:
(3.38)A(v)=p-1[∑l=1n-1∫Ω|∇vl|pdx+∫Ω(|∇vn|γ+∫∂Ω|vn|γdσ)p/γdx],D1(v)=q1-1[∑l=1n-1∫Ωρl|vl+vn|q1-1(vl+vn)dx-∑l=1n∫∂Ωd̂l|vl|q1dσ],
where
(3.39)1<γ<p,1<q1<p̂,q1≠p,ρl∈L∞(Ω),ρl>0,d̂l∈L∞(∂Ω),d̂l>0.
Let us consider the system:
(3.40)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+ρi|ui+un|q1-1inΩasi=1,…,n-1,-div⁡[(|∇un|γ+∫∂Ω|un|γdσ)(p/γ)-1|∇un|γ-2∇un]=λnbn|un|p-2un+∑l=1n-1ρl|ul+un|q1-1inΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui-d̂i|ui|q1-2uion⁡∂Ωasi=1,…,n-1,(|∇un|γ+∫∂Ω|un|γdσ)(p/γ)-1|∇un|γ-2∂un∂ν=μnb̂n|un|p-2un-[∫Ω(|∇un|γ+∫∂Ω|un|γdσ)(p/γ)-1dx]|un|γ-2un-d̂n|un|q1-2unon⁡∂Ω.
Pointing out that V+(D1)≠∅, we advance the conditions
(3.41)∫Ω(∑l=1n-1ρl)dx<∫∂Ωd̂dσ(d̂=min⁡⁡{d̂1,…,d̂n}),(3.42)∫Ωbldx>0,∫∂Ωb̂ldσ>0asl=1,…,n-1,(3.43)bn≥0,b̂n≥0.
Taking into account that
(3.44)(3.41)⟹D1(c1,…,cn-1,0)<0∀(c1,…,cn-1)∈Rn-1∖{0},
we have (Propositions 3.2 and 3.4)
(3.45)(3.41)and(3.43)⟹(withλn,μn≤0,λn+μn<0∃δ1*>0:(i14)holdsif|λl|,|μl|≤δ1*asl=1,…,n-1),(3.46)(3.41)–(3.43)⟹(withλn,μn≤0,λn+μn<0∃δ2*>0:(i15)holdsifλl,μl∈[0,δ2*]asl=1,…,n-1andλl+μl>0forsomel).Proposition 3.13 . (see ([1], Theorems 2.1 and 4.1; Remark 2.1); Proposition A.4; [5, 6]).
Under assumption (3.39), we have(i)
When (3.41) and (3.43) hold, with λℓ,μℓ as in (3.45) system (3.40) has at least one weak solution u0 (u0=τ0v0,τ0=const.>0,v0∈Sλμ∩V+(D1)), and it results in
(3.47)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),ui0>0asi=1,…,n-1,un0∈L∞(Ω),un0≥0,un0≢0;(ii)
When (3.41)–(3.43) hold, with λℓ,μℓ as in (3.46) system (3.40) has at least one weak solution u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλμ-∩S(D1)), and it results in u̅i≢0 as i=1,…,n.
Consequently, when (3.41)–(3.43) hold, with λℓ,μℓ as in (3.46) and min{δ1*,δ2*} instead of δ2* system (3.40) has at least two different weak solutions.About the properties ofui0 and u̅i expressed by Proposition 3.13, it is necessary to remark that ifu=(u1,…,un)isanontrivialweaksolutiontosystem (3.40), then ui≢0 as i=1,…,n. In fact,(3.48)un≡0⟹ui≡0asi=1,…,n-1,ui≡0forsomei∈{1,…,n-1}⟹un≡0.Application 3.14.
Letn=2 and for any v=(v1,v2)∈W:
(3.49)A(v)=p-1∑l=12∫Ω|∇vl|pdx,Dj(v)=qj-1∫Ωρj|∑l=12djl|vl|γj|qj/γjdxasj=1,…,m-1,Dm(v)=qm-1∫∂Ωρm(∑l=12|vl|γm)qm/γmdσ,
where
(3.50)1<γj<qjasj=1,…,m,p<q1<⋯<qm<p̂,ρj∈L∞(Ω),ρj<0,djl∈L∞(Ω)∖{0},ρm∈L∞(∂Ω).
Let us consider the system:
(3.51)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+∑j=1m-1ρj|∑l=12djl|ul|γj|(qj/γj)-2(∑l=12djl|ul|γj)dji|ui|γj-2uiinΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui+ρm(∑l=12|ul|γm)(qm/γm)-1|ui|γm-2uion⁡∂Ωasi=1,2.
Let us introduce the conditions:
(3.52)ρm+≢0(⟹V+(Dm)≠∅(PropositionA.2)),(3.53)∫∂Ωρmdσ<0(⟹Dm(c1,c2)<0∀(c1,c2)∈R2∖{0}),(3.54)∫Ωbldx>0,∫∂Ωb̂ldσ>0asl=1,2,
we have (Propositions 3.3 and 3.5)
(3.55)(3.52)and(3.53)⟹(∃δ1*>0:(i14)holdsif|λl|,|μl|≤δ1*asl=1,2),(3.56)(3.53)and(3.54)⟹(∃δ2*>0:(i15)holdsifλl,μl∈[0,δ2*]asl=1,2,λl+μl>0forsomel).Proposition 3.15 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5]).
Under assumption (3.50), we have(i)
When (3.52) and (3.53) hold, with λℓ,μℓ as in (3.55) system (3.51) has at least two weak solutions u0 and -u0(u0=τ0v0,τ0=const.>0,v0∈Sλμ∩V+(Dm)), and it results in
(3.57)ui0∈L∞(Ω)∩Cloc1,αi0(Ω),ui0≥0asi=1,2;(ii)
When (3.53) and (3.54) hold, with λℓ,μℓ as in (3.56) system (3.51) has at least two weak solutions u̅ and -u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλμ-∩S(Dm)), and it results in
(3.58)u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω),u̅i≥0asi=1,2.
Consequently, when (3.52)–(3.54) hold, with λℓ,μℓ as in (3.56), and min{δ1*,δ2*} instead of δ2* system (3.51) has at least four different weak solutions.Proposition 3.16.
Under the assumptionp≤2γj and dj1·dj2<0 as j=1,…,m-1, we have(i)
ifγj0<p for some j0∈{1,…,m}, then ui0>0 as i=1,2;(ii)
ifγj0<γm≤p for some j0∈{1,…,m-1}, thenu̅i>0asi=1,2.Proof.
First of allui0 is a weak supersolution to the equation:
(3.59)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+∑j=1m-1ρj|dj1(u10)γj+dj2(u20)γj|(qj/γj)-2dji2|ui|2γj-2uiinΩ.
Also, u̅i has a similar property. Then [6] it is sufficient to verify that
(3.60)ui0≢0,(3.61)u̅i≢0.
About (3.60), let us prove (Remark 1.1) that
(3.62)(i16h)holdsash=1,2withF=Sλμ∩V+(Dm).
Let v=(v1,v2)∈V+(Dm)∩Sλμ. Let, for example, v1≡0. Let
(3.63)K⊆Ωacompactset:|K|N>0,v2≢0inK,Ω′anopenset:K⊆Ω′,Ω′̅⊆Ω,Γ⊆∂Ω:σ(Γ)>0,ρm|v2|>0on⁡Γ.
Since Propositions A.1 and A.2, there exists φ∈C0∞(RN), with 0≤φ≤1 and suppφ⊆Ω′∪(RN∖Ω′̅), such that
(3.64)∫Ωρj|dj2|v2|γj|(qj/γj)-2|v2|γjφγjdj1dj2dx>0asj=1,…,m-1,∫∂Ωρm|v2|qm-γmφγmdσ>0,δ=p-1[∫Ω|∇φ|pdx-λ1∫Ωb1φpdx-μ1∫∂Ωb̂1φpdσ]>0.
Then with v(s)=((1-s)1/pδ-1/pφ,s1/pv2), we have
(3.65)Hλμ(v(s))=1∀s∈[0,1],∃s0∈[0,1[:Dm(v(s))>0∀s∈[s0,1],lim⁡s→1-ddsDj0(v(s))=-∞,lim⁡s→1-ddsDj(v(s))<+∞asj≠j0.
Passing to (3.61), let us introduce the function ψ(t,v)=ptp-1Hλμ(v)-∑j=1mqjtqj-1Dj(v), and let us remember that ([1], Theorem  4.2)
(3.66)∀v∈Vλμ-∩S(Dm)∃∣t(v)>0:ψ(t(v),v)=0,Ẽ̃(v̅)=inf⁡⁡{Ẽ̃(v):v∈Vλμ-∩S(Dm)},
where Ẽ̃(v)=(t(v))pHλμ(v)-∑j=1m(t(v))qjDj(v).
Reasoning by contradiction, let us set, for example,v̅1≡0 and set  v(s)=((1-s)1/γmv̅2,s1/γmv̅2). Since
(3.67)Dm(v(s))=-1∀s∈[0,1],∃s0∈[0,1[:Hλμ(v(s))<0∀s∈[s0,1],lim⁡s→1-ddsDj0(v(s))=-∞,lim⁡s→1-ddsDj(v(s))<+∞asj∈{1,…,m-1}∖{j0},
as in Proposition 2.24, we get the contradiction:
(3.68)Ẽ̃(v̅)≤Ẽ̃(v(s))<Ẽ̃(v̅)∀s∈[s1,1[(s0≤s1<1).Application 3.17.
Letn>2 and set for each v=(v1,…,vn)∈W:
(3.69)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,Dj(v)=qj-1∑l≠jl=1n∫Ωρj|djj|vj|γj+djl|vl|γj|qj/γjdxasj=1,…,n,Dn+1(v)=qn+1-1∫∂Ωρn+1(∑l=1n|vl|γn+1)qn+1/γn+1dσ,
where
(3.70)1<γj<qjasj=1,…,n+1,p<q1<⋯<qn+1<p̂,ρj∈L∞(Ω),ρj<0,djl∈L∞(Ω)∖{0},ρn+1∈L∞(∂Ω).
Let us consider the system:
(3.71)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+∑l≠iρi|dii|ui|γi+dil|ul|γi|(qi/γi)-2(dii|ui|γi+dil|ul|γi)dii|ui|γi-2ui+∑j≠iρj|djj|uj|γj+dji|ui|γj|(qj/γj)-2(djj|uj|γj+dji|ui|γj)dji|ui|γj-2uiinΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui+ρn+1(∑l=1n|ul|γn+1)(qn+1/γn+1)-1|ui|γn+1-2uion⁡∂Ωasi=1,…,n.
Let us make the assumptions:
(3.72)ρn+1+≢0,∫∂Ωρn+1dσ<0,∫Ωbldx>0,∫∂Ωb̂ldσ>0asl=1,…,n.
About Neumann’s problem (3.71), we have an existence result similar to the one of Proposition 3.15 related to system (3.51). About the positive sign of the components of the weak solutions u0 and u̅ to system (3.71), as in Proposition 3.16, we show.Proposition 3.18.
Under the assumptionp≤2γj as j=1,…,n and djj·djℓ<0 as j,ℓ∈{1,…,n} with ℓ≠j, we have(i)
if eitherγn+1<p  or γj<p for all j∈{1,…,n}∖{j0}  for some  j0,  then  ui0>0  as  i=1,…,n;(ii)
ifγj<γn+1≤p  for all j∈{1,…,n}∖{j0}   for some  j0,  then  u̅i>0  as  i=1,…,n.The following remark deals also with Application 3.14.Remark 3.19.
Making in (3.50) (resp. (3.70)) the change
(3.73)q1<⋯<qm<p[resp.q1<⋯<qn+1<p],
system (3.51) (resp. (3.71)) has at least the two weak solutions u̅and-u̅ ([1], Theorem 4.2; Remark 4.4). The components of u̅ keep the properties that Propositions 3.15 and 3.16 (Proposition 3.15 and Proposition 3.18 resp.) underline.Application 3.20.
Let for eachv=(v1,…,vn)∈W:
(3.74)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,Dj(v)=qj-1∫∂Ωρj|dj1|vj|γj+dj2|vn|γj|qj/γjdσasj=1,…,n-1,Dn(v)=(∫∂Ω|vn|γ̂ndσ)(∫Ωρn|vn|γndx),
where
(3.75)1<γj<qj<p̂asj=1,…,n-1,1<γn<p̃,1<γ̂n<p̂,p<q1<⋯<qn-1<qn=γn+γ̂n,ρj∈L∞(∂Ω),ρj<0,dj1,dj2∈L∞(∂Ω)∖{0},ρn∈L∞(Ω).
Let us consider the system:
(3.76)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2inΩasi=1,…,n-1,-div⁡(|∇un|p-2∇un)=λnbn|un|p-2un+γn(∫∂Ω|un|γ̂ndσ)ρn|un|γn-2uninΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui+ρi|di1|ui|γi+di2|un|γi|(qi/γi)-2×(di1|ui|γi+di2|un|γi)di1|ui|γi-2uion⁡∂Ωasi=1,…,n-1,|∇un|p-2∂un∂ν=μnb̂n|un|p-2un+∑j=1n-1ρj|dj1|uj|γj+dj2|un|γj|(qj/γj)-2×(dj1|uj|γj+dj2|un|γj)dj2|un|γj-2un+γ̂n(∫Ωρn|un|γndx)|un|γ̂n-2unon⁡∂Ω.
Let us introduce the conditions:
(3.77)ρn+≢0(⟹V+(Dn)≠∅(PropositionsA.1andA.2)),(3.78)∫Ωρndx<0(⟹Dn(0,….,0,cn)<0∀cn∈Rn∖{0}),(3.79)bl≥0,b̂l≥0asl=1,…,n-1,(3.80)∫Ωbndx>0,∫∂Ωb̂ndσ>0.
We have (Propositions 3.2 and 3.4)
(3.81)(3.77)–(3.79)⟹(withλl,μl≤0,λl+μl<0asl=1,…,n-1∃δ1*>0:(i14)holdsif|λn|,|μn|≤δ1*),(3.82)(3.78)–(3.80)⟹(withλl,μl≤0,λl+μl<0asl=1,…,n-1∃δ2*>0:(i15)holdsifλn,μn∈[0,δ2*]andλn+μn>0).Proposition 3.21 (see ([1], Theorems 2.2 and 4.2; Remarks 2.3 and 4.4); Proposition A.4; [5, 6]).
Under assumption (3.75), we have(i)
When (3.77)–(3.79) hold, with λℓ,μℓ as in (3.81), system (3.76) has at least two weak solutions u0 and -u0(u0=τ0v0,τ0=const.>0,v0∈Sλμ∩V+(Dn)), and it results in ui0≥0(i=1,…,n),  un0≢0. If  γn<p̂, then(3.83)ui0∈L∞(Ω)∩Cloc1,αi0(Ω)(i=1,…,n),ui0≢0⟹ui0>0(i=1,…,n-1);(ii)
When (3.78)–(3.80) hold, with λℓ,μℓ as in (3.82), system (3.76) has at least two weak solutions u̅ and -u̅(u̅=τ̅v̅,τ̅=const.>0,v̅∈Vλμ-∩S(Dn)), and it results in u̅i≥0(i=1,…,n),  u̅n≢0. If γn<p̂, then(3.84)u̅i∈L∞(Ω)∩Cloc1,α̅i(Ω)(i=1,…,n),u̅i≢0⇒u̅i>0(i=1,…,n-1).
Consequently, when (3.77)–(3.80) hold, with λℓ,μℓ as in (3.82), and min{δ1*,δ2*} instead of δ2* system (3.76) has at least four different weak solutions. Obviously, un0>0 and u̅n>0 if p≤γn<p̂.The following proposition gives a sufficient condition to(3.85)ui0>0asi=1,…,n-1,(3.86)u̅i>0asi=1,…,n-1.Proposition 3.22.
Letγn<p̂. If γj<p and dj1·dj2<0 as j=1,…,n-1, then (3.85) and (3.86) hold.Proof.
Since(3.87)(v1,…,vn)∈V+(Dn)⟹(∃Γ⊆∂Ω:σ(Γ)>0,|vn|>0on⁡Γ),
using Propositions A.1 and A.2, we can verify that
(3.88)(i16h)holdsash=1,…,n-1withF=Sλμ∩V+(Dn),
from which (Remark 1.1) we get (3.85).
Let us prove (3.86). Reasoning by contradiction, let us set, for example, v̅1≡0. If v(s)=((1-s)1/pv̅n,v̅2,…,v̅n), we have
(3.89)Dn(v(s))=-1∀s∈[0,1],∃s0∈[0,1[:Hλμ(v(s))<0∀s∈[s0,1],lim⁡s→1-ddsD1(v(s))=-∞.
Then as in Proposition 3.16, we get a contradiction.Remark 3.23.
Making in (3.75) the change:
(3.90)1<γj<qjasj=1,…,n-1,1<γn,1<γ̂n,q1<⋯<qn=γn+γ̂n<p,
system (3.76) has at least the two weak solutions u̅and-u̅ ([1], Theorem 4.2; Remark 4.4). The components of u̅, all bounded, are locally Hölderian with their first derivatives. If dj1·dj2<0 as j=1,…,n-1, then (3.86) holds.Application 3.24.
Let for eachv=(v1,…,vn)∈W:
(3.91)A(v)=p-1∑l=1n∫Ω(|∇vl|γ+∫Ω|vl|γdx)p/γdx,Dj(v)=qj-1∫∂Ω(∑l=1n|vl|γ)qj/γdσasj=1,…,m-1,Dm(v)=qm-1∫Ω(∑l=1n|vl|γ)qm/γdx,
where
(3.92)1<γ<q1<⋯<qm<p.
Let us consider the system:
(3.93)-div⁡[(|∇ui|γ+∫Ω|ui|γdx)(p/γ)-1|∇ui|γ-2∇ui]=λibi|ui|p-2ui-(∫Ω(|∇ui|γ+∫Ω|ui|γdx)(p/γ)-1dx)|ui|γ-2ui+(∑l=1n|ul|γ)(qm/γ)-1|ui|γ-2uiinΩ,(|∇ui|γ+∫Ω|ui|γdx)(p/γ)-1|∇ui|γ-2∂ui∂ν=μib̂i|ui|p-2ui+∑j=1m-1(∑l=1n|ul|γ)(qj/γ)-1|ui|γ-2uion⁡∂Ωasi=1,…n.
We advance the condition:
(3.94)bl≥0,b̂l≥0asl=1,…n,
and we note that (Proposition 3.1)
(3.95)(3.94)⟹((i13)holdsifλl,μl≤0,λl+μl<0asl=1,…,n).Proposition 3.25.
Under conditions (3.92) and (3.94), with λℓ,μℓ as in (3.95), system (3.93) has at least two weak solutions u0 and -u0(u0=τ0v0,τ0=const.>0,v0∈S+(D1,…,Dm)), and it results in
(3.96)ui0∈L∞(Ω),ui0≥0,ui0≢0asi=1,…,n.Proof.
We recall that ([1], Section 2), set ψ(t,v)=ptp-1Hλμ(v)-∑j=1mqjtqj-1Dj(v), we have
(3.97)∀v∈V+(D1,…,Dm)∃∣t(v)>0:ψ(t(v),v)=0,thefunctionalt(v)isC1inV+(D1,…,Dm).
We introduce the functional Ẽ̃(v)=(t(v))pHλμ(v)-∑j=1m(t(v))qjDj(v) which is C1 in V+(D1,…,Dm). We still remember that ([1], Theorem 2.3; Remark 2.5)
(3.98)∃v0∈S+(D1,…,Dm),withvi0≥0asi=1,…,n,suchthatẼ̃(v0)=inf⁡⁡{Ẽ̃(v):v∈S+(D1,…,Dm)},u0=t(v0)v0isaweaksolutiontosystem(3.93).
The property ui0∈L∞(Ω) is due to Proposition A.4. Let us verify that ui0≢0 as i=1,…,n. Reasoning by contradiction, let us set, for example, v10≡0 and v20≢0. As v(s)=((1-s)1/γv20,s1/γv20,v30,…,vn0), we have
(3.99)∑j=1mDj(v(s))=1∀s∈[0,1],[ddsHλμ(v(s))]s=1>0.
Then, since (d/ds)Ẽ̃(v(s))=(t(v(s)))p(d/ds)Hλμ(v(s)),there exists s0∈[0,1[ such that (d/ds)Ẽ̃(v(s))>0foralls∈[s0,1], from which the contradiction:
(3.100)Ẽ̃(v0)≤Ẽ̃(v(s))<Ẽ̃(v0),∀s∈[s0,1[.Application 3.26.
Let for eachv=(v1,…,vn)∈W:
(3.101)A(v)=p-1∑l=1n∫Ω|∇vl|pdx,D1(v)=q1-1∫Ωρ|∑l=1ndlvl|q1dx,D2(v)=(∫Ω[∑l=1nd̃l|vl|γ]dx)(∫∂Ωρ̂[∑l=1nd̂l|vl|γ̂]dσ),
where
(3.102)1<γ<p̃,1<γ̂<p̂,1<q1<min⁡⁡{p̃,q2=γ+γ̂},p<q2,ρ,dl∈L∞(Ω)∖{0},ρ≤0,ρdl≢0assomel,d̃l∈L∞(Ω)∖{0},d̃l≥0,ρ̂∈L∞(Ω)∖{0},d̂l=const.>0.
Let as ℓ=1,…,nFℓ=fℓ+f̂ℓ, where fℓ∈Lp′(Ω)(p′=p/(p-1)) and f̂ℓ∈(W1-(1/p),p(∂Ω))*  (dual space  of  W1-(1/p),p(∂Ω)).  Let 〈〈F,v〉〉=∑ℓ=1n〈Fℓ,vℓ〉forallv=(v1,…,vn)∈W. Let us consider the system:
(3.103)-div⁡(|∇ui|p-2∇ui)=λibi|ui|p-2ui+ρ|∑l=1ndlul|q1-2(∑l=1ndlul)di+γ(∫∂Ωρ̂[∑l=1nd̂l|ul|γ̂]dσ)d̃i|ui|γ-2ui+fiinΩ,|∇ui|p-2∂ui∂ν=μib̂i|ui|p-2ui+γ̂(∫Ω[∑l=1nd̃l|ul|γ]dx)ρ̂d̂i|ui|γ̂-2ui+f̂ion⁡∂Ωasi=1,…,n.
Let us introduce the conditions:
(3.104)(ρ̂)+≢0(⟹V+(D2)≠∅(PropositionA.1andA.2)),∫∂Ωρ̂dσ<0(⟹D2(c)<0∀c∈Rn∖{0}),
and let us note that (Proposition 3.3)
(3.105)(3.104)⟹(∃δ*>0:(i14)holdsif|λl|,|μl|≤δ*asl=1,…,n).Proposition 3.27.
Under assumptions (3.102) and (3.104), if F≢0 and ∥F∥* is sufficiently small, then with λℓ,μℓ as in (3.105), system (3.103) has at least one weak solution ũ(ũ=τ̃ṽ,τ̃=const.>0,ṽ∈Sλμ∩V+(D2)). When γ<p≤q1, it results in
(3.106)ũh≢0evenifFh≡0.Proof.
The existence ofũ is due to ([1], Theorem 3.2). About (3.106), it is sufficiently (Remark 1.1) to verify that
(3.107)(i16h)holdsash=1,…,nwithF=Sλμ∩V+(D2).
Let v=(v1,…,vn)∈V+(D2)∩Sλμ with, for example, v1≡0. Let ψ=∑ℓ≠1dℓvℓ. Let 𝕂⊆Ω be a compact set having positive measure such that
(3.108)d̃1>0inKifρd1ψ≡0,eitherρd1ψ>0inKorρd1ψ<0inKifρd1ψ≢0.
Proposition A.1 lets us choose φ∈C0∞(RN) satisfying the following conditions:
(3.109)δ=p-1[∫Ω|∇φ|pdx-λ1∫Ωb1|φ|pdx]>0,∫Ωd̃1φγdx>0ifρd1ψ≡0,∫Ωρd1|ψ|q1-2ψφdx>0ifρd1ψ≢0.
Then with v(s)=((1-s)1/pδ-1/pφ,s1/pv2,…,s1/pvn), we have
(3.110)Hλμ(v(s))=1∀s∈[0,1],D2(v(s))>0∀s∈[s0,1](0≤s0<1),lim⁡s→1-⁡ddsD1(v(s))∈R,lim⁡s→1-⁡ddsD2(v(s))=-∞ifρd1ψ≡0,lim⁡s→1-⁡ddsD1(v(s))=-∞,lim⁡s→1-⁡ddsD2(v(s))∈Rifρd1ψ≢0.
Now we replace conditions (3.104) with the following:
(3.111)ρ̂≥0,bl≥0,b̂l≥0asl=1,…,n.Proposition 3.28.
Under assumptions (3.102) and (3.111), if F≢0 and ∥F∥* is sufficiently small, then with λℓ,μℓ≤0 andλℓ+μℓ<0 as ℓ=1,…,n system (3.103) has at least two different weak solution u1 and u2 (ui=τivi,τi=const.>0,v1∈Sλμ∩V+(F),v2∈Sλμ∩V+(D2)). When γ<p≤q1, it results in
(3.112)uh2≢0evenifFh≡0.Proof.
The existence ofu1 and u2 is due to ([1], Theorems 3.1, 3.2, and 3.3; Remark 3.1). Relation (3.112) is proved as in Proposition 3.27.



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*Source: 760854-2012-08-15.xml*