# On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

Authors: Rafik Guefaifia; Salah Mahmoud Boulaaras; Adel Abd Elaziz El-Sayed; Mohamed Abdalla; Bahri-Belkacem Cherif
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system havingp-Laplacian fractional-order differentiation.


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## Body

## 1. Introduction

One of the main applications of fractional calculus science is the fractional-order differential equations (FDEs). Various natural phenomena are modeled mathematically through the FDEs, and this is evident in numerous areas of physics, engineering, chemistry, and other fields. The fractional-order partial differential equations have several applications in many fields such as engineering, biophysics, physics, mechanics, chemistry, and biology (see [1–7]). More and more efforts have been made in the fractional calculus field especially in FDEs (see, for instance, [2, 5, 8–14, 27–39]). Solution existence for a lot of boundary value problems and several nonlinear elementary problems is studied via a huge number of techniques and nonlinear mathematical tools (see [7, 15–23]): the theory of critical point, fixed-point theory, technique of monochromatic iterative, theory degree of coincidence, and the change methods. Motivated by multiple works involved in this domain, we concentrate in this paper on the existence of several infinite solutions to the following fractional-order differentiation system:
(1)tDTαiΦp0Dtαiut=λFuit,u1t,u2t,⋯,unt+μGuit,u1t,u2t,⋯,unt+hiui,a.e.t∈0,T,ui0=uiT=0,for 1≤i≤n, αi∈0;1, 0Dtαi and tDTαi are the left and right αi fractional-order derivatives of the Riemann-Liouville operator, respectively,
(2)Φps=sp−2s,p>1.λ, μ are positive and nonnegative real parameters, respectively, F0F,G:0,T×ℝn⟶ℝ are continuous functions according to t∈0,T for any x1,x2,⋯,xn∈ℝn and are C1 with respect to x1,x2,⋯,xn∈ℝn for a.e. t∈0,T,Ft,0,0,⋯,0=0 and Gt,0,0,⋯,0=0 for it is to say t∈0,T. Also, Fui and Gui indicates partial derivatives of F and G according to ui, respectively, and hi:ℝ⟶ℝ are p−1-order of Lipschitz continuous functions with Li>0 constants of Lipschizian, 1≤i≤n, i.e.,
(3)hix1−hix2≤Lix1−x2p−1.In the last few months, we treated the same area of this study, in [18], by using variational methods together with a critical point theory due to Bonano and Marano. We got at least three weak solutions for the following nonlinear dual-Laplace systems with respect to two parameters:
(4)tDTαi1witp−2ϕpwit0DTαiuit+μuitp−2uit=λFuit,u1t,u2t,⋯,unta.e.t∈0,T,ui0=uiT=0.In addition, in [24], by using the variational method and Ricceri’s critical point theorems, the existence of three weak solutions has been used to investigate the following class of perturbed nonlinear fractional p-Laplacian differential systems:
(5)tDTα1w1tp−2Φpw1t0Dtαut+μμtp−2μt=λFut,ut,υt+δGut,ut,υta.e.t∈0,T,tDTβ1w2tp−2Φpw2t0Dtβυt+μυtp−2υt=λFυt,ut,υt+δGυt,ut,υta.e.t∈0,T,u0=uT=0,υ0=υT=0,where some necessary conditions on the primitive function of nonlinear terms Fu and Fv have been considered. Then, in [25], the same last methods have been used for problem (5), the existence of multiplicity of weak solutions for the following perturbed nonlinear fractional differential systems:
(6)tDTαiait0DTαiuit=λFuit,u1t,u2t,⋯,unt+μGuit,u1t,u2t,⋯,unt+h1uia.e.0,T,ui0=uiT=0,where Lipschitz nonlinearity order of p−1 has been used.Most recently, in [26], the authors proved the existence of infinitely multiple solutions of the following perturbed nonlinear fractional p-Laplacian differential systems:
(7)tDTαΦp0Dtαut=λFut,ut,υt+h1u1,a.e.t∈0,T,tDTβΦp0Dtβυt=λFυt,ut,υt+h2u2,a.e.t∈0,T,u0=uT=0,υ0=υT=0,where one control parameter with the variational method has been used.Motivated by recently mentioned papers, the main contribution of this article is to use two control parameters and variational method to study a class of a nonlinear perturbed fractional-orderp-Laplacian differential system which is defined in (6), where we can prove that the studied system admits sequences of weak different solutions, strongly converge to zero.


## 2. Preliminaries

In this section, we introduce some notations, lemmas that are required for the subsequential. Then, a variational framework is constructed; then, the critical point theory is applied to explore the existence of infinite solutions for the system given in (6).We denoteYX the class of all functionals ϕ:X⟶ℝ, where X is real Banach space which has the following properties.Ifwn is a sequence in X converge weakly to w∈X and limn⟶∞infϕwn≤ϕw, thus wn has a subsequence that strongly converge to w.As an example, suppose a uniformly convexX with S:0,+∞⟶ℝ is an increasing, continuous strictly function, then the functional w⟶Sw∈YX.Definition 1 (see [4]).
Letu be a defined function on a,b. The left and right Riemann-Liouville fractional derivatives of order α>0, respectively, are given as
(8)aDtαut≔dndtnaDtα−nut=1Γn−αdndtn∫att−sn−α−1usds,tDbαut≔−1ndndtntDbα−nut=−1nΓn−αdndtn∫tbt−sn−α−1usds,where the right-hand sides are pointwise defined over a,b, ∀t∈a,b, n−1≤α<n and n∈ℕ.The gamma function,Γα, is given by
(9)Γα≔∫0+∞zα−1e−zdz.SettingACna,b,ℝ the space of functions u:a,b⟶ℝ, where
(10)u∈Cn−1a,b,ℝ,un−1∈ACa,b,ℝ.As familiar, we denoteCn−1a,b,ℝ the mappings set indicates n−1-times continuously differentiable on a,b.Actually, we imply(11)ACa,b,ℝ≔AC1a,b,ℝ.Definition 2 (see [22]).
Let0<αi≤1 (1≤i≤n, 1<p<∞), we introduce the space of the fractional-order derivative as follows:
(12)Eαip=ut∈Lp0,T,ℝ0Dtαiut∈Lp0,T,ℝ,u0=uT=0,then, the norm of Eαip can be defined ∀u∈Eαip, as the following
(13)uαi=∫0Tutpdt+∫0T0Dtαiutpdt1/p.Lemma 3 (see [3]).
Let0<αI≤11≤i≤n,1<p<∞. For all ui∈Eαip, we have
(14)uiLp≤TαiΓαi+10DtαiuiLp.Also, ifαi>p and 1/p+1/q=1, then
(15)ui∞≤Tαi−1/pΓαiαi−1q+11/q0DtαiuiLp.Hence, the operatorEαip according to the norm can be considered
(16)uiαi=∫0T0Dtαiutpdt1/p,∀ui∈Eαip,for1≤i≤n, that is equivalent to (13).Definition 4.
Suppose a Cartesian productX of n spaces Eαp; that is to say,
(17)X=Eα1p×Eα2p×⋯×Eαnp,provided with the norm
(18)uX=∑i=1nuiαi,u=u1,u2,⋯,un∈X,1≤i≤n,where uiαi is defined in (16).Clearly,X is embedded compact in C00,T,ℝn.Lemma 5 (see [23]).
For0<αi≤11≤i≤n and 1<p<∞. The space of the fractional-order derivative X is a reflexive separable Banach space.Lemma 6 (see [16]).
Assume that1/p<αi≤1 and un be the sequence that weakly converges to u in Eαp, i.e., un⇀u. Then, un have a strong convergence to u in C0,T,ℝ, i.e., un−u∞⟶0, as n⟶+∞.Definition 7 (see [3]).
We indicate a weak solution for system (6), any u=u1,u2,⋯,un∈X such that
(19)∫0T∑i=1nΦp0Dtαiut0Dtαivitdt−λ∑i=1n∫0TFuit,u1t,u2t,⋯,untvitdt−μ∫0T∑i=1nGuit,u1t,u2t,⋯,untvitdt−∫0T∑i=1nhiuitvitdt=0,∀vi=v1,v2,⋯,vn∈X.We define(20)Hix=∫0xhizdz,Θix=∫0THixsds,i=1,2,∀t∈0,T,&x∈ℝ.Lemma 8.
Suppose thathi:ℝ⟶ℝ satisfy (3) and Hix,Θix,1≤i≤n defined through (20). Then, Θu:X⟶ℝ be the functional that is described by
(21)Θu=Θu1,u2,⋯,un=∑i=1nΘiui=∑i=1n∫0THiuitdtis a weakly continuous sequentially Gâteaux differentiable functional on X together with a compact derivative
(22)Θ′u1,u2,⋯,unv1,v2,⋯,vn=∑i=1n∫0Thiuitvitdt,for every u=u1,u2,⋯,un,v=v1,v2,⋯,vn∈X.Proof.
Assume thatuk=u1k,u2k,⋯,unk⊂X,1≤k≤n,uk=u1k,u2k,⋯,unk⇀u=u1,u2,⋯,u in X as k⟶+∞. It follows from Lemma 8 that uk=u1k,u2k,⋯,unk converges uniformly to u=u1,u2,⋯,u on 0,T. Therefore, there exist constants ci>0, 1≤i≤n such that uik∞≤ci,1≤i≤n.
Then,(23)Hiuikt−Hiut≤Li∫utuiktsp−1ds≤Lipuiktp+utp≤Lipcip+ut∞p,for any n∈ℕ, & t∈0,T. Moreover, Hiuikt⟶Hiuit,1≤i≤n,∀t∈0,T, and via the convergence theorem in the Lebesgue sense, we have
(24)Θu1k,u2k,⋯,unk=∑i=1n∫0THiuiktdt⟶∑i=1n∫0THiuitdt,for1≤i≤n,=Θu1,u2,⋯,un=Θu.In the following, a Gâteaux differentiability ofΘ will be implemented. Let u1,x∈Eα1p and s≠0, then we claim
(25)Θ1u1+sx−Θ1us−∫0Th1u1txtdt≤∫0TH1u1+sx−H1u1s−h1u1txtdt=∫0Th1u1t+sζtxt−h1u1txtdt≤L1x∞ps,where ∀t∈0,T, 0<ζt<1. Hence, Θ1:Eα1p⟶ℝ is a Gâteaux differentiable at any u1∈Eα1p.Analogously, we have thatΘ2:Eα2p⟶ℝ at any u2∈Eα2p is a differentiable in Gâteaux sense.Therefore,Θ:X⟶ℝ to each u=u1,u2,⋯,un∈X with derivative is differentiable with a Gâteaux description
(26)Θ′u1,u2,⋯,unv1,v2,⋯,vn=∑i=1n∫0Thiuitvitdt,∀v=v1,v2,⋯,vn∈X.For any three elementsu1,u2,⋯,un,v1,v2,⋯,vn, and w1,w2,⋯,wn of X, it is easy to see that
(27)Θ′u1,u2,⋯,un−Θ′v1,v2,⋯,vnw1,w2,⋯,wn=∑i=1n∫0Thiui−hiviwitdt≤∑i=1nLi∫0Tui−vip−1witdt≤∑i=1nLiTαi−1/pΓαiαi−1q+11/qui−vi∞p−1wiα,which implies(28)Θ′u1,u2,⋯,un−Θ′v1,v2,⋯,vnX≤T∗∑i=1nui−vi∞p−1wiα,where(29)T∗≔maxLiTαi−1/pΓαiαi−1q+11/q,1≤i≤n.Consequently, the operatorΘ′:X⟶X∗ is compact.Below, we recall Theorem 2.5 of [23] which is an essential tool in our paper.Lemma 9 (see [23], Theorem 2.5).
Let a real reflexive Banach spaceX. Also, suppose two Gâteaux differentiable functionals ϕ,Ψ:X⟶ℝ such that ϕ is sequentially weakly lower semicontinuous, strongly continuous, and coercive where sequentially weakly upper semicontinuous achieved for Ψ. ∀r>infXϕ, put
(30)φr=infu∈ϕ−1−∞,rsupv∈ϕ−1−∞,rΨv−Ψur−ϕu,(31)γ≕limr⟶+∞infφr,δ≔limr⟶infXϕ+infφr.Then,(a)
For everyr>infXϕ & λ∈0,1/φr, the functional constraint of Iλ=ϕ−λΨ to ϕ−1−∞,r allows a global minimum, which is a critical point (local minimum) for Iλ in the space X(b)
Ifγ<+∞ & λ∈0,1/γ, the subsequent alternative holds:(b1)The functional ϕ−λΨ has a global minimum or(b2)∃ a sequence un of critical points (local minima) for Iλ such that limn⟶+∞ϕun=+∞(c)
Ifδ<+∞ & λ∈0,1/δ, the next alternative exists:(c1)∃ a global minimum of ϕ that is a local minimum of Iλ or(c2)∃ a sequence un of pairwise disjoint critical points (local minima) for Iλ that one converges weakly to a global minimum of ϕ together with limn⟶+∞ϕun=infXϕ


## 3. Principle Results

This section deals with stating and proving our main results. For assistance, suggest(32)k′≔min1≤i≤n1−LiTpαiΓαi+1p,ρ≔max1≤i≤n1+LiTpαiΓαi+1p,k=max1≤i≤nTpαi−1Γαipαi−1q+1p/q.For a given constantθ∈1/p,0, set
(33)Pαi,θ=1pθTp∫0θTtp1−αidt+∫θT1−θTt1−αi−t−θT1−αipdt+∫1−θTTt1−αi−t−θT1−αi−t−1−θT1−αip,Δ≔min1≤i≤nPαi,θ,Δ′≔max1≤i≤nPαi,θ.For anyρ>0, we set
(34)Ωϱ=x=x1,x2,⋯,xn∈ℝn:∑i=1n1pxip≤ϱ.Theorem 10.
Let1/p<αi≤1 for 1≤i≤n. Suppose that there exists θ∈0,1/p such that
(h1)Ft,x1,⋯,xn≥0 for each t,x1,⋯,xn∈0,θT∪1−θT,T×ℝn
(h2)(35)limξ⟶+∞inf∫0Tsupx1,x2,⋯,xn∈ΩξFt,x1,⋯,xndtξp<k′pkpnpρΔ′limξ⟶+∞sup∫θ1−θTFt,Γ2−α1ξ,⋯,Γ2−αnξdtξp.
Then, for eachλ∈Λ≔λ1,λ2 where
(36)λ1≔ρΔ′limξ⟶+∞sup∫θ1−θTFt,Γ2−α1ξ,⋯,Γ2−αnξdt/ξp,λ2≔k′/pkpnplimξ⟶+∞inf∫0Tsupx1,x2,⋯,xn∈ΩξFt,x1,⋯,xndt/ξp,for each nonnegative function G:0,T×ℝn⟶ℝ achieving the constrain
(37)G∞≔limξ⟶+∞sup∫0Tsupx1,x2,⋯,xn∈ΩξGt,x1,⋯,xndtξp<+∞,and for every μ∈0,μG,λ where
(38)μG,λ≔k′pkpnpG∞1−λpkpnpk′limξ⟶+∞inf∫0Tsupx1,x2,⋯,xn∈ΩξFt,x1,⋯,xndtξp,system (1) has an unbounded sequence of weak solutions in space X.Proof.
The main aim here is applying Lemma6 (see [16]) over system (1). For this purpose, fix λ∈Λ, and let G be a function that satisfies our hypotheses. Since λ¯<λ2, we claim
(39)μG,λ¯=k′pkpnpG∞1−λ¯pkpnpk′limξ⟶+inf∞∫0Tsupx1,x2,⋯,xn∈ΩξFt,x1,⋯,xndtξp>0.Now, fixμ¯∈0,μG,λ¯. Set
(40)Jt,ξ1,⋯,ξn≔Ft,ξ1,⋯,ξn+μ¯λ¯Gt,ξ1,⋯,ξn,for everyt∈0,T and ξ=ξ1,⋯,ξn∈ℝn. We construct the mappings ϕ,Ψ:X⟶ℝ as follows:
(41)ϕu≔∑i=1nuiαipp−Θu,Ψu≔∫0TJt,u1t,⋯,untdt,∀u=u1,⋯,un∈X and determine
(42)Iλ¯,μ¯u≔ϕu−λΨu,u∈X.Let us prove thatϕ & Ψ satisfy the required constrains. Since X is compactly embedded in C0,T,ℝn, it is well known that Ψ is well-defined Gateaux differentiable functional whose Gateaux derivative at u∈X is the functional Ψ′, given by
(43)Ψ′uv=∫0T∑i=1nJt,u1t,⋯,untvtdt,∀v=v1,⋯,vn∈X. Moreover, Ψ is sequentially weakly continuous.The functionalΦ is a Gateaux differentiable functional with the differential at u∈X,
(44)ϕ′uv=∫0T∑i=1n0Dtαiuitp−20Dtαiuit.0Dtαivitdt−∫0T∑i=1nhiuitvitdt,for everyv∈X. Moreover, ϕ is sequentially weakly lower semicontinuous, strongly continuous, and coercive functional on X.Obviously, the weak solutions of system (1) are precisely the critical points of the functional Iλ¯≤μ¯. Furthermore, since (3) holds for every x1,⋯,xn∈ℝ and h10=⋯=hn0=0, one has ∣hix∣≤Lixp−1,1≤i≤n∀x∈ℝ. It obtained from (14) and (15) the following:
(45)ϕu≥∑i=1nuiαipp−∫0T∑i=1nHiuitdt≥∑i=1nuiαjpp−∑i=1nLip∫0Tuitpdt≥1p−LiTpαjpΓαi+1puiαip≥k′p∑i=1nuiαi′p,∀u∈X, and as a result for this, ϕ is coercive.Now, allow us to verify thatλ¯<1/γ. Assume ξk is a positive number sequence such that ξk⟶∞ as k⟶∞ and
(46)limk⟶+∞∫0Tsupx1,x2,⋯,xn∈ΩξkFt,x1,⋯,xndtξkp=limξ⟶+∞inf∫0Tsupx1,x2,⋯,xn∈ΩξFt,x1,⋯,xndtξ2.Putrk≔k′ξkp/pkpnp∀k∈ℕ. Since maxt∈0,T∣uit∣≤kuiαi for all ui∈E0αi0,T and 1≤i≤n, we have
(47)supt∈0,T∑i=1nuit∣p≤kp∑i=1nuiαip,for each u=u1,⋯,un∈X. So, from (45) and (47), we have
(48)ϕ−1−∞,rk≔u∈X:k′2∑i=1nuiαip<rk⊆u∈X:∑i=1nuixp≤pkpk′rkforeacht∈0,T⊆u∈X:∑i=1nuit≤ξkforeacht∈0,T.Consequently, taking into the description thatϕ0,⋯,0=Ψ0,⋯,0=0, for all k big enough, one has
(49)φrk=infu∈ϕ−1∞,rksupu∈ϕ−1−∞¯rkΨv−Ψurk−ϕu,supu∈ϕ−1−∞,rkΨvrk≤∫0Tsupx1′⋯′xn∈ΩξkJt,x1,⋯,xndtk′ξkp/pkpnp=∫0Tsupx1′⋯′xn∈ΩξkFt,x1,⋯,xn+μ¯/λ¯Gt,x1,⋯,xndtk′ξkp/pkpnp≤∫0Tsupx1′⋯′xn∈ΩξkFt,x1,⋯,xndtk′ξkp/pkpnp+μ¯λ¯∫0Tsupx1′⋯′xn∈ΩξkGt,x1,⋯,xndtk′ξkp/pkpnp.Moreover, from assumption (h2) and (37), one has
(50)limk⟶∞inf∫0Tsupx1,⋯,xn∈ΩξkFt,x1,⋯,xndtk′/pkpnpξkp+limk⟶∞μ¯λ¯∫0Tsupx1,⋯,xn∈ΩξkGt,x1,⋯,xndtk′ξkp/pkpnp<+∞,which implies
(51)limk⟶∞inf∫0Tsupx1,⋯;xn∈ΩξkJt,x1,⋯,xndtξkp<+∞.Therefore,(52)γ≤limk⟶∞infφrk≤pkpnpk′limk⟶∞inf∫0Tsupx1,⋯,xn∈ΩξkJt,x1,⋯,xndtξkp<+∞.The assumptionμ¯∈0,μG,λ¯ immediately yields λ¯<1/γ.The succeeding step is to confirm that for a fixedλ¯, the functional Iλ¯,μ¯ has no global minimum. Let us verify that Iλ¯,μ¯ is unbounded from below. Since
(53)1λ¯<1ρΔ′limξ⟶+∞sup∫θT1−θFt,Γ2−α1ξ,⋯,Γ2−αnξdtξp≤limξ⟶+∞sup∫θT1−θTJt,Γ2−α1ξ,⋯,Γ2−αnξdtξp,consider ηk is a real sequence and τ is a positive constant such that ηk⟶∞ as k⟶∞, and
(54)1λ¯<τ<1ρΔ′∫θT1−θTFt,Γ2−α1ηk′⋯′Γ2−αnηkdtηkp,∀k∈ℕ huge adequate. ∀k∈ℕ, and θ∈0,1/p define wk=w1k,⋯,wnk by setting
(55)ωikt≔Γ2−αiηkθTt,t∈0,θT,Γ2−αiηk,t∈θT,1−θT,Γ2−αiηkθTT−t,t∈1−θT,T,for 1≤i≤n. Clearly, ωik0=ωikT=0 and ωik∈Lp0,T for 1≤i≤n. A direct calculation shows that
(56)0Dtαiωikt=ηkθTt1−αi,t∈0,θT,ηkθTt1−αi−t−θT1−αi,t∈θT,1−θT,ηkθTt1−αi−t−θT1−αi−t−1−θT1−αi,t∈1−θT,T,for 1≤i≤n. Furthermore,
(57)∫0T0Dtαiωiktpdt=∫0θT+∫θT1−θT+∫1−θTT0Dtαiωiktpdt=ηkpθTp∫0θTtp1−αidt+∫θT1−θTt1−αi−t−θT1−αipdt+∫1−θTTt1−αi−t−θT1−αi−t−1−θT1−αipdt=pPα,θηkp,for 1≤i≤n. Thus, ωk∈X, and in particular,
(58)ωikαip=∫0T0Dtαiωiktpdt=pPα,θηkp.On the other hand, similar to (45), we have
(59)ϕωk=∑i=1nωikαipp−Θω≤ρp∑i=1nωikαip=ρ∑i=1nPiαi′θηkp≤ρΔ′ηkp.Bearing in mind assumption (A) and since G is nonnegative, then using Ψ definition, we conclude that
(60)Ψwk≥∫θT1−θTFt,Γ2−α1ηk,⋯,Γ2−αnηkdt.So, according to (54), (55), and (60),
(61)Iλ¯,μ¯wk≤ρΔ′ηkp−λ¯∫θT1−θTFt,Γ2−α1ηk,⋯,Γ2−αnηkdt<ρΔ′1−λτηkp,for everyk∈ℕ large enough. Hence, Iλ¯,μ¯ is unbounded from below and so has no global minimum. Therefore, applying Lemma 6 (b), we deduce that there is a sequence uk=u1k,⋯,unk⊂X of critical points of Iλ¯,μ¯ such that
(62)limk⟶∞u1k,⋯,unk=+∞.Here, the outcome is produced.Remark 11.
Under the conditions(63)limξ⟶+∞inf∫0Tsupx1,⋯,xn∈ΩξFt,x1,⋯,xndtξp=0,limξ⟶+∞sup∫θT1−θFt,Γ2−α1ξ,⋯,Γ2−αnξξp=+∞,from Theorem 10, we claimed ∀λ>0 and μ∈0,k′/pkpnpG∞ system (1) admits infinitely many weak solutions in the space X. Also, if G∞=0, then the result holds ∀λ>0 and μ≥0.Here, we point out the following conclusion of Theorem10 beside μ=0.Corollary 12.
Supposeθ∈0,1/p such that hypothesis (h1) holds. Assume
(B1)limξ⟶+∞inf∫supx1,⋯,xn∈ΩξFt,x1,⋯,xndt/ξp<k′/pkpnp
(B2)limξ⟶+∞sup∫θT1−θTFt,Γ2−α1ξ,⋯,Γ2−αnξdt/ξp>ρΔ′
Then, the system(64)tDTαiΦp0Dtαiuit=Fuit,u1t,⋯,unt+hiuia.e.t∈0,T,ui0=uiT=0,for 1≤i≤n, has an unbounded sequence of weak solutions in X.Follow the same steps of Theorem10 proving but alternatively of (b) of Lemma 6 applying conclusion (c), the following result will be obtained.Theorem 13.
Suppose that all of Theorem10 assumptions hold except for hypothesis (h2). Assume thatassumptions hold except for hypothesis (h2). Assume that
(65)limξ⟶0+inf∫supx1,⋯;xn∈ΩξFt,x1,⋯,xndtξP<k′pkpnpρΔ′limξ⟶0+sup∫θT1−θTFt,Γ2−α1ξ,⋯′Γ2−αnξdtξp.
Then, for eachλ∈λ3,λ4, where
(66)λ3≔ρΔ′limξ⟶0+sup∫θT1−θTFt,Γ2−α1ξ,⋯,Γ2−αnξdt/ξp,λ4≔k′/pkpnplimξ⟶0+inf∫0Tsupx1,⋯;xn∈ΩξFt,x1,⋯,xndt/ξp,for every nonnegative function G:0,T×ℝn⟶ℝ satisfying the condition
(67)G0≔limξ⟶0+sup∫0Tsupx1,⋯;xn∈ΩξGt,x1,⋯,xndtξp<+∞,
and for everyμ∈0,μG,λ′ where
(68)μG,λ′≔k′pkpnpG01−λpkpnpk′limξ⟶0+inf∫0Tsupx1,⋯;xn∈ΩξFt,x1,⋯,xndtξp,
a sequence of weak solutions for system (system (1) exists, and it strongly converges to zero in the space X.Proof.
Fixλ∈λ3,λ4 and let G be a function satisfying (67). Since λ<λ4, one has
(69)μG,λ¯′=k′2k2n2G01−λ¯2k2n2k′limξ⟶0+inf∫0Tsupx1,⋯;xn∈ΩξFt,x1,⋯,xndtξp>0.Fixμ¯∈0,μG,λ¯′ and put
(70)Jt,ξ1,⋯,ξn≔Ft,ξ1,⋯,ξn+μ¯λ¯Gt,ξ1,⋯,ξn,for every t∈0,T and ξ=ξ1,⋯,ξn∈ℝn. We take Φ,Ψ, and Iλ¯,μ¯ as Theorem 10 proof. We verify that λ<1/γ. For this, let ξk be a sequence of positive number such that ξk⟶0+ as k⟶∞ and
(71)limk⟶+∞∫0Tsupx1,⋯;xn∈ΩξKFt,x1,⋯,xndtξkp=limξ⟶0+inf∫0Tsupx1,⋯;xn∈ΩξFt,x1,⋯,xndtξp.Through the evidenceinfXΦ=0 in addition to δ definition, we claim δ=limr⟶0+infφr. Then, like in display (52) of Theorem 10 proof, δ<+∞ can be proven, and hence, λ¯<1/δ.Letλ¯ be fixed. We conclude that at zero, there is no local minimum for the functional Iλ¯,μ¯. For this purpose, a sequence of positive numbers ηk is supposed such that ηk⟶0+ when k⟶∞ and choosing τ>0 such that
(72)1λ¯<τ<1ρΔ′∫θT1−θTFt,Γ2−α1ηk′⋯′Γ2−αnηkdtηkp,for every large enough k∈ℕ. Assume a sequence ωk=ω1k,⋯,ωnk in the space X together with ωιk given in (55). Remark that λτ>1. Hence, like in (62) show, the following can be obtained:
(73)Iλ¯,μ¯ωk≤ρΔ′ηkp−λ¯∫θT1−θTFt,Γ2−α1ηk,⋯,Γ2−αnηkdt<1−λ¯τηkpρΔ′<0,∀ large enough k∈ℕ. Where Iλ¯,μ¯0=0, this means at the point zero there is no local minimum for the functional Iλ¯,μ¯.Therefore, part (c) of Lemma6 ensures that there exists a sequence uk=u1k,⋯,unk in the space X of critical points for Iλ¯,μ¯ that is convergent weakly to the point zero. According to the established truth X−⟶C0,T,ℝn is compact, we conclude that the critical points strongly converge to zero, and the proof is performed.Remark 14.
According to the conditions(74)limξ⟶0+inf∫0Tsupx1,⋯;xn∈ΩξFt,x1,⋯,xndtξp=0,limξ⟶0+sup∫θT1−θTFt,Γ2−α1ξ′⋯′Γ2−αnξdtξp=+∞,ensures that ∀λ>0 and μ∈0,k′/pkpnpG0 system 1,1 admits infinitely many weak solutions in the sapce X. Moreover, if G0=0, the conclusion exists ∀λ>0 and μ≥0.Now, the following example will be presented for illustrating the above result.Example 15.
Consider the system(75)tD10.75Φ30Dt0.75x1t=λFx1t,x1t,x2t+μGx1t,x1t,x2t+h1x1ta.e.t∈0,1,tD10.8Φ30Dt0.8x2t=λFx2t,x1t,x2t+μGx2t,x1t,x2t+h2x2ta.e.t∈0,1,x10=x20=x11=x21=0,where h1x1=1/4sinx12 and h2x2=1/9x22. Moreover, for all t,x1,x2∈0,1×ℝ2, let F:0,1×ℝ2⟶ℝ be defined as
(76)Ft,x1′x2≔0,forallt,x1,x2∈0,1×02,atx121−sinln∣x1∣+btx221−coslnx,forallt,x1,x2∈0,1×ℝ−02,where a,b:0,1⟶ℝ are nonnegative continuous functions. Let θ=1/4. We observe that
(77)limξ⟶0+inf∫01sup1/3x13+x23≤ξFt,x1′x2dtξ3=0,limξ⟶0+sup∫1/43/4Ft,Γ1.25ξ,Γ0.2ξdtξ3=+∞.Now, letG: 0,T×ℝ2⟶ℝ be a function defined by
(78)Gt,x1′x2=1−cosx1x2.By definition,G∈C1ℝ2 and
(79)limξ⟶0+sup∫01sup1/3x13+x23≤ξGt,x1′x2dtξ3=0<∞.All hypotheses of Remark14 are satisfied. Then, for all λ,μ∈0,+∞×0,+∞, system (75) admits a sequence of weak solutions which strongly converges to 0 in E0.753×E0.83.



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*Source: 5510387-2021-03-12.xml*