# Multiple Solutions for Ellipticpx,qx-Kirchhoff-Type Potential Systems in Unbounded Domains

Authors: Nabil Chems Eddine; Abderrahmane El Hachimi
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

In this paper, we establish the existence of at least three weak solutions for a parametric double eigenvalue quasi-linear ellipticpx,qx-Kirchhoff-type potential system. Our approach is based on a variational method, and a three critical point theorem is obtained by Bonano and Marano.


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## Body

## 1. Introduction

The aim of this paper is to show the existence of at least three weak solutions for the following class of nonlocal quasi-linear elliptic systems inℝN:(1)−M1LpuΔpxu−axupx−2u=λFux,u,v,inℝN,−M2LqvΔqxv−bxvqx−2v=λFvx,u,v,inℝN,where N≥2, p and q∈C∗ℝN≔r∈CℝN:1<r−=infx∈ℝNrx<rx<r+=supx∈ℝNrx<N,∀x∈ℝN, λ is a positive real parameter, and a, b∈L∞ℝN such that a≔essinfx∈ℝNax>0 and b≔essinfx∈ℝNbx>0. M1 and M2 are bounded continuous functions, F belongs to C1ℝN×ℝ2 and satisfies adequate growth assumptions, and Furespectively,Fv denotes the partial derivative of F with respect to u (respectively, v). Here, we denote Δpxu≔div∇upx−2∇u the so-called px-Laplacian operator, and for u∈C∗ℝN,(2)Lru≔∫ℝN1rx∇urx+axurxdx.System (1) is a generalization of the elliptic equation associated with the following Kirchhoff equation, introduced by Kirchhoff in [1]:(3)ρ∂2u∂t2−ρ0h+E2L∫0L∂u∂x2dx∂2u∂x2=0,where ρ, ρ0, E, and L are constants. This equation extends classical D’Alembert’s wave equation by considering the effects of the changes on the length of the strings during the vibrations. A distinguishing feature of equation (3) is that the equation contains a nonlocal coefficient ρ0/h+E/2L∫0L∂u/∂x2dx which depends on the average 1/2L∫0L∂u/∂x2dx, and hence, the equation is no longer a pointwise equation. The parameters in equation (3) have the following meanings: E is Young’s modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross section, and ρ0 is the initial tension.Thepx-Laplacian operator possesses more complicated nonlinearities than p-Laplacian operator mainly due to the fact that it is not homogeneous. The study of various mathematical problems involving variable exponents has received a strong rise of interest in recent years. We can, for example, refer to [2–12]. This great interest may be justified by their various physical applications. In fact, there are applications concerning nonlinear elasticity theory [13], electrorheological fluids [14, 15], stationary thermorheological viscous flows [16], and continuum mechanics [17]. It also has wide applications in different research fields, such as image processing model [18] and the mathematical description of the filtration process of an ideal barotropic gas through a porous medium [19].The existence and multiplicity of solutions for the elliptic systems involving thepx-Kirchhoff model have been studied by many authors, where the nonlinear source F has different mixed growth conditions. We refer the reader to see [20–22] and the references therein for an overview on this subject. In connection to our context, the author obtained in [23] the existence and multiplicity of solutions for the vector-valued elliptic system:(4)−M1∫ℝN1pxupxdiv∇upx−2∇u=∂F∂ux,u,v,inΩ,−M2∫ℝN1qxvqxdiv∇vqx−2∇v=∂F∂vx,u,v,inΩ,u=v=0,in∂Ω,where Ω is a bounded domain in ℝN, with smooth boundary ∂Ω, p and q∈C∗Ω=r∈CΩ:1<r−=infx∈Ωrx<rx<r+=supx∈Ωrx<N,∀x∈Ω, and M1t and M2t are continuous functions such that M1t=M2t. The author applies a direct variational approach and the theory of variable exponent Sobolev spaces.On the contrary, by using the mountain pass theorem, the authors in [24] showed the existence of nontrivial solutions for system (1) when p,q∈CℝN2N≥2, M1t and M2t are continuous functions such that M1t=M2t, ax=bx=0, λ=1, and F∈C1ℝN×ℝ2,ℝ verifies some mixed growth conditions.The goal of this work is to establish the existence of a definite interval in whichλ lies such that system (1) admits at least three weak solutions by applying the following very recent abstract critical point result of Bonanno and Marano [25], which is a more precise version of Theorem 3.2 of [26].Lemma 1 (see [25], Theorem 3.6).
LetX be a reflexive real Banach space; Φ:X⟶ℝ be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X∗; and Ψ:X⟶ℝ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that(5)Φ0=Ψ0=0.
Assume that there existe>0 and x¯∈X, with e<Φx¯, such that(a1)
supΦx≤eΨx/e<Ψx¯/Φx¯(a2)
For eachλ∈Λe≔]Φx¯/Ψx¯,e/supΦx≤eΨx[, the functional Φ−λΨ is coercive
Then, for eachλ∈Λe, the functional Φ−λΨ has at least three distinct critical points in X.The rest of the paper is organized as follows. Section2 contains some basic preliminary knowledge of the variable exponent spaces and some results that we shall use here. Finally, in Section 3, we state and establish our main result.


## 2. Preliminaries and Basic Notations

First, we introduce the definitions of Lebesgue–Sobolev spaces with variable exponents. The details can be found in [27–29]. Denote ℳℝN as the set of all measurable real functions on ℝN. Set(6)C+ℝN=p∈CℝN:infx∈ℝNpx>1.For anyp∈C+ℝN, we define(7)p−≔infx∈ℝNpx,p+≔supx∈ℝNpx.For anyp∈C+ℝN, we define the variable exponent Lebesgue space as(8)LpxℝN=u∈ℳℝN:∫ℝNuxpxdx<∞,endowed with the Luxemburg norm(9)upx≔|u|LpxℝN=infμ>0:∫ℝNuxμpxdx≤1.Leta∈MℝN be such that ax>0, for a.e x∈ℝN. Define the weighted variable exponent Lebesgue space LapxℝN:(10)LapxℝN=u∈ℳℝN:∫ℝNaxuxpxdx<∞,with the norm(11)upx,ax≔uLapxℝN=infμ>0:∫ℝNaxuxμpxdx≤1.From now on, we suppose thata∈L∞ℝN with a≔essinfx∈ℝNax>0. Then, obviously, Lapx is a Banach space (see [30] for details).On the contrary, the variable exponent Sobolev spaceW1,pxℝN is defined as follows:(12)W1,pxℝN=u∈LpxℝN:∇u∈LpxℝNand is endowed with the norm(13)u1,px≔uW1,pxℝN=upx+∇upx,∀u∈W1,pxℝN.Next, the weighted variable exponent Sobolev spaceWa1,pxℝN is defined as(14)Wa1,pxℝN≔u∈LapxℝN:∇u∈LapxℝNand is endowed with the norm(15)ua≔infμ>0:∫ℝN∇uxμpx+axuxμpxdx≤1,∀u∈Wa1,pxℝN.Note that·a and ·1,px are equivalent norms in Wa1,pxℝN. Moreover, when p−>1, it is well known that LpxℝN, W1,pxℝN, and Wa1,pxℝN are separable, reflexive, and uniformly convex Banach spaces.Now, we display some facts that we shall use later.Proposition 1 (see [27, 28]).
The conjugate space ofLpxΩ is Lp′xΩ, where(16)1px+1p′x=1.
Moreover, for anyu,v∈LpxΩ×Lp′xΩ, we have(17)∫Ωuvdx≤1p−+1p′−upxvp′x≤2upxvp′x.Proposition 2 (see [27, 28]).
Denoteρu≔∫ℝNupxdx, for all u∈LpxℝN. We have(18)minupxp−,upxp+≤ρu≤maxupxp−,upxp+,and the following implications are true:(i)
upx<1resp.=1,>1⟺ρu<1resp.=1,>1(ii)
upx>1⟹upxp−≤ρu≤upxp+(iii)
upx<1⟹upxp+≤ρu≤upxp−Denoteρau≔∫ℝN∇uxpx+axuxpxdx, for all u∈Wa1,pxℝN. From Proposition 2, we have(19)uap−≤ρau≤uap+,ifua≥1,(20)uap+≤ρau≤uap−,ifua≤1.Proposition 3 (see [31]).
Letpx and qx be measurable functions such that p∈L∞ℝN and 1≤px,qx<∞ almost everywhere in ℝN. If u∈LqxℝN, u≠0, then we have(21)upxqx≤1⟹upxqxp−≤upxqx≤upxqxp+,upxqx≥1⟹upxqxp+≤upxqx≤upxqxp−.
In particular, ifpx=p is constant, then(22)upqx=upqxp.For allx∈ℝN, denote(23)p∗x=NpxN−px,forpx<N,+∞,forpx≥N,the critical Sobolev exponent of px.Proposition 4 (see [27, 31]).
Letp∈C+0,1ℝN, the space of Lipschitz-continuous functions defined on ℝN. There exists a positive constant c such that(24)up∗x≤cua,∀u∈Wa1,pxℝN.Proposition 5 (see [27, 31]).
Assume thatp∈CℝN satisfies px>1 for each x∈ℝN. If q∈CℝN is such that 1<qx<p∗x, for each x∈ℝN, then there exists a continuous and compact embedding W1,pxℝN⟶LqxℝN.In the following, we shall use the product space(25)X≔Wa1,pxℝN×Wb1,qxℝN,equipped with the norm(26)u,v≔maxua,vb,∀u,v∈X,where ⋅a (respectively, ⋅b) is the norm in Wa1,pxℝN (respectively, Wb1,qxℝN) defined above. We denote X∗ as the dual space of X equipped with the usual dual norm.Definition 1.
u
,
v
∈
X is called a weak solution of system (1) if(27)M1Lpu∫ℝN∇uxpx−2∇u∇φ+axupx−2uφdx+M2Lqu∫ℝN∇vxqx−2∇v∇ψ+bxvqx−2vψdx−λ∫ℝNFux,u,vϕdx−λ∫ℝNFvx,u,vψdx=0,for all φ,ψ∈X, where Lru is defined in (2).
We denoteEλ as the energy functional associated with problem (1):(28)Eλ⋅≔Φ⋅−λΨ⋅,where Φ,Ψ:X⟶ℝ are defined as follows:(29)Φu,v=Φ1Lpu+Φ2Lqv,Ψu,v=∫RNFx,u,vdx,where(30)Φ1Lpu=M1^Lpu,Φ2Lqv=M2^Lqv.for any w=u,v in X, with(31)Mi^t≔∫0tMisds,for allt≥0,i=1,2.
Note that we have the following formula:(32)Fx,u,v=∫0u∂F∂sx,s,vds+∫0v∂F∂sx,0,sds+Fx,0,0.
It is well know thatEλ∈C1X,ℝ and that critical points of Eλ correspond to weak solutions of problem (1).


### 2.1. Hypotheses

In this paper, we use the following assumptions:(H1)
F∈C1ℝN×ℝ2,ℝ and Fx,0,0=0.(H2)
There exist positive functionsai and bi i=1,2 such that(33)∂F∂ux,u,v≤a1xuμ1−1+a2xvμ2−1,∂F∂vx,u,v≤b1xuν1−1+b2xvν2−1,where1<μ1,μ2,ν1,ν2<infpx,qx and px, qx>N/2, for all x∈ℝN, and the weight functions a1,b2 (respectively, a2,b1) belong to the generalized Lebesgue spaces LαiℝN (respectively, LβℝN), with(34)α1x=pxpx−1,α2x=qxqx−1,βx=p∗xq∗xp∗xq∗x−p∗x−q∗x.(H3)
Mi:ℝ+⟶ℝ are continuous and increasing functions such that 0<m0≤Mit≤m1, for all t≥0, i=1,2.(H4)
There existe>0 and w1,w2∈X such that the following conditions are satisfied:(C1)
m0/p+minw1ap−,w1ap++m0/q+minw1bq−,w1bq+>e,(C2)
1/e∫ℝNsupξ1,ξ2∈Kse/m0Fx,ξ1,ξ2dx<1/m1maxw1ap−,w1ap++maxw2bq−,w2bq+∫ℝNFx,w1,w2dx,where(35)Kt≔ξ1,ξ2∈ℝ2:minξ1p∗xp∗−,ξ1p∗xp∗++minξ2q∗xq∗−,ξ2q∗xq∗+≤t,(36)s=minp+mincpxp∗−,cpxp∗+,q+mincqxq∗−,cqxq∗+,with t>0 and cpx and cqx representing the constants defined in Proposition 4.


## 2.1. Hypotheses

In this paper, we use the following assumptions:(H1)
F∈C1ℝN×ℝ2,ℝ and Fx,0,0=0.(H2)
There exist positive functionsai and bi i=1,2 such that(33)∂F∂ux,u,v≤a1xuμ1−1+a2xvμ2−1,∂F∂vx,u,v≤b1xuν1−1+b2xvν2−1,where1<μ1,μ2,ν1,ν2<infpx,qx and px, qx>N/2, for all x∈ℝN, and the weight functions a1,b2 (respectively, a2,b1) belong to the generalized Lebesgue spaces LαiℝN (respectively, LβℝN), with(34)α1x=pxpx−1,α2x=qxqx−1,βx=p∗xq∗xp∗xq∗x−p∗x−q∗x.(H3)
Mi:ℝ+⟶ℝ are continuous and increasing functions such that 0<m0≤Mit≤m1, for all t≥0, i=1,2.(H4)
There existe>0 and w1,w2∈X such that the following conditions are satisfied:(C1)
m0/p+minw1ap−,w1ap++m0/q+minw1bq−,w1bq+>e,(C2)
1/e∫ℝNsupξ1,ξ2∈Kse/m0Fx,ξ1,ξ2dx<1/m1maxw1ap−,w1ap++maxw2bq−,w2bq+∫ℝNFx,w1,w2dx,where(35)Kt≔ξ1,ξ2∈ℝ2:minξ1p∗xp∗−,ξ1p∗xp∗++minξ2q∗xq∗−,ξ2q∗xq∗+≤t,(36)s=minp+mincpxp∗−,cpxp∗+,q+mincqxq∗−,cqxq∗+,with t>0 and cpx and cqx representing the constants defined in Proposition 4.


## 3. The Main Results

We will use the three critical point theorem obtained by Bonano and Marano together with the following lemmas to get our main results.Lemma 2.
The functionalΦ is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous, coercive whose Gâteaux derivative admits a continuous inverse on X∗.Proof.
It is well known that the functionalΦ is well defined and is continuously Gâteaux differentiable functional whose derivative at the point u,v∈X is the functional Φ′u,v given by(37)Φ′u,v,φ,ψ=Φ1′u,φ+Φ2′v,ψ,where(38)Φ1′u,φ=M1Lpu∫ℝN∇uxpx−2∇u∇φ+axupx−2uφdx,Φ2′v,ψ=M2Lqu∫ℝN∇vxqx−2∇v∇ψ+bxvqx−2vψdx,for every φ,ψ∈X and Lru is defined in (19).
Let us show thatΦ is coercive. By using (19) and (20), we have for all u,v∈X,(39)Φu,v=M1^Lpu+M2^Lqv≥m0p+∫ℝN∇uxpxdx+axuxpx+m0q+∫ℝN∇vxqxdx+bxvxqxdx≥m0p+minuap−,uap++m0q+minvbq−,vbq+.
This shows thatΦu,v⟶+∞ as u,v⟶+∞, that is, Φ is coercive on X.
Now, in order to show that the operatorΦ′:X⟶X∗ is strictly monotone, it suffices to prove that Φ is strictly convex.
Forr∈C∗ℝN, the functional Lr:Wa1,rxℝN⟶ℝ defined in (2) is clearly a Gâteaux derivative at any u∈Wa1,rxℝN, and his derivative is given by(40)Lr′u,φ=∫ℝN∇uxrx−2∇u∇φ+axurx−2uφdx,for all φ∈Wa1,rxℝN.
Taking into account the inequality (see, e.g., Chapter I in [32]) for γ>1, there exists a positive constant Cγ such that(41)αγ−2α−βγ−2β,α−β≥Cγα−βγ,ifγ≥2,Cγα−β2α+β2−γ,α,β≠0,0,if1<γ<2,for any α,β∈ℝN. Therefore, we have(42)Lp′u1−Lp′u2,u1−u2>0,for all u1≠u2∈Wa1,pxℝN which means that Lp′ is strictly monotone. So, by ([33], Proposition 25.10), Lp is strictly convex. Moreover, since the Kirchhoff function M1 is nondecreasing, M1^ is convex in 0,+∞. Thus, for every u1,u2∈Wa1,pxℝN with u1≠u2 and every ]s,t∈0,1[ with s+t=1, one has(43)M1^Lsu1+tu2<M1^sLu1+tLu2≤sM1^Lu1+tM1^Lu2.
This shows thatΦ1 is strictly convex in Wa1,pxℝN. Similarly, we have that Φ2 is strictly convex in Wb1,qxℝN. Hence, Φ is strictly convex in X, and so Φ′=Φ1′+Φ2′ is strictly monotone.
It is clear thatΦ′ is an injection since Φ′ is a strictly monotone operator in X. Moreover, since we have(44)limu,v⟶+∞Φ′u,v,u,vu,v≥limu,v⟶+∞m0∫ℝN∇upx+axupxdx+∫ℝN∇vqx+bxvqxdxu,v=+∞,then, we deduce that Φ′ is coercive (see (19)). Thus, Φ′ is a surjection. Now, since Φ′ is hemicontinuous in X, then by applying (Proposition 4.2, [22]), we conclude that Φ′ admits a continuous inverse on X∗. Moreover, the monotonicity of Φ′ on X∗ ensures that Φ is sequentially lower semicontinuously on X (see [33], Proposition 25. 20). The proof of the lemma is complete.Lemma 3 (see [8]).
Under assumptionsH1 and H2, the functional Ψ is well defined and is of class C1 on X. Moreover, its derivative is given by(45)Ψ1u,vφ,ψ=∫ℝN∂F∂ux,u,vφ+∂F∂vx,u,vψdx,∀u,v,φ,ψ∈X.
Moreover,Ψ1 is compact from X to X∗.Theorem 1.
Under assumptionsH1−H4, system (1) admits at least three distinct weak solutions in X for each(46)λ∈m1maxw1ap−,w1ap++maxw2bq−,w2bq+∫ℝNFx,w1x,w1xdx,e∫ℝNsupξ1,ξ2∈Kse/m0Fx,ξ1,ξ2dx.Proof.
By Lemma2, Φ is coercive, and by the definitions of Φ and Ψ and from hypothesis H1, we have Φ0,0=Ψ0,0=0. Moreover, the required hypothesis Φx¯>e follows from condition C1 and the definition of Φ by choosing x¯=w1,w2. On the contrary, by applying Proposition 4 for u,v∈X, we have(47)1sminup∗p−,up∗p++minvs∗q−,vq∗q+≤1p+minuap−,uap++1q+minvbq−,vbq+,with s=minp+mincpxp∗−,cpxp∗+,q+mincqxq∗−,cqxq∗+, defined in (36). Now, from (47), we obtain for e>0(48)Φ−1−∞,e=x=u,v∈X:Φu,v≤e⊆u,v∈X:m0p+minuap−,uap++m0q+minvbq−,vbq+≤e⊆u,v∈X:minup∗p−,up∗p++minvs∗q−,vq∗q+≤sem0=Ksem0,where K⋅ is defined in (35). Then,(49)supu,v∈Φ−1−∞,eΨu=supu,v∈Φ−1−∞,e∫ℝNFx,u,vdx≤∫ℝNsupξ1,ξ2∈Kse/m0Fx,ξ1,ξ2dx.
Therefore, from conditionC2, we have(50)supu,v∈Φ−1−∞,eΨu≤e∫ℝNFx,w1x,w1xdxm1maxw1ap−,w1ap++maxw2bq−,w2bq+≤eΨw1,w2Φw1,w2,from which condition a1 of Lemma 1 follows.
To show that the functionalEλ=Φ−λΨ is coercive, we use inequality (3.8). For all u,v∈X, we have in virtue of H1 and H2(51)Eλu,v=M1^Lpu+M2^Lqv−λ∫ℝNFx,ux,vxdx≥m0p+∫ℝN∇uxpxdx+axuxpx+m0q+∫ℝN∇vxqxdx+bxvxqxdx−∫ℝN∫0u∂F∂sx,s,vds+∫0v∂F∂sx,0,sds+Fx,0,0dx≥m0p+ρau+m0q+ρbv−∫ℝNa1xuμ1+a2xvμ2−1u+b2xvν2dx≥m0p+ρau+m0q+ρbv−a1α1xuμ1px+a2βxvμ2−1q∗xup∗x+b2α2xvν2qx.
Using Young’s inequality, we obtain(52)Eλu,v≥m0p+uap−+m0q+vbq−−λa1α1xupxμ1+a2βxμ2−1μ2vqxμ2+1μ2upxμ2+b2α2xvqxν2≥m0p+uap−+m0q+vbq−−ca1α1xupxμ1+a2βxvqxμ2+a2βxupxμ2+b2α2xvqxν2.
This shows thatΦ−λΨ⟶+∞ as u,vX⟶∞ since we have 1<μ1,μ2,ν1,ν2<infpx,qx, that is, Φ−λΨ is coercive on X, for every parameter λ, in particular, for every λ∈Λe≔]Φw1,w2/Ψw1,w2,e/supΦu,v≤eΨu,v[. Then, condition a2 in Lemma 1 also holds. Now, all the hypotheses of Lemma 1 are satisfied. Note that the solutions of the equation Φ′u,v−λΨ1u,v=0 are exactly the weak solutions of (1). Thus, for each(53)λ∈m1maxw1ap−,w1ap++maxw2bq−,w2bq+∫ℝNFx,w1x,w1xdx,e∫ℝNsupξ1,ξ2∈Kse/m0Fx,ξ1,ξ2dx,system (1) admits at least three weak solutions in X.



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