# Multiplicity of Solutions for Nonlocal Elliptic System of(p,q)-Kirchhoff Type

Authors: Bitao Cheng; Xian Wu; Jun Liu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff
type −[M1(∫Ω|∇u|p)]p−1Δpu=λFu(x,u,v), in Ω, −[M2(∫Ω|∇v|q)]q−1Δqv=λFv(x,u,v), in Ω, u=v=0, on ∂Ω. Under bounded condition on M and some novel and periodic condition on F, some new
results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by
means of Bonanno's multiple critical points theorems without the Palais-Smale condition
and Ricceri's three critical points theorem, respectively.


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## Body

## 1. Introduction and Preliminaries

We are concerned with the following nonlocal elliptic system of(p,q)-Kirchhoff type:
(1.1)-[M1(∫Ω|∇u|p)]p-1Δpu=λFu(x,u,v),inΩ,-[M2(∫Ω|∇v|q)]q-1Δqv=λFv(x,u,v),inΩ,u=v=0;on⁡∂Ω,
where  Ω⊂RN(N≥1) is a bounded smooth domain, λ∈(0,+∞), p>N,q>N, Δp is the p-Laplacian operator
(1.2)Δpu=div⁡(|∇u|p-2∇u),
and Mi:R+→R,i=1,2, are continuous functions with bounded conditions.(M)
There are two positive constantsm0,m1  such that
(1.3)m0≤Mi(t)≤m1,∀t≥0,i=1,2.
Furthermore, F:Ω×R×R→R is a function such that F(x,s,t) is measurable in x for all (s,t)∈R×R and F(x,s,t) is C1 in (s,t) for a.e. x∈Ω, and Fu denotes the partial derivative of F with respect to u. Moreover, F(x,s,t) satisfies the following.(F1)
F(x,0,0)=0 for a.e. x∈Ω.(F2)
There exist two positive constantsγ<p,β<q and a positive real function α(x)∈L∞(Ω) such that(1.4)|F(x,s,t)|≤α(x)(1+|s|γ+|t|β),fora.e.x∈Ωand  all(s,t)∈R×R.The system (1.1) is related to a model given by the equation of elastic strings (1.5)ρ∂2u∂t2-(P0h+E2L∫0L|∂u∂x|2dx)∂2u∂x2=0
which was proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where the parameters in (1.5) have the following meanings: ρ is the mass density, P0 is the initial tension, h is the area of the cross-section, E is the Young modulus of the material, and L is the length of the string. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.Later, (1.5) was developed to the form (1.6)utt-M(∫Ω|∇u|2)Δu=f(x,u)inΩ,
where M:R+→R is a given function. After that, many people studied the nonlocal elliptic boundary value problem (1.7)-M(∫Ω|∇u|2)Δu=f(x,u)inΩ,u=0on∂Ω,
which is the stationary counterpart of (1.6). It is pointed out in [2] that (1.7) models several physical and biological systems, where u describes a process which depends on the average of itself (e.g., population density). By using the methods of sub and supersolutions, variational methods, and other techniques, many results of (1.7) were obtained, we can refer to [2–12] and the references therein. In particular, Alves et al. [2, Theorem  4] supposes that M satisfies bounded condition (M) and f(x,t) satisfies condition AR, that is, for some ν>2 and R>0 such that (AR)0<νF(x,t)≤f(x,t)t,∀|t|≥R,x∈Ω,
where F(x,t)=∫0tf(x,s)ds; one positive solutions for (1.7) was obtained. It is well known that condition AR plays an important role for showing the boundedness of Palais-Smale sequences. More recently, Corrêa and Nascimento in [13] studied a nonlocal elliptic system of p-Kirchhoff type (P)-[M1(∫Ω|∇u|p)]p-1Δpu=f(u,v)+ρ1(x),inΩ,-[M2(∫Ω|∇v|p)]p-1Δpv=g(u,v)+ρ2(x),inΩ,∂u∂η=∂v∂η=0,on∂Ω,
where η is the unit exterior vector on ∂Ω, and Mi,ρi(i=1,2),f,g satisfy suitable assumptions, where a special and important condition: periodic condition on nonlinearity was assumed. They obtained the existence of a weak solution for the nonlocal elliptic system of p-Kirchhoff type P under Neumann boundary condition via Ekeland's Variational Principle.In the present paper, our objective is to consider the nonlocal elliptic system of(p,q)-Kirchhoff-type (1.1), instead of the nonlocal elliptic system of p-Kirchhoff type and single Kirchhoff type equation. Under bounded condition on M and some novel conditions without PS condition and periodic condition on F, we will prove the existence of two solutions and three solutions of system (1.1) by means of one multiple critical points theorem without the Palais-Smale condition of Bonanno in [14] and an equivalent formulation [15, Theorem  2.3] of Ricceri's three critical points theorem [16, Theorem  1], respectively.In order to state our main results, we need the following preliminaries.LetX=W01,p(Ω)×W01,q(Ω) be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm (1.8)‖(u,v)‖=‖u‖p+‖v‖q,
where ∥·∥p and ∥·∥q denote the norms of W01,p(Ω) and W01,q(Ω), respectively. That is, (1.9)‖u‖p=(∫Ω|∇u|p)1/p,‖v‖q=(∫Ω|∇v|q)1/q
for all u∈W01,p(Ω) and v∈W01,q(Ω).Sincep>N and q>N, W01,p(Ω) and W01,q(Ω) are compactly embedded in C0(Ω̅). Let (1.10)C=max⁡{sup⁡u∈W01,p(Ω)∖{0}max⁡x∈Ω̅⁡{|u(x)|p}‖u‖pp,sup⁡v∈W01,q(Ω)∖{0}max⁡x∈Ω̅⁡{|v(x)|q}‖v‖qq};
then we have C<+∞. Furthermore, it is known from [17] that (1.11)sup⁡u∈W01,p(Ω)∖{0}max⁡x∈Ω̅⁡{|u(x)|p}‖u‖p≤N-1/Pπ(Γ(1+N2))1/N(p-1p-N)1-1/P|Ω|(1/N)-(1/P),
where Γ denotes the Gamma function and |Ω| is the Lebesgue measure of Ω. Additionally, (1.11) is an equality when Ω is a ball.Recall that(u,v)∈X is called a weak solution of system (1.1) if (1.12)[M1(∫Ω|∇u|p)]p-1∫Ω|∇u|p-2∇u∇φ+[M2(∫Ω|∇v|q)]q-1∫Ω|∇v|q-2∇v∇ψ-λ∫ΩFu(x,u,v)φ(x)dx-λ∫ΩFv(x,u,v)ψ(x)dx=0,
for all (φ,ψ)∈X. Define the functional I:X→R given by (1.13)I(u,v)=1pM̂1(∫Ω|∇u|p)+1qM̂2(∫Ω|∇v|q)-λ∫ΩF(x,u,v)dx
for all (u,v)∈X, and where (1.14)M̂1(t)=∫0t[M1(s)]p-1ds,M̂2(t)=∫0t[M2(s)]q-1ds,∀t≥0.
By the conditions (M) and (F2), it is easy to see that I∈C1(X,R) and a critical point of I corresponds to a weak solution of the system (1.1).Now, givingx0∈Ω and choosing R2>R1>0 such that B(x0,R2)⊆Ω, where B(x,R)={y∈RN:|y-x|<R}. Next we give some notations.(1.15)α1=α1(N,p,R1,R2)=C1/P(R2N-R1N)1/PR2-R1(πN/2Γ(1+N/2))1/P,α2=α2(N,q,R1,R2)=C1/q(R2N-R1N)1/qR2-R1(πN/2Γ(1+N/2))1/q.
Moreover, let a,c be positive constants, denote(1.16)y(x)=aR2-R1(R2-{∑i=1N(xi-x0i)2}1/2),∀x∈B(x0,R2)∖B(x0,R1),A(c)={(s,t)∈R×R:|s|p+|t|q≤c},g(c)=∫Ωsup⁡(s,t)∈A(c)F(x,s,t)dx,k(a)=∫B(x0,R2)∖B(x0,R1)F(x,y(x),y(x))dx+∫B(x0,R1)F(x,a,a)dx,h(c,a)=k(a)-g(c),M+=max⁡{m1p-1p,m1q-1q},M-=min⁡{m0p-1p,m0q-1q}.Now we are ready to state our main results for the system (1.1)Theorem 1.1.
Assume that(F1)-(F2) hold and there are three positive constants a,c1,c2 with c1<(aα1)p+(aα2)q≤1<c2 such that
(1.17)M+g(c1)<M-h(c1,a),M+g(c2)<M-h(c1,a).
Then, for each (1.18)λ∈(M+[(aα1)p+(aα2)q]Ch(c1,a),M-c1[(aα1)p+(aα2)q]Cmin⁡{1g(c1),1g(c2)}),
there exists a positive real number ρ such that the system (1.1) has at least two weak solutions (ui,vi)∈X(i=1,2) whose norms in C0(Ω) are less than some positive constant ρ.Theorem 1.2.
Assume that(F1)-(F2) hold and there are two positive constants a,b, with (aα1)p+(aα2)q>bM+/M- such that
(F3)F(x,s,t)≥0 for a.e. x∈Ω∖B(x0,R1) and all (s,t)∈[0,a]×[0,a];
(F4)[(aα1)p+(aα2)q]|Ω|sup(x,s,t)∈Ω×A(bM+/M-)F(x,s,t)<b∫B(x0,R1)F(x,a,a)dx.
Then there exist an open intervalΛ⊆[0,+∞] and a positive real number ρ such that, for each λ∈Λ, the system (1.1) has at least three weak solutions wi=(ui,vi)∈X(i=1,2,3) whose norms ∥wi∥ are less than ρ.


## 2. Proofs of Main Results

Before proving the results, we state one multiple critical points theorem without the Palais-Smale condition of Bonanno in [13] and an equivalent formulation [14, Theorem  2.3] of Ricceri's three critical points theorem [15, Theorem  1], which are our main tools.Theorem 2.1 (see [14, Theorem  2.1]).
LetX be a reflexive real Banach space, and let Ψ,Φ:X→R be two sequentially weakly lower semicontinuous functions. Assume that Ψ is (strongly) continuous and satisfies lim∥u∥→∞Ψ(u)=+∞. Assume also that there exist two constants r1 and r2 such that(j)
infXΨ<r1<r2;(jj)
φ1(r1)<φ2(r1,r2);(jjj)
φ1(r2)<φ2(r1,r2);where
(2.1)φ1(r)=inf⁡u∈Ψ-1(-∞,r)Φ(u)-inf⁡u∈Ψ-1(-∞,r)w¯⁡Φ(u)r-Ψ(u),φ2(r1,r2)=inf⁡u∈Ψ-1(-∞,r1)sup⁡v∈Ψ-1[r1,r2)Φ(u)-Φ(v)Ψ(v)-Ψ(u).
Then, for each
(2.2)λ∈(1φ2(r1,r2),min⁡{1φ1(r1),1φ2(r2)}),
the functional Ψ+λΦ has two local minima which lie in Ψ-1(-∞,r1) and Ψ-1[r1,r2), respectively.Theorem 2.2 (see [15, Theorem  2.3]).
LetX be a separable and reflexive real Banach space. Ψ:X→R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X*; Φ:X→R is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Suppose that(i)
lim∥u∥→∞(Ψ(u)+λΦ(u))=+∞ for each λ>0;(ii)
There are a real numberr, and u0,u1∈X such that Ψ(u0)<r<Ψ(u1);(iii)
infu∈Ψ-1(-∞,r]Φ(u)>((Ψ(u1)-r)Φ(u0)+(r-Ψ(u0))Φ(u1))/(Ψ(u1)-Ψ(u0)).
Then there exist an open intervalΛ⊆[0,+∞] and a positive real number ρ such that, for each λ∈Λ, the equation Ψ′(u)+λΦ′(u)=0 has at least three weak solutions whose norms in X are less than ρ.First, we give one basic lemma.Lemma 2.3.
Assume that(M) and (F2) hold; let
(2.3)Ψ(u,v)=1pM̂1(∫Ω|∇u|p)+1qM̂2(∫Ω|∇v|q),Φ(u,v)=-∫ΩF(x,u,v)dx,
for all (u,v)∈X. Then Ψ and Φ are continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the Gâteaux derivative of Ψ admits a continuous inverse on X* and the Gâteaux derivative of Φ is compact.Proof.
By condition(M), it is easy to see that Ψ is continuously Gâteaux differentiable. Moreover, the Gâteaux derivative of Ψ admits a continuous inverse on X*. Thanks to  p>N,q>N, and (F2), Φ is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative is compact. Next We will prove that Ψ is a sequentially weakly lower semicontinuous functional. Indeed, for any (un,vn)∈X with (un,vn)⇀(u,v) in X, then un⇀u in W01,p(Ω) and vn⇀v in W01,q(Ω). Therefore,
(2.4)liminf⁡n→∞‖un‖p≥‖u‖p,liminf⁡n→∞‖vn‖q≥‖v‖q
due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of M̂1 and M̂2, we conclude that
(2.5)M̂1(∫Ω|∇u|p)≤M̂1(liminf⁡n→∞∫Ω|∇un|p)≤liminf⁡n→∞M̂1(∫Ω|∇un|p),M̂2(∫Ω|∇v|q)≤M̂2(liminf⁡n→∞∫Ω|∇vn|q)≤liminf⁡n→∞M̂2(∫Ω|∇vn|q),
Consequently, Ψ is a sequentially weakly lower semicontinuous functional.Proof of Theorem1.1.
Let(2.6)Ψ(u,v)=1pM̂1(∫Ω|∇u|p)+1qM̂2(∫Ω|∇v|q),Φ(u,v)=-∫ΩF(x,u,v)dx
for all (u,v)∈X. Under condition (M), by a simple computation, we have
(2.7)M-(‖u‖pp+‖v‖qq)≤Ψ(u,v)≤M+(‖u‖pp+‖v‖qq).
Therefore, (2.7) implies that
(2.8)lim⁡∥(u,v)∥→+∞Ψ(u,v)=+∞.
Put(2.9)r1=M-c1C[(aα1)p+(aα2)q],r2=M-c2C[(aα1)p+(aα2)q].
Denote(2.10)φ1(r)=inf⁡(u,v)∈Ψ-1(-∞,r)Φ(u,v)-inf⁡(u,v)∈Ψ-1(-∞,r)w¯⁡Φ(u,v)r-Ψ(u,v),φ2(r1,r2)=inf⁡(u,v)∈Ψ-1(-∞,r1)sup⁡(u1,v1)∈Ψ-1[r1,r2)Φ(u,v)-Φ(u1,v1)Ψ(u1,v1)-Ψ(u,v),
and Ψ-1(-∞,r)w¯ is the closure of Ψ-1(-∞,r) in the weak topology.
Set(2.11)w0(x)={0,x∈Ω̅∖B(x0,R2),aR2-R1(R2-{∑i=1N(xi-x0i)}1/2),x∈B(x0,R2)∖B(x0,R1),a,x∈B(x0,R1).
Then (u0,v0)∈X, where u0(x)=v0(x)=w0(x) and
(2.12)‖u0‖pp=‖w0‖pp=(aα1)pC,‖v0‖qq=‖w0‖qq=(aα1)qC.
Consequently, (2.7) and (2.12) imply that
(2.13)r1<Ψ(u0,v0)<r2.
Furthermore, (2.13) implies that
(2.14)φ2(r1,r2)=inf⁡(u,v)∈Ψ-1(-∞,r1)sup⁡(u1,v1)∈Ψ-1[r1,r2)Φ(u,v)-Φ(u1,v1)Ψ(u1,v1)-Ψ(u,v)≥inf⁡(u,v)∈Ψ-1(-∞,r1)Φ(u,v)-Φ(u0,v0)Ψ(u0,v0)-Ψ(u,v).
On the other hand, by (F1), (1.17), and, (2.11), one has
(2.15)∫ΩF(x,u0,v0)dx=k(a)>h(c1,a)>M+M-g(c1)>g(c1)=∫Ωsup⁡(s,t)∈A(c1)F(x,s,t)dx.
For each (u,v)∈X with Ψ(u,v)≤r1, and x∈Ω, by (2.7), we conclude
(2.16)|u(x)|p+|v(x)|q≤C(‖u‖pp+‖v‖qq)≤Cr1M-=c1[(aα1)p+(aα2)q]≤c1.
Therefore, the combination of (2.15) and (2.16) implies
(2.17)Φ(u,v)-Φ(u0,v0)Ψ(u0,v0)-Ψ(u,v)=∫ΩF(x,u0,v0)dx-∫ΩF(x,u,v)dxΨ(u0,v0)-Ψ(u,v)≥∫ΩF(x,u0,v0)dx-∫Ωsup⁡|u(x)|p+|v(x)|q≤c1⁡F(x,u,v)dxΨ(u0,v0)-Ψ(u,v)≥∫ΩF(x,u0,v0)dx-∫Ωsup⁡|u(x)|p+|v(x)|q≤c1⁡F(x,u,v)dxΨ(u0,v0)≥∫ΩF(x,u0,v0)dx-∫Ωsup⁡|u(x)|p+|v(x)|q≤c1⁡F(x,u,v)dxM+(‖u0‖pp+‖v0‖qq)=CM+[(aα1)p+(aα2)q]h(c1,a).
By (2.14) and (2.17), we have
(2.18)φ2(r1,r2)≥CM+[(aα1)p+(aα2)q]h(c1,a).
Similarly, for every (u,v)∈X such that Ψ(u,v)≤r, where r is a positive real number, and x∈Ω, one has
(2.19)|u(x)|p+|v(x)|q≤C(‖u‖pp+‖v‖qq)≤CrM-.
By virtue of Ψ being sequentially weakly lower semicontinuous, then Ψ-1(-∞,r)w¯=Ψ-1(-∞,r]. Consequently,
(2.20)φ1(r)=inf⁡(u,v)∈Ψ-1(-∞,r)Φ(u,v)-inf⁡(u,v)∈Ψ-1(-∞,r)w¯⁡Φ(u,v)r-Ψ(u,v)≤Φ(0,0)-inf⁡(u,v)∈Ψ-1(-∞,r)w¯⁡Φ(u,v)r-Ψ(0,0)≤-inf⁡(u,v)∈Ψ-1(-∞,r)w¯⁡Φ(u,v)r≤∫Ωsup⁡|u(x)|p+|v(x)|q≤Cr/M-F(x,u,v)dxr.
It implies that
(2.21)φ1(r1)≤g(c1)r1=CM-c1[(aα1)p+(aα2)q]g(c1)<CM+[(aα1)p+(aα2)q]h(c1,a),(2.22)φ1(r2)≤g(c2)r2=CM-c2[(aα1)p+(aα2)q]g(c2)<CM+[(aα1)p+(aα2)q]h(c1,a).
By (2.18)–(2.22), we conclude
(2.23)φ1(r1)≤φ2(r1,r2),φ1(r2)≤φ2(r1,r2).
Therefore, the conditions (j), (jj), and (jjj) in Theorem 2.1 are satisfied. Consequently, by Lemma 2.3 and above facts, the functional Ψ+λΦ has two local minima (u1,v1),(u2,v2)∈X, which lie in Ψ-1(-∞,r1) and Ψ-1[r1,r2), respectively. Since I=Ψ+λΦ∈C1, (u1,v1),(u2,v2)∈X are the solutions of the equation
(2.24)Ψ′(u,v)+λΦ′(u,v)=0.
Then (u1,v1),(u2,v2)∈X are the weak solutions of system (1.1).
SinceΨ(ui,vi)<r2,i=1,2, by (1.10) and (2.7),
(2.25)|ui(x)|p+|vi(x)|q≤Cr2M-≤c2,i=1,2;
which implies there exists a positive real number ρ such that the norms of (ui,vi)∈X(i=1,2) in C0(Ω) are less than some positive constant ρ. This completes the proof of Theorem 1.1.Proof of Theorem1.2.
Let(2.26)Ψ(u,v)=1pM̂1(∫Ω|∇u|p)+1qM̂2(∫Ω|∇v|q),Φ(u,v)=-∫ΩF(x,u,v)dx
for all (u,v)∈X. By (F2) and (2.7), we have
(2.27)Ψ(u,v)+λΦ(u,v)=1pM̂1(∫Ω|∇u|p)+1qM̂2(∫Ω|∇v|q)-λ∫ΩF(x,u,v)dx≥M-(‖u‖pp+‖v‖qq)-λ∫Ωα(x)(1+|u(x)|γ+|v(x)|β)dx≥M-(‖u‖pp+‖v‖qq)-λ‖α‖∞(|Ω|+k1‖u‖pγ+k2‖v‖qβ),
where k1,k1 are positive constants. Since γ<p,β<q, (2.27) implies that
(2.28)lim⁡∥(u,v)∥→+∞(Ψ(u,v)+λΦ(u,v))=+∞.
The  same as in (2.11), defining a function w0(x), and letting u0(x)=v0(x)=w0(x), then (2.12) is also satisfied. Choosing r=bM+/C, by (2.7), (2.12), and (aα1)p+(aα2)q>bM+/M-, we conclude
(2.29)Ψ(u0,v0)≥M-(‖u0‖pp+‖v0‖qq)=M-C[(aα1)p+(aα2)q]>M-CbM+M-=r.
By (F3) and the definitions of u0 and v0, one has
(2.30)|Ω|sup⁡(x,s,t)∈Ω×A(bM+/M-)F(x,s,t)<b(aα1)p+(aα2)q∫B(x0,R1)F(x,a,a)dx=bM+C∫B(x0,R1)F(x,a,a)dxM+((aα1)p+(aα2)q)/C≤bM+C∫Ω∖B(x0,R1)F(x,u0,v0)dx+∫B(x0,R1)F(x,u0,v0)dxM+(‖u0‖pp+‖v0‖qq)≤bM+C∫ΩF(x,u0,v0)dxΨ(u0,v0).
For every (u,v)∈X such that Ψ(u,v)≤r, and x∈Ω, one has
(2.31)|u(x)|p+|v(x)|q≤C(‖u‖pp+‖v‖qq)≤CrM-=CM-bM+C=bM+M-.
By the combination of (2.30) and (2.31), we have
(2.32)sup⁡(u,v)∈Ψ-1(-∞,r)(-Φ(u,v))=sup⁡{(u,v)∣Ψ(u,v)≤r}∫ΩF(x,u,v)dx≤sup⁡{(u,v)||u(x)|p+|v(x)|q≤bM+/M-}∫ΩF(x,u,v)dx≤∫Ωsup⁡(s,t)∈A(bM+/M-)F(x,s,t)dx≤|Ω|sup⁡(x,s,t)∈Ω×A(bM+/M-)F(x,s,t)≤bM+C∫ΩF(x,u0,v0)dxΨ(u0,v0)=r-Φ(u0,v0)Ψ(u0,v0).
Therefore,
(2.33)inf⁡(u,v)∈Ψ-1(-∞,r)Φ(u,v)>rΦ(u0,v0)Ψ(u0,v0).
Note that Φ(0,0)=Ψ(0,0)=0, we conclude that
(2.34)inf⁡(u,v)∈Ψ-1(-∞,r]Φ(u,v)>(Ψ(u0,v0)-r)Φ(0,0)+(r-Ψ(0,0))Φ(u0,v0)Ψ(u0,v0)-Ψ(0,0).
Hence, by Lemma 2.3 and above facts, Ψ and Φ satisfy all conditions of Theorem 2.2; then the conclusion directly follows from Theorem 2.2.



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