# Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

Authors: Yuping Cao; Chuanzhi Bai
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.


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## Body

## 1. Introduction

This paper is concerned with the multiplicity of solutions to the following elliptic systems of Kirchhoff type involving the nonlocal integrodifferential operators:(1)-M1(∫R2n‍|u(x)-u(y)|2K1(x-y)dxdy)LK1u=f(x,v)inΩ,-M2(∫R2n‍|v(x)-v(y)|2K2(x-y)dxdy)LK2v=g(x,u)inΩ,u=v=0inRn∖Ω,
where Ω⊂Rn (n≥2) is a bounded domain with smooth boundary ∂Ω and f,g:Ω×R→R are two continuous functions. Mi:R+→R+ (i=1,2) are two continuous functions whose properties will be introduced later. LKi (i=1,2) are the nonlocal operators defined by(2)LKiu(x)=12∫Rn‍(u(x+y)+u(x-y)-2u(x))Ki(y)dy,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhx∈Rn,i=1,2; here Ki:Rn∖{0}→(0,+∞) is a function such that
(3)mKi∈L1(Rn),i=1,2,wherem(x)=min⁡{|x|2,1};
there exist θi and si∈(0,1) (i=1,2) such that
(4)Ki(x)≥θi|x|-(n+2si),foranyx∈Rn∖{0};(5)Ki(x)=Ki(-x),∀x∈Rn∖{0}.
A typical example for Ki is given by Ki(x)=|x|-(n+2si) (i=1,2). In this case LKi is the fractional Laplace operator -(-Δ)si, where -(-Δ)si is defined by
(6)-(-Δ)siu(x)=12∫Rn‍u(x+y)+u(x-y)-2u(x)|y|n+2sidy,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhx∈Rn;
here si∈(0,1) and n>2si (i=1,2). The fractional Laplacian -(-Δ)si is a classical linear integrodifferential operator of order 2si which gives the standard Laplacian when si=1 (see [1]).Denote byXi the linear space of Lebesgue measurable functions u:Rn→R such that
(7)themap(x,y)↣(u(x)-u(y))2Ki(x-y)isinL1(Q,dxdy),
where Q=(Rn×Rn)∖O and O=(CΩ)×(CΩ)⊂Rn×Rn. The space Xi is endowed with the norm
(8)∥u∥Xi=∥u∥L2(Rn)+(∫Rn×Rn‍|u(x)-u(y)|2Ki(x-y)dxdy)1/2,11111111111111111111111111111111i=1,2.The spaceZi denotes the closure of C0∞(Ω) in Xi. By Lemmas 6 and 7 in [2], the space Zi is a Hilbert space which can be endowed with the norm defined as
(9)∥u∥Zi=(∫Q‍|u(x)-u(y)|2Ki(x-y)dxdy)1/2,i=1,2.
Since u=0 a.e. in Rn∖Ω, we have that the integral in (8) and (9) can be extended to all R2n.LetE=Z1×Z2 be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm
(10)∥(u,v)∥=∥u∥Z1+∥v∥Z2.Denote by0<λ1<λ2≤⋯≤λk≤⋯ the eigenvalues of the following nonlocal operator eigenvalue problem:
(11)-LK1u=λuinΩu=0inRn∖Ω.
Similarly, denote by 0<μ1<μ2≤⋯≤μk≤⋯ the eigenvalues of the following nonlocal operator eigenvalue problem:
(12)-LK2v=μvinΩ,v=0inRn∖Ω.We say that(u,v)∈E is a weak solution of system (1) if, for every (ϕ,ψ)∈E, one has
(13)M1(∥u∥Z12)∫R2n‍(u(x)-u(y))(ϕ(x)-ϕ(y))×K1(x-y)dxdy+M2(∥v∥Z22)×∫R2n‍(v(x)-v(y))(ψ(x)-ψ(y))K2(x-y)dxdy-∫Ω‍f(x,v)ϕ(x)dx-∫Ω‍g(x,u)ψ(x)dx=0.The fractional Laplacian and nonlocal operators of elliptic type arise in both pure mathematical research and concrete applications, since these operators occur in a quite natural way in many different contexts. For an elementary introduction to this topic, see [2] and the references therein. Recently, some elliptic boundary problems driven by the nonlocal integrodifferential operator LK have been studied in the works [3–8].Recently, problems involving Kirchhoff type operators have been studied in many papers; we refer to [9–13] in which the authors have used the variational method and topological method to get the existence of solutions.In this paper, motivated by the above mentioned works, we will use Morse theory to investigate the multiplicity of solutions of problem (1). To the best of our knowledge, there is no effort being made in the literature to study the existence of solutions for problem (1). This paper will make some contribution to this research field.In order to establish solutions for problem (1), we make the following assumptions.(H1)
Mi:R+→R+ (i=1,2) are two continuous functions, and there exist constants m1, m2, M1, M2>0 such that
(14)mi≤Mi(t)≤Mi,i=1,2,∀t≥0.(H2)
f(x,v) and g(x,u) are two continuous functions with the subcritical growth; that is, there exist some positive constants C1, C2 such that
(15)|f(x,v)|≤C1(1+|v|p-1),|g(x,u)|≤C2(1+|u|q-1),hhhhhhh∀x∈Ω,u,v∈Rhold, where1<p<2s1*=2n/(n-2s1), 1<q<2s2*=2n/(n-s2).(H3)
There existsr>0, λ-∈(λ1,λ2) and μ-∈(μ1,μ2) such that M1λ1<m1λ-, M2μ1<m2μ-, and |u|, |v|≤r implies
(16)12M2μ1v2≤F(x,v)≤12m2μ-v2,12M1λ1u2≤G(x,u)≤12m1λ-u2,hhhhhhhhhhhhhhhhha.e.x∈Ω.(H4)
lim⁡|v|→∞(F(x,v)/v2)<(1/2)m2μ1, lim⁡|u|→∞(G(x,u)/u2)<(1/2)m1λ1, uniformly for all a.e. x∈Ω¯.The main result of this paper is as follows.Theorem 1.
If (H1)–(H4) hold, then the problem (1) has at least two nontrivial weak solutions in E.


## 2. Preliminaries

For each(u,v)∈E, we define the functional J:E→R as follows:
(17)J(u,v)=12M1^(∥u∥Z12)+12M2^(∥v∥Z22)-∫ΩF(x,v)dx-∫ΩG(x,u)dx,
where
(18)Mi^(t)=∫0tMi(τ)dτ,i=1,2,t≥0,F(x,v)=∫0v‍f(x,s)ds,G(x,u)=∫0u‍g(x,s)ds.It is easy to check that(u,v) is a weak solution of problem (1) which is equivalent to being a critical point of the functional J.First let us recall the definition of the local linking which plays an important role in our paper.Definition 2.
LetX be a Banach space with a direct sum decomposition X=X1⊕X2. The functional f∈C1(X,R) has a local linking at 0 with respect to (X1,X2) if there is r>0 such that
(19)f(u)≥0,∀u∈X1with∥u∥≤rf(u)≤0,∀u∈X2with∥u∥≤r.Lemma 3.
Assume that (H1) and (H4) hold; then the functionalJ is coercive in E; that is, J(u,v)→+∞ as ∥(u,v)∥→∞.Proof.
From (H4) and the continuity of the potentialsF and G we have that, for some ϵ>0, there exists a positive constant C3 such that
(20)F(x,t)≤m22(μ1-ϵ)|t|2+C3,G(x,t)≤m12(λ1-ϵ)|t|2+C3,hhhhhhh∀t∈R,a.e.x∈Ω.
Thus, by the Sobolev inequality [1] and (H1), for (u,v)∈E, we obtain
(21)J(u,v)≥m12∥u∥Z12+m22∥v∥Z22-m1(λ1-ϵ)2∫Ω‍u2dx-m2(μ1-ϵ)2∫Ω‍v2dx-2C3|Ω|≥m12(1-λ1-ϵλ1)∥u∥Z12+m22(1-μ1-ϵμ1)∥v∥Z22-2C3|Ω|⟶+∞,
as ∥(u,v)∥→∞. Hence, we have that J is coercive in E.Lemma 4.
If (H1), (H2), and (H4) hold, thenJ satisfies the (P.S.) condition.Proof.
Let{zn=(un,vn)} be a (PS) sequence of J; then {(un,vn)} must be bounded by Lemma 3. Passing to a subsequence if necessary, there exists z=(u,v)∈E such that (un,vn)⇀(u,v) weakly in E. Thus, there exists a strictly decreasing subsequence ϵn, lim⁡n→∞ϵn=0, such that
(22)|J′(un,vn)(un-u,0)|≤ϵn∥un-u,0∥.
In particular,
(23)|∫R2n‍M1(∥un∥Z12)×∫R2n‍(un(x)-un(y))×((un-u)(x)-(un-u)(y))K1(x-y)dxdy-∫Ω‍f(x,vn)(un-u)dx|≤ϵn∥(un-u,0)∥.
Since the potential F satisfies (H2) and by remark (3.2.24) in [14] we have
(24)∫Ω‍f(x,vn)(un-u)dx⟶0.
Combining (23) with (24), we obtain
(25)m1|∫R2n‍(un(x)-un(y))((un-u)(x)-(un-u)(y))×K1(x-y)dxdy∫R2n‍|≤|M1(∥un∥Z12)∫R2n‍(un(x)-un(y))×((un-u)(x)-(un-u)(y))×K1(x-y)dxdy∫R2n‍(∥un∥Z12)|⟶0.
On the other hand, we have
(26)lim⁡n→∞∫R2n‍(u(x)-u(y))((un-u)(x)-(un-u)(y))×K1(x-y)dxdy=0.
Adding (25) to (26), we conclude that
(27)0=lim⁡n→∞[∫R2n‍(un(x)-un(y))2K1(x-y)dxdy-∫R2n‍(u(x)-u(y))2K1(x-y)dxdy],
which implies ∥un∥Z12→∥u∥Z12. So, ∥un∥Z1→∥u∥Z1.
Similarly, we can obtain that∥vn∥Z2→∥v∥Z2. The uniform convexity of E yields that {zn} converges strongly to z in E.
Thanks to the fact thatL2s1*(Ω)↪Lp1(Ω) (2<p1<2s1*) continuously, we get by Lemma 6 in [2] and (4) that
(28)∥u∥Lp1(Ω)≤|Ω|(2s1*-p1)/(p12s1*)∥u∥L2s1*(Ω)≤|Ω|(2s1*-p1)/(p12s1*)×c1(∫R2n‍|u(x)-u(y)|2|x-y|n+2s1dxdy)1/2≤C4∥u∥Z1,
where C4=|Ω|(2s1*-p1)/(p12s1*)c1/θ1. Similarly, for 2<q1<2s2*, there exists a constant C5>0 such that
(29)∥v∥Lq1(Ω)≤C5∥v∥Z2.
In the following, setU=span⁡{φ1}×span⁡{ψ1}:=〈φ1〉×〈ψ1〉, where φ1>0 with ∥φ1∥Z1=1 is the corresponding eigenfunction of λ1 and ψ1>0 with ∥ψ1∥Z2=1 is the corresponding eigenfunction of μ1. Eigenvalues λ1 and μ1 are as in (11) and (12), respectively. Taking
(30)V={(u,v)∈E:u∈〈φ1〉⊥,v∈〈ψ1〉⊥},
we can easily know that V is complementary subspace of U. Hence we have the following direct sum:
(31)E=U⊕V.
If(u,v)∈U, from Proposition 9 in [4], we get
(32)∥u∥Z12=λ1∫Ω‍|u(x)|2dx,∥v∥Z22=μ1∫Ω‍|v(x)|2dx.
Moreover, if (u,v)∈V, by Proposition 9 in [4], we have
(33)∥u∥Z12≥λ2∫Ω‍|u(x)|2dx,∥v∥Z22≥μ2∫Ω‍|v(x)|2dx.Lemma 5.
Assume that (H1)–(H3) hold. Then the functionalJ has a local linking at the origin with respect to E=U⊕V.Proof.
(i) Let(u,v)∈U. Since
(34)∥(u,v)∥⟶0⟹∫Ω‍|u(x)|2dx⟶0,∫Ω‍|v(x)|2⟶0
by (32), we have that, for given r>0, there is some ρ>0 small enough such that
(35)(u,v)∈U,∥(u,v)∥≤ρ⟹|u(x)|≤r,|v(x)|≤r,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhha.e.x∈Ω.
Now on U, we have by (H1) and (H3) that, for (u,v)∈U with ∥(u,v)∥≤ρ,
(36)J(u,v)=12M1^(∥u∥Z12)+12M2^(∥v∥Z22)-∫ΩF(x,v)dx-∫ΩG(x,u)dx≤M12λ1∫Ω|u|2dx+M22μ1∫Ω‍|v|2dx-∫ΩF(x,v)dx-∫ΩG(x,u)dx=∫|u|≤r‍(12M1λ1|u|2-G(x,u))dx+∫|v|≤r‍(12M2μ1|v|2-F(x,v))dx≤0.
(ii) Let(u,v)∈V. By (33), similar to (34) and (35), we obtain by (H1)–(H3) that, for (u,v)∈V with ∥(u,v)∥≤ρ,
(37)J(u,v)=12M1^(∥u∥Z12)+12M2^(∥v∥Z22)-12m1λ-∫Ωu2dx-12m2μ-∫Ωv2dx-∫{|v|≤r}‍(F(x,v)-12m2μ-|v|2)dx-∫{|v|>r}‍(F(x,v)-12m2μ-|v|2)dx-∫{|u|≤r}‍(G(x,u)-12m1λ-|u|2)dx-∫{|u|>r}‍(G(x,u)-12m1λ-|u|2)dx≥m12(1-λ-λ2)∥u∥Z12+m22(1-μ-μ2)∥v∥Z22-∫{|v|>r}‍(F(x,v)-12m2μ-|v|2)dx-∫{|u|>r}‍(G(x,u)-12m1λ-|u|2)dx≥m12(1-λ-λ2)∥u∥Z12+m22(1-μ-μ2)∥v∥Z22-C6∫{|v|>r}‍|v|p2dx-C7∫{|u|>r}‍|u|q2dx≥m12(1-λ-λ2)∥u∥Z12+m22(1-μ-μ2)∥v∥Z22-C8∥u∥Z1p2-C9∥v∥Z2q2,(by(28)-(29)),
where Ci (i=6,…,9) are positive constants, 2<p2<2s1*, and 2<q2<2s2*. Thus, (37) implies that J(u,v)>0 for 0<∥(u,v)∥≤ρ with ρ>0 is small enough. The proof is complete.LetX be a real Banach space and f∈C1(X,R). Suppose p is an isolated critical point of f with f(p)=c and U is a neighborhood of p, containing the unique critical point; the group
(38)Cq(f,p)=Hq(fc∩U,fc∩U∖{p}),q=0,1,2,…,
is called the qth critical group of f at p, where fc={u∈X:f(u)≤c} and Hq(·,·) is the qth singular relative homology group with integer coefficients.Lemma 6 (see [15]).
LetE be a Banach space and f:E→R a C1-functional satisfying the (P.S) condition. Assume that f has a local linking to the decomposition E=U⊕V near the origin, where dim⁡U=m<∞. If 0∈E is the unique critical point of f in Bρ, then
(39)Cm(f,0)=Hm(fc∩Bρ,fc∩Bρ∖{0})≠0.


## 3. The Proof of Theorem1

We say thatu is a homological nontrivial critical point of f if at least one of its critical groups is nontrivial. By [16], we have the following abstract critical point theorem.Lemma 7 (see [16]).
LetX be a real Banach space and let Φ∈C1(X,R) satisfy the (P.S) condition and be bounded from below. If Φ has a critical point that is homologically nontrivial and is not the minimizer of Φ, then Φ has at least three critical points.From the proof of Lemma3, we can conclude that (0,0)∈E is the unique critical point of our J in a ball that is small enough. Since dim⁡U=dim⁡〈φ1〉×〈ψ1〉=2<∞, by Lemmas 5 and 6, we have the following lemma.Lemma 8.
Let (H1)–(H3) hold. Then(0,0) is a critical point of J and C2(J,(0,0))≠0.Proof of Theorem1.
By Lemmas3 and 4, J is coercive and satisfies the (P.S) condition. Hence J is bounded below. By Lemma 8, (0,0)∈E is homologically nontrivial critical point of J but not a minimizer. Then the conclusion follows from Lemma 7.



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