# Boundary Value Problem for a Second-Order Difference Equation with Resonance

Authors: Zhenguo Wang; Zhan Zhou
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study the existence and multiplicity of nontrivial solutions of a second-order discrete boundary value problem with resonance and sublinear or superlinear nonlinearity. The main methods are based on the Morse theory and the minimax methods. In addition, some examples are given to illustrate our results.


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## Body

## 1. Introduction

Letℤ and ℝ be the sets of integers and real numbers, respectively. For a,b∈ℤ, ℤa,b denotes the discrete interval a,a+1,…,b if a≤b.In this paper, we consider the existence and multiplicity of nontrivial solutions for the following discrete Dirichlet boundary value problem:(1)ΔpkΔuk−1+qkuk+fk,uk=0,k∈ℤ1,T,u0=uT+1=0,where the forward difference operator is defined by Δuk=uk+1−uk; pk and qk are real valued on ℤ, and pk is nonzero; and fk,·∈C1ℝ,ℝ satisfies fk,0=0 for each k∈ℤ1,T. Clearly, problem (1) has the trivial solution u=0.In the recent years, the existence of solutions for nonlinear difference equations has been widely studied by many authors. We note that these results were usually obtained by means of critical point theory, for example, the existence of ground state solutions [1], homoclinic orbits [2–5], the boundary value problem, and periodic solutions [6–8].The boundary value problem (1) may be regarded as a discrete analogue of the following boundary value problem:(2)ptu′t′+qtut+ft,ut=0,0<t<1,u0=u1=0,which has been successfully applied to the modeling of astrophysics, gas dynamics, and chemically reacting system, see [9–11].By various methods and techniques, many authors studied the similar second-order difference equation under various boundary value conditions. For example, Agarwal in [10] considered the existence of solutions for the boundary value problem (1) by the contraction mapping principle and Brouwer fixed point theorems in Euclidean space. Yu and Guo in [12] first used the critical point theory to study the following discrete boundary value problem:(3)ΔpkΔuk−1+qkuk=fk,uk,k∈ℤ1,T,u0=A,uT+1=B,where A and B are the constants, and they proved the existence of solutions of problem (3) when the nonlinear term f is sublinear or superlinear. We refer the readers to [13–16] and the reference therein for more information.It is well known that the resonance exists in many real-world applications, and the equations with resonance have been extensively studied in various fields [17–21]. In fact, it is more difficult to study the boundary value problem under the case of resonance because the resonance case can change the local geometric properties of the critical points [17]. In this paper, we study the existence and multiplicity of nontrivial solutions for problem (1) at resonance by means of the interaction between the nonlinearity and the spectrum of the symmetric matrix P+Q, where P+Q denotes a matrix whose elements are given by pk and qk for k∈ℤ1,T.Note that in [12], the authors obtained the existence of one solution for (3) via the variational methods or the saddle point theorem. In the case where A=B, the solution could be trivial, which was not considered in [12]. In this paper, we will show that the boundary value problem (1) has at least one nontrivial solution. In fact, our Theorem 1 and Corollary 1 extend and complement the existence results given in [12] and establish the existence of multiple solutions.Our results are obtained by combining the Morse theory [22–24], critical group computation, and the minimax methods including the local linking [17, 25]. We prove the existence of nontrivial solutions for problem (1) at resonance by the relationship between these nonlinear techniques and methods. In order to apply these ideas to identify the unknown critical point, we compute the corresponding critical groups of functional J. Obviously, when we consider the existence of nontrivial solutions of problem (1) at resonance, we need to overcome much more difficulties than those in the literature [12].We consider theT-dimensional Banach space:(4)S=u:ℤ0,T+1⟶Rsuch thatu0=uT+1=0.The spaceS is a Hilbert space with the inner product(5)u,v=∑k=1Tukvk,∀u,v∈S,endowed with the norm(6)u=u,u=∑k=1Tuk21/2.Now, we define theC1-functional J on S as follows:(7)Ju=∑k=1T+112pkΔuk−12−qku2k−∑k=1TFk,uk,for every u∈S, where Fk,t=∫0tfk,sds, k,t∈ℤ1,T×ℝ. We can compute the Frećhet derivative as(8)J′u,v=∑k=1T+1pkΔuk−1Δvk−1−qkukvk−fk,ukvk=−∑k=1TΔpkΔuk−1+qkuk+fk,ukvk,for all u,v∈S. It is clear that the critical points of J are the solutions of problem (1).For convenience, we identifyu∈S with u=u1,u2,…,uT∈ℝT. Thus, we rewrite Ju and J′u,v as(9)Ju=12uτP+Qu−∑k=1TFk,uk,J′u,v=uτP+Qv−∑k=1Tfk,ukvk,further,(10)J″uv,w=wτP+Qv−∑k=1Tf′k,ukvkwk,where uτ denotes the transpose of u and P and Q are the T×T symmetric matrices given by(11)P=p1+p2−p20⋯00−p2p2+p3−p3⋯000−p3p3+p4⋯00⋯⋯⋯⋯⋯⋯000⋯pT−1+pT−pT000⋯−pTpT+pT+1,Q=−q100⋯000−q20⋯0000−q3⋯00⋯⋯⋯⋯⋯⋯000⋯−qT−10000⋯0−qT.LetE be a real Banach space and J∈C1E,ℝ. For c∈ℝ, we write Jc=u∈E:Ju≤c and K=u∈E:J′u=0.In order to obtain the critical points of the functionalJ on E, we recall some basic concepts and results of the Morse theory.Definition 1.
(see [22]). J is said to satisfy the Palais–Smale condition (PS condition for short), if any sequence un⊂E for which Jun is bounded and J′un⟶0 as n⟶∞ possesses a convergent subsequence in E.
In the Morse theory, functionalJ is always required to satisfy the deformation condition (D), which was introduced by Bartsch and Li [26]. Bartolo et al. [27] proved that J satisfies the deformation condition (D) if J satisfies the PS condition.
Assume thatJ∈C1E,ℝ, and let u0∈K be an isolated critical point of J with Ju0=c∈ℝ, and U is a neighborhood of u0, containing the unique critical point, then we call(12)CqJ,u0=HqJc∩U,Jc∩U\u0,q∈ℤ,the q-th critical group of J at u0, where Hq·,· stands for the q-th singular relative homology group with integer coefficients [22, 23].
We say thatu0 is a homological nontrivial critical point of J, if at least one of its critical groups is nontrivial.
LetJ∈C1E,ℝ satisfy the PS condition and all critical values of J be greater than some α∈ℝ, then the group(13)CqJ,∞=HqE,Jα,q∈ℤ,is called the q-th critical group of J at infinity [26].
Assume#K<∞ and J satisfies the PS condition. The Morse-type numbers of the pair E,Jα are defined by Mq=MqE,Jα=∑u∈KdimCqJ,u, and the Betti numbers of the pair E,Jα are βq=dimCqJ,∞. By the Morse theory [22, 23], the following relations hold:(14)∑j=0q−1q−jMj≥∑j=0q−1q−jβj,∀q∈ℤ,(15)∑q=0∞−1qMq=∑q=0∞−1qβq.
By (14), it follows that Mq≥βq for all q∈ℤ. Hence, when βq≠0 for some q∈ℤ and J satisfies the PS condition, then J must have a critical point u0 satisfying CqJ,u0≇0. If J has only one critical point u0, then CqJ,∞≅CqJ,u0, ∀q∈ℤ. If CqJ,∞≇CqJ,u0 for some q, then J must have another critical point u1≠u0. Further, if uandv are two critical points of J and CqJ,u≇CqJ,v for some q, then u≠v.
We need the following lemmas to prove the main results.Lemma 1.
(see [17]). Let E be a real Banach space and J∈C1E,ℝ satisfy the (D) condition and be bounded from below. If J has a critical point that is homological nontrivial and is not the minimizer of J, then J has at least three critical points.
Obviously, we need to show thatJ has a homological nontrivial critical point, which is not the global minimizer in this lemma.Lemma 2.
(see [17, 25]). Assume that J has a critical point u=0 with J0=0. If J has a local linking at 0 with respect to E=V⊕W and h=dimV<∞, i.e., there exists γ>0 such that(16)Ju≤0,u∈V,u≤γ,Ju>0,u∈W,0<u≤γ.Then,ChJ,0n≅0. Moreover, if h=μ0 or h=μ0+ν0, then CqJ,0≅δq,hℤ, where μ0 and ν0 denote the Morse index and the nullity of J at 0, respectively.From Lemma2, if u=0 is a trivial critical point and J has a local linking structure at 0, then 0 is a homological nontrivial critical point of J.Lemma 3 (see [26]).
LetE be a Banach space and J∈C1E,ℝ satisfy the PS condition. Suppose E splits as E+⊕E− such that J is bounded from below on E+ and Ju⟶−∞ for u∈E− as u⟶∞. Then, ChJ,∞≇0 for h=dimE−.Lemma 4.
(see [28]). Let Y⊂B⊂A⊂X be tophological spaces and q∈ℤ. If(17)HqA,B≠0,HqX,Y≠0,then(18)Hq+1X,A≠0orHq−1B,Y≠0.


## 2. Sublinear Case

In this section, we will consider the case wherefk,t is sublinear in t for each k∈ℤ1,T. Let the spectrum of the symmetric matrix P+Q consist of isolated eigenvalues of finite multiplicity denoted by(19)λ1≤λ2≤⋯≤λl≤λl+1≤⋯≤λT.It follows that(20)λ1u2≤uτP+Qu≤λTu2,for every u∈S.We assume the following conditionsG1 and G2 are satisfied:(i)
G1 There exist 1<θ<2andM>0 such that(21)0<tfk,t≤θFk,t,t≥M,∀k∈ℤ1,T.(ii)
G2 There exist δ>0, λl≠λl+1 for some l<T, and λ¯∈λl,λl+1 such that(22)λlt2≤2Fk,t≤λ¯t2,t≤δ,∀k∈ℤ1,T.Theorem 1.
Assume that the matrixP+Q is positive definite and fk,t satisfies G1 and G2. Then, problem (1) has at least two nontrivial solutions in S.To prove Theorem1, we need a series of lemmas.Lemma 5.
If the matrixP+Q is positive definite and G1 holds. Then, J satisfies the PS condition.Proof.
For any sequenceun⊂S, with Jun is bounded and J′un⟶0 as n⟶+∞, there exists a positive constant C∈ℝ such that Jun≤C. By G1, we have(23)−Cθ−un≤θJun−J′unun=θ2−1unτP+Qun−∑k=1TθFk,unk−fk,unkunk=θ2−1unτP+Qun−∑unk<MθFk,unk−fk,unkunk−∑unk≥MθFk,unk−fk,unkunk≤θ2−1λ1un2−L,where L=mink,t∈ℤ1,T×−M,MθFk,t−fk,tt. Then, for any n∈ℤ,(24)1−θ2λ1un2−un≤Cθ−L.
Since1<θ<2, thus un is bounded in S and the Bolzano–Weierstrass theorem implies that un has a convergent subsequence.Lemma 6.
If the matrixP+Q is positive definite and G1 holds. Then, J is coercive on S, that is, Ju⟶+∞ as u⟶+∞.Proof.
If the matrixP+Q is positive definite, then λi>0, ∀i∈ℤ1,T. Since G1 is satisfied, it implies that there exist constants a1>0 and a2>0 such that(25)Fk,t≤a1tθ+a2,for allk∈ℤ1,T,t∈ℝ,then(26)Ju=12uτP+Qu−∑k=1TFk,uk≥12λ1u2−a1∑k=1Tukθ−a2T.
Since(27)∑k=1Tukθ≤T1−θ/2uθ,we have(28)Ju≥12λ1u2−a1T1−θ/2uθ−a2T⟶+∞,asu⟶∞.
The proof is complete.
LetSλl denote the eigenspace of λl for some l∈ℤ1,T−1, V denote the subspace of S spanned by the eigenfunctions corresponding to the eigenvalues λ1,…,λl−1, and W denote the subspace of S spanned by the eigenfunctions corresponding to the eigenvalues λl+1,…,λT, then we are given an orthogonal decomposition(29)S=V⊕Sλl⊕W.Lemma 7.
Assume thatG2 holds. Then, J has a local linking at the origin with respect to S=V⊕Sλl⊕W and h=dimV⊕Sλl<T.Proof.
Letγ=δ, for each u∈V⊕Sλl and u≤δ implies uk≤δ∀k∈ℤ1,T. By G2, we have(30)Ju=12uτP+Qu−∑k=1TFk,uk≤λl2u2−∑k=1TFk,uk=∑k=1Tλl2uk2−Fk,uk≤0.
For eachu∈W, 0<u≤δ implies uk≤δ∀k∈ℤ1,T. We have the following estimates:(31)Ju=12uτP+Qu−∑k=1TFk,uk≥λl+12u2−∑k=1TFk,uk=λl+12u2−λ¯2u2−∑k=1TFk,uk−λ¯2uk2≥λl+1−λ¯2u2>0.
This completes the proof.Proof.
Proof of Theorem 1. By Lemmas5 and 6, J satisfies the PS condition and is coercive, hence J is bounded from below. Combining Lemma 7 with Lemma 2, the trivial solution u=0 is homological nontrivial and is not a minimizer. We apply Lemma 1 to conclude that problem (1) has at least two nontrivial solutions in S.Remark 1.
(i)
The conclusion of Theorem1 holds, if the condition G2 is replaced by the following condition:G2′ There exist δ>0, λl≠λl+1 for some l<T, and λ¯∈λl,λl+1 such that(32)λlt2≤tfk,t≤λ¯t2,fort≤δ,∀k∈ℤ1,T.(ii)
The conditionG2 means that problem (1) is resonant near 0 at the eigenvalue λl from the right side, and G2 contains the situation limsupt⟶02Fk,t/t2=λ∈λl,λl+1.(iii)
Iffk,0=0 for each k∈ℤ1,T, then problem (1) admits the trivial solution u=0. In Theorem 1, we find two nontrivial solutions for (1). From G1 and G2, we see that the existence of nontrivial solutions for problem (1) depends on the behaviors of the term fk,t or Fk,t at 0 and at infinity ∀k∈ℤ1,T.Example 1.
LetT=2, and we consider the boundary value problem (1) with(33)fk,t=ft=t,ift<1,tt,ift≥1,for all k∈ℤ1,2. Then,(34)Fk,t=Ft=t22,ift<1,t−12,ift≥1.
Fixθ=3/2, pk=1, and qk=0∀k∈ℤ1,2, then the matrix P+Q,(35)P+Q=2−1−12,is positive definite and admits two distinct eigenvalues given by λ1=1andλ2=3.
For eachk∈ℤ1,2, we have(36)32Fk,t−tfk,t=12t−34⟶+∞,ast⟶+∞,thus the condition G1 holds.
Letδ=1, we see that(37)t2=2Fk,t<3t2,t≤1,∀k∈ℤ1,2,then, the condition G2 holds. Clearly, the conditions of Theorem 1 are satisfied, and problem (1) admits at least two nontrivial solutions in S.
In fact, ifuk≥1 for k∈ℤ1,2, then problem (1) is(38)−2u1+u2+u1u1=0,u1−2u2+u2u2=0,u0=u3=0.
We can show thatu0=0,u1=1,u2=1,u3=0 and u0=0,u1=−1,u2=−1,u3=0 are the only two nontrivial solutions of problem (38) in S.Theorem 2.
Assume that the matrixP+Q is negative definite, and the conditions G1 and G2 hold. Then, problem (1) possesses at least one nontrivial solution.Proof.
We now show that we can find at least one nontrivial critical point of functionalJ.
Since the matrixP+Q is negative definite, in this case, λi<0 for every i∈ℤ1,T and the inequality (20) holds. If u∈S satisfies u≥M, by G1, we have(39)Ju≤λT2u2⟶−∞,asu⟶∞.
We notice that if−J is continuity and coercive, then J has at least a local maximum u1∈S by the above inequality. This means that CqJ,u1≅δq,Tℤ. If u1=0, then CqJ,0≅δq,Tℤ, noticing Lemma 7, this contradicts Lemma 2. Hence, 0 is different from u1.Theorem 3.
Assume that matrixP+Q is negative semidefinite, and(40)G3tfk,t>0,fort≠0,k∈ℤ1,T.Then, problem (1) has no nontrivial solution.Proof.
If we assume that (1) has a nontrivial solution, then Ju has a nonzero critical point u¯. Since J′u¯=0 and the matrix P+Q is negative semidefinite, one has(41)u¯τP+Qu¯=∑k=1Tfk,u¯ku¯k≤0.
This contradicts withG3. The proof is complete.
If the matrixP+Q is just nonsingular, we may suppose that its first positive eigenvalue is λl>0. We denote its eigenvalues as follows:(42)λ1≤λ2≤⋯≤λl−1<λl≤λl+1≤⋯≤λT.
Then, we have(43)λ1u2≤uτP+Qu≤λlu2,∀u∈V⊕Sλl,λl+1u2≤uτP+Qu≤λTu2,∀u∈W.Theorem 4.
If the matrixP+Q is just nonsingular and 2≤l<T, we assume that G1 and the following conditions hold:(i)
G4limt⟶0fk,t/t=λl,∀k∈ℤ1,T(ii)
G5 There exists δ>0 such that 2Fk,t≥λlt2 for t≤δ, ∀k∈ℤ1,T
Then, problem (1) has at least one nontrivial solution.Proof.
We show thatJ has a local linking at 0. By G4, let ε∈0,λl+1−λl, then there exists γ∈0,δ such that(44)2Fk,t−λlt2<εt2,t≤γ,∀k∈ℤ1,T.
SinceG5 holds, then we obtain(45)0≤2Fk,t−λlt2<εt2,t≤γ,∀k∈ℤ1,T.
For everyu∈W and 0<u≤γ, one has uk≤γ∀k∈ℤ1,T. Then,(46)Ju=12uτP+Qu−∑k=1TFk,uk≥λl+12u2−λl+ε2∑k=1Tuk2=λl+1−λl−ε2u2>0.
For everyu∈V⊕Sλl, u≤γ implies uk≤γ∀k∈ℤ1,T. Thus, we get(47)Ju=12uτP+Qu−∑k=1TFk,uk≤λl2u2−λl2∑k=1Tuk2=0.
We can apply Lemma2 to conclude that J has a local linking at 0, then ChJ,0≇0, h=dimV⊕Sλl, and 0 is not local maximum in S.
Letζ be the element of Sλl, and owing to G4, we have(48)J″0ζ,ζ=∑k=1Tλl−f′k,0ζk2=0,therefore, ν0=dimSλl≥1 and CqJ,0≅δq,l−1+ν0ℤ.
We see thatJ satisfies the PS condition from G1, and we have(49)Ju≤λl−12u2⟶−∞,asu⟶∞,for every u∈V.
On the other hand, we have(50)Ju≥12λlu2−a1T1−θ/2uθ−a2T⟶+∞,asu⟶∞,for every u∈Sλl⊕W. Hence, J is bounded from below on Sλl⊕W.
It can be easily seen that the functionalJ satisfies all the assumptions of Lemma 3; hence, we have(51)Cl−1J,∞≇0.
We note thatl−1≠l−1+ν0, and then J has a critical point u1≠0 such that Cl−1J,u1≇0. Thus, J has at least one nontrivial solution u1.


## 3. Superlinear Case

In this section, we will study the case wherefk,t is superlinear in t at infinity for each k∈ℤ1,T. We give the following assumption:(i)
G6 There exist β∈2,∞andM1>0 such that(52)tfk,t≥βFk,t>0,fort≥M1,∀k∈ℤ1,T.We apply a similar method in [18] to look for nontrivial critical points of J in S.In order to prove the main results of this section, we need to make use of the following lemmas.Lemma 8.
IfG6 holds, then J satisfies the PS condition.Proof.
Suppose that the eigenvalues of the matrixP+Q are λ1,λ2,…,λT, let λmax=maxλ1,λ2,…,λT>0. Using G6, there exist constants a3>0anda4>0 such that(53)Fk,t≥a3tβ−a4,for allk∈ℤ1,T,t∈ℝ.
For any sequenceun⊂E, Jun≤C for some C∈ℝ and J′un⟶0 as n⟶+∞, then we have(54)−C≤Jun=12unτP+Qun−∑k=1TFk,unk≤12λmaxun2−a3∑k=1Tunkβ+a4T.
Since(55)∑k=1Tunk2≤T1−2/β∑k=1Tunkβ2/β,then,(56)∑k=1Tunkβ≥T1−β/2unβ.
We have(57)−C≤Jun≤12λmaxun2−a3T1−β/2unβ+a4T.
For anyn∈ℤ,(58)a3T1−β/2unβ−12λmaxun2≤a4T+C.
Note thatβ>2, thus un is bounded in S, and the Bolzano–Weierstrass theorem implies that un has a convergent subsequence.Lemma 9.
Assume thatG6 holds. Then, there exists a constant D > 0 such that(59)Ja^≅ST−1,fora^−3TM12D,where ST−1 is the unit sphere in S.Proof.
FromG6, we obtain(60)Fk,t≥a3tβ,fort≥M1.
Letu∈ST−1, then(61)Jsu=s22uτP+Qu−∑k=1TFk,suk≤λmax2s2−a3sβ∑k=1Tukβ≤λmax2s2−a3sβT1−β/2,we have Jsu⟶−∞ as s⟶+∞.
LetD=maxfk,t for k,t∈ℤ1,T×−M1,M1. In view of G6, we have(62)∑k=1TFk,uk−12∑k=1Tukfk,uk=∑uk>M1Fk,uk+∑uk≤M1Fk,uk−12∑uk>M1ukfk,uk−12∑uk≤M1ukfk,uk≤1β−12∑uk>M1ukfk,uk+∑uk≤M1Fk,uk+12∑uk<M1ukfk,uk≤1β−12∑uk>M1ukfk,uk+3TM12D.
Leta¯<−3TM1/2D, s>0, and(63)Jsu=s22uτP+Qu−∑k=1TFk,suk≤a¯,foru∈ST−1.
We can compute the derivative ofJsu with respect to s and obtain(64)dJsuds=suτP+Qu−∑k=1Tukfk,suk≤2s∑k=1TFk,suk−12∑k=1Tsukfk,suk+a¯≤2s1β−12∑suk>M1sukfk,suk+3TM12D+a¯≤2s1β−12∑suk>M1sukfk,suk<0.
By the implicit function theorem, there exists a uniqueφ∈CST−1,ℝ such that s=φu and Jφuu=a^≤a¯ for u∈ST−1. If u≠0, let φ¯u=1/uφu/u, then φ¯∈CS\0,ℝ and Jφ¯uu=a^, for u∈S\0. Further, if Ju=a^, then φ¯u=1.
Now, we define a function(65)Φu=φ¯u,ifJu≥a^,1,ifJu≤a^.
Obviously,Φ:S\0⟶ℝ is a continuous function.
We set a mapμ:0,1×S\0⟶S\0 as μs,u=1−su+sΦuu since Ja^ is a strong deformation retract of S\0, that is, Ja^≅S\0≅ST−1.Theorem 5.
Assume that the conditionsG2 and G6 hold. Then, problem (1) has at least one nontrivial solution.Proof.
Since the conditionsG2 and G6 hold, it follows from Lemma 2, Lemma 7, and ([23], Theorem 4.2) that there exists a ε>0 such that(66)HhJε,J−ε=ChJ,0≠0,where h=dimV⊕Sλl.
By Lemma9, we have(67)HhS,Ja^≅HhS,ST−1=0.
Using Lemma4,(68)Hh+1S,Jε≠0orHh−1J−ε,Ja^≠0.
Hence,J has a critical point u≠0 for which(69)Ju>εora^<Ju<−ε.
The proof is complete.
Note that the conclusion of Theorem5 holds if the symmetric matrix P+Q is positive definite or negative definite or just nonsingular.Example 2.
LetT=3, and we study the boundary value problem (1) with(70)fk,t=ft=2t,ift<14,tt,if14≤t≤1,t2t,ift>1,for k∈ℤ1,3.
Then,(71)Fk,t=Ft=t2,ift<14,23t3/2−148,if14≤t≤1,t44+1948,ift>1.
Letβ=3, pk=1, and qk=0 for each k∈ℤ1,3. The matrix P+Q,(72)P+Q=2−10−12−10−12,admits three distinct eigenvalues given by(73)λ1=2−2,λ2=2,λ3=2+2.
Clearly, we have(74)2t2=2Fk,t<2+2t2,t≤14,∀k∈ℤ1,3,and(75)tfk,t−3Fk,t=t44−1916⟶+∞,ast⟶+∞,∀k∈ℤ1,3.
This means that the conditionsG2 and G6 hold.
All conditions of Theorem5 are verified, and then problem (1) admits one nontrivial solution in S.Corollary 1.
LetG2 and G6 hold and dimV⊕Sλl±1≠T. Then, problem (1) possesses at least two nontrivial solutions.Proof.
We notice that conditionG6 implies (53). Combining with (56), we have(76)Ju=12uτP+Qu−∑k=1TFk,uk≤λT2u2−a3∑k=1Tukβ+a4T≤λT2u2−a3T1−β/2uβ+a4T⟶−∞asu⟶∞.
Hence,−J is coercive, and J has a local maximum at some u1∈S. This means CqJ,u1≅δq,Tℤ. Note that the condition G2 holds, so u1≠0 by Lemma 7. Now, dimV⊕Sλl±1≠T, combining with (68) and (69), there exists another nontrivial critical point u2≠u1 of J.Example 3.
LetT=3, and we consider the following boundary value problem (1) with(77)fk,t=ft=−2t,ift<14,−tt,if14≤t≤1,−t2t,ift>1,for k∈ℤ1,3.
Then,(78)Fk,t=Ft=−t2,ift<14,−23t3/2+148,if14≤t≤1,−t44−1948,ift>1.
Letpk=1 and qk=7/2 for each k∈ℤ1,3. Then, the matrix(79)P+Q=−32−10−1−32−10−1−32,is negative definite, and the eigenvalues are given by(80)λ1=−3+222,λ2=−32,λ3=22−32.
Letδ=1/4 and β=8, then we have(81)−3+222t2<2Fk,t<−32t2,t≤14,∀k∈ℤ1,3,and(82)tfk,t−8Fk,t=t4+196⟶+∞,ast⟶+∞,∀k∈ℤ1,3.
Then, the conditionsG2 and G6 hold. Moreover, dimSλ1=1, and obviously, dimSλ1±1≠3. Then, problem (1) admits at least two nontrivial solutions in S by Corollary 1.


## 4. Conclusion

In this paper, we have studied the existence of one or multiple nontrivial solutions of the boundary value problem (1) for two cases: (i) f is sublinear and (ii) f is superlinear. Examples are provided to illustrate the applicability of our results.We point out that our work is different from the previous works and the results of this paper improve, extend, and complement some related results in the literature [12]. Since our solutions are obtained by the variational approach combining with the Morse theory, functional J is required to satisfy the PS conditions, and some computations on the critical group of a local linking-type critical point are used to deal with the resonance at zero.



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*Source: 7527030-2020-06-20.xml*