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Theory and Modern Applications

Multiplicity results on discrete boundary value problems with double resonance via variational methods

Abstract

The existence of solutions for a class of difference equations with double resonance is studied via variational methods, and multiplicity results are derived.

1 Introduction

Let Z denote the set of integers, and for a,bZ with a<b, define Z[a,b]={a,a+1,,b}. For a given positive integer N2, consider the following discrete boundary value problem:

(BP) { Δ 2 x ( k 1 ) = f ( k , x ( k ) ) , k Z [ 1 , N ] , x ( 0 ) = x ( N + 1 ) = 0 ,

where Δ is the forward difference operator defined by Δx(k)=x(k+1)x(k) and Δ 2 x(k)=Δ(Δx(k)) for kZ. Throughout this paper, we always assume that f:Z[1,N]×RR is C 1 -differentiable with respect to the second variable and satisfies f(k,0)0 for kZ[1,N], which implies that (BP) has a trivial solution x(k)=0, k=0,1,,N+1. We investigate the existence of nontrivial solutions of (BP).

In different fields of research, such as computer science, mechanical engineering, control systems, population biology, economics and many others, the mathematical modeling of important questions leads naturally to the consideration of nonlinear difference equations. The dynamic behaviors of nonlinear difference equations have been studied extensively in [1, 2]. Recently, many authors considered the solvability of nonlinear difference equations via variational methods. For example, on the second-order difference equations, the boundary value problems are studied in [37] and the existence of periodic solutions is investigated in [810].

As a natural phenomenon, resonance exists in the real world from macrocosm to microcosm. In a system described by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity. It is known from [1] that the eigenvalue problem

{ Δ 2 x ( k 1 ) = λ x ( k ) , k Z [ 1 , N ] , x ( 0 ) = x ( N + 1 ) = 0 ,

possesses N distinct eigenvalues λ l =4 sin 2 (lπ/2(N+1)), l=1,2,,N. Many authors considered the complete resonance situation in the sense that for some hZ[1,N],

lim | t | f ( k , t ) t = λ h ,kZ[1,N]

via different methods in critical point theory such as Morse theory [6], index theory [9] and minimax methods [7]. The assumption that

( f ) there exists some hZ[1,N1] such that

λ h lim inf | t | f ( k , t ) t lim sup | t | f ( k , t ) t λ h + 1 for kZ[1,N]

characterizes problem (BP) as double resonance between two consecutive eigenvalues at infinity. In the case of resonance, one needs to impose various conditions on the nonlinearity of f near infinity to ensure the global compactness. In fact, many results on differential equations with double resonance have been obtained (see [1113]). As to discrete boundary value problems with double resonance, however, there are few results published. In [10], the existence of periodic solutions to a second-order difference equation with double resonance, as is described in ( f ), is investigated.

Motivated by the study in [10], we consider problem (BP) with double resonance indicated in ( f ). To control the double resonance, a selectable restriction on the nonlinearity of f is that

( f ) there exists some hZ[1,N1] such that

i ( i ) r ̲ h ( f ) : = lim inf | t | | t | ( f ( k , t ) t λ h ) > 0 ( ii ) r ¯ h ( f ) : = lim sup | t | | t | ( f ( k , t ) t λ h + 1 ) < 0 for kZ[1,N],

which has completely the same form as its counterpart in [10]. However, instead of ( f ), in this paper we assume that

(f) there exists some hZ[1,N1] such that

i ( i ) R ̲ h ( f ) : = lim inf | t | t 2 ( f ( k , t ) t λ h ) > 0 ( ii ) R ¯ h ( f ) : = lim sup | t | t 2 ( f ( k , t ) t λ h + 1 ) < 0 for kZ[1,N].

Remark 1.1 It is easy to see that, as a restriction on the nonlinearity of f, (f) is more relaxed than ( f ) (see Examples 1.1-1.3 and Remark 1.3). In addition, (f), as well as ( f ), implies ( f ).

A sequence {x(0),x(1),,x(N+1)} is said to be a positive (negative) solution of (BP) if it satisfies (BP) and x(k)>0 (<0) for kZ[1,N].

Theorem 1.1 Assume that (f) holds. Then (BP) has at least four nontrivial solutions in which one is positive and one is negative in each of the following two cases:

  1. (i)

    hZ[2,N1] and f (k,0)< λ 1 for kZ[1,N];

  2. (ii)

    hZ[1,N2] and f (k,0)> λ N for kZ[1,N].

To state the following theorems, we further assume that

(f0) there exists t 0 0 such that f(k, t 0 )=0 for kZ[1,N].

Theorem 1.2 Assume that (f0) and (f) hold with hZ[2,N1]. If there exists mZ[2,N1] with mh such that λ m < f (k,0)< λ m + 1 for kZ[1,N], then (BP) has at least four nontrivial solutions.

Let f (k,t) denote the derivative of f(k,t) with respect to the second variable. In the case where (BP) is also resonant at the origin, that is, there exists mZ[1,N] such that f (k,0) λ m for kZ[1,N], we assume that

( F 0 ± )  ± 0 t ( f ( k , s ) λ m s ) ds0  for |t|>0 small and kZ[1,N].

Theorem 1.3 Assume that (f0) and (f) hold with hZ[2,N1]. If there exists mZ[1,N] such that f (k,0) λ m for kZ[1,N], then (BP) has at least four nontrivial solutions in each of the following two cases:

  1. (i)

    ( F 0 + ) with m2 and mh;

  2. (ii)

    ( F 0 ) with m3 and mh+1.

Remark 1.2 In view of the proofs in Section 4, we see that if t 0 >0 (<0) in (f0), two of the solutions derived in Theorems 1.2, 1.3 are positive (negative).

Set hZ[1,N1] and define g:RR by

g(t)= λ h t+( λ h + 1 λ h )t ( 1 + t 2 ) 1 ( sin 2 t + t 2 cos 2 t ) .

By calculation, we get R ̲ h (g)= R ¯ h (g)= λ h + 1 λ h >0 and g (0)= λ h . Define g 1 (t)=g(t)+αt ( 1 + t 4 ) 1 , g 2 (t)=g(t)+(αt+β t 2 ) ( 1 + t 4 ) 1 and g 3 (t)=g(t)+(αt+β t 3 ) ( 1 + t 6 ) 1 , tR, where α and β are constants. Obviously, R ̲ h ( g i )= R ̲ h (g)>0, R ¯ h ( g i )= R ¯ h (g)<0 and g i (0)= λ h +α, i=1,2,3. The following examples are presented to illustrate the applications of the above results.

Example 1.1 Consider (BP) with f(k,t) g 1 (t), (k,t)Z[1,N]×R. We have f (k,0)= λ h +α for kZ[1,N]. If hZ[2,N1] and α< λ 1 λ h or hZ[1,N2] and α> λ N λ h , then by Theorem 1.1, (BP) has at least four nontrivial solutions in which one is positive and one is negative.

Example 1.2 Set h,mZ[2,N1] with mh. Let α( λ m λ h , λ m + 1 λ h ) and β<( λ h + λ h + 1 +α). Consider (BP) with f(k,t) g 2 (t), (k,t)Z[1,N]×R. We have f (k,0)= λ h +α> λ m >0 and f(k,1)=( λ h + λ h + 1 +α+β)/2<0 for kZ[1,N], which implies that there exists t 0 (0,1) such that f(k, t 0 )=0 for kZ[1,N]. By Theorem 1.2 and Remark 1.2, (BP) has at least four nontrivial solutions in which two are positive.

Example 1.3 Set hZ[2,N1], α= λ m λ h and β<max{( λ m + λ h + 1 ),2( λ h + 1 λ h )} for some mZ[3,N]{h+1}. Consider (BP) with f(k,t) g 3 (t), (k,t)Z[1,N]×R. We have f (k,0)= λ h +α= λ m >0 and f(k,1)=( λ m + λ h + 1 +β)/2<0 for kZ[1,N], which implies that there exists t 0 (0,1) such that f(k, t 0 )=0 for kZ[1,N]. Moreover,

0 t ( f ( k , s ) λ m s ) ds= ( β 4 + λ h + 1 2 λ h 2 ) t 4 +o ( t 4 ) (t0).

By Theorem 1.3(ii) and Remark 1.2, (BP) has at least four nontrivial solutions in which two are positive.

Remark 1.3 It is easy to see that r ̲ h ( g i )= r ¯ h ( g i )=0, i=1,2,3, that is, the restriction imposed here is more relaxed than that in [10].

The paper is organized as follows. In Section 2 we give a simple revisit to Morse theory, and in Section 3 we give some lemmas. The main results will be proved in Section 4.

2 Preliminary results on critical groups

Let H be a Hilbert space and Φ C 2 (H,R) be a functional satisfying the Palais-Smale condition ((PS) in short), that is, every sequence { x n }H such that {Φ( x n )} is bounded and Φ ( x n )0 as n has a convergent subsequence. Denote by H q (X,Y) the q th singular relative homology group of the topological pair (X,Y) with integer coefficients. Let u 0 be an isolated critical point of Φ with Φ( u 0 )=c, cR, and U be a neighborhood of u 0 . For qN{0}, the group

C q (Φ, u 0 ):= H q ( Φ c U , Φ c U { u 0 } )

is called the q th critical group of Φ at u 0 , where Φ c ={uH:Φ(u)c}.

If the set of the critical points of Φ, denoted by K:={uH: Φ (u)=0}, is finite and a<infΦ(K), the critical groups of Φ at infinity are defined by (see [14])

C q (Φ,):= H q ( H , Φ a ) ,qN{0}.

For qN{0}, we call β q :=dim C q (Φ,) the Betti numbers of Φ and define the Morse-type numbers of the pair (H, Φ a ) by

M q := M q ( H , Φ a ) = u K dim C q (Φ,u).

With the above notations, we have the following facts (2.a)-(2.f) [[15], Chapter 8].

(2.a) If C μ (Φ,)0 for some μN{0}, then there exists x 0 K such that C μ (Φ, x 0 )0;

(2.b) If K={ x 0 }, then C q (Φ,) C q (Φ, x 0 );

(2.c) j = 0 ( 1 ) j M j = j = 0 ( 1 ) j β j .

If x 0 K and Φ ( x 0 ) is a Fredholm operator and the Morse index μ 0 and nullity v 0 of x 0 are finite, then we have

(2.d) C q (Φ, x 0 )0 for qZ[ μ 0 , μ 0 + ν 0 ];

(2.e) If C μ 0 (Φ, x 0 )0, then C q (Φ, x 0 ) δ q , μ 0 Z, and if C μ 0 + ν 0 (Φ, x 0 )0, then C q (Φ, x 0 ) δ q , μ 0 + ν 0 Z;

(2.f) If m:=dimH<+, then C q (Φ, x 0 ) δ q , 0 Z when x 0 is local minimizer of Φ, while C q (Φ, x 0 ) δ q , m Z when x 0 is the local maximizer of Φ.

We say that Φ has a local linking at x 0 K if there exists the direct sum decomposition: H= H + H and ϵ>0 such that

Φ ( x ) > Φ ( x 0 ) if  x x 0 H + , 0 < x x 0 ϵ , Φ ( x ) Φ ( x 0 ) if  x x 0 H , x x 0 ϵ .

The following results are due to Su [13].

(2.g) Assume that Φ has a local linking at x 0 K with respect to H= H + H and k=dim H <+. Then

C q ( Φ , x 0 ) δ q , μ 0 Z if  k = μ 0 , C q ( Φ , x 0 ) δ q , μ 0 + v 0 Z if  k = μ 0 + v 0 .

We say that Φ satisfies the Cerami condition ((C) in short) if every sequence { x n }H such that {Φ( x n )} is bounded and (1+ x n ) Φ ( x n )0 as n has a convergent subsequence. The following lemma derives from [[11], Proposition 3.2].

Lemma 2.1 [11]

Let H be a Hilbert space, and { Φ s C 1 (H,R)|s[0,1]} are a family of functionals such that Φ s and s Φ s are locally Lipschitz continuous. Assume that Φ 0 and Φ 1 satisfy (C). If there exists M>0 such that

inf s [ 0 , 1 ] , x > M ( 1 + x ) Φ s ( x ) >0and inf s [ 0 , 1 ] , x M Φ s (x)>,

then

C q ( Φ 0 ,)= C q ( Φ 1 ,).

Remark 2.1 The deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition [16]. Therefore, if the (PS) condition is replaced by the (C) condition, (2.a)-(2.g) stated above still hold.

3 Compactness and critical group at infinity

In this section, we are going to prove the compactness of the associated energy functionals and to calculate the critical groups at infinity. First of all, let us introduce the variational structure for problem (BP).

3.1 Variational structure

The class E of functions x:Z[0,N+1]R such that x(0)=x(N+1)=0, equipped with the inner product , and norm as follows:

x,y= k = 1 N x(k)y(k),x= ( k = 1 N | x ( k ) | 2 ) 1 / 2 for x,yE,

is linearly homeomorphic to R N . Denote θ= ( 0 , 0 , , 0 ) T R N . Throughout this paper, we always identify xE with x= ( x ( 1 ) , x ( 2 ) , , x ( N ) ) T R N .

Set f(x)= ( f ( 1 , x ( 1 ) ) , , f ( N , x ( N ) ) ) T , xE and

A= ( 2 1 0 0 0 0 1 2 1 0 0 0 0 0 0 1 2 1 0 0 0 0 1 2 ) N × N .

Then we can equivalently rewrite (BP) as a nonlinear algebraic system

Ax=f(x),xE.
(3.1)

Denote E l =ker(A λ l I), l=1,,N, where I is the identity operator. Thus dim E l =1, l=1,2,,N. Set

E = l = 1 h 1 E l , E + = ( l = 1 h + 1 E l ) , E v = E E + ,

then E has the decomposition E= E h E h + 1 E v . In the rest of this paper, the expression x= x h + x h + 1 + x v for xE always means x E , =h,h+1,v.

Define a functional J:ER by

J(x)= 1 2 Ax,x k = 1 N F ( k , x ( k ) ) for xE,

where F(k,t)= 0 t f(k,s)ds, (k,t)Z[1,N]×R. Then the Fréchet derivative of J at xE, denoted by J (x), can be described by (see [3])

J ( x ) , y =Ax,y k = 1 N f ( k , x ( k ) ) y(k)for yE.
(3.2)

Remark 3.1 From (3.2) we see that xE is a critical point of J if and only if x is a solution of (3.1) (or equivalently (BP)). In addition, J is C 2 -differentiable with

J ( x ) y , z =Ay,z k = 1 N f ( k , x ( k ) ) y(k)z(k)for y,zE.
(3.3)

3.2 Compactness of related functionals

Define a family of functionals J s :ER, s[0,1] by

J s (x)= 1 2 Ax,x 1 s 4 ( λ h + λ h + 1 ) x 2 s k = 1 N F ( k , x ( k ) ) for xE,

then the Fréchet derivative of J s at xE, denoted by J s (x), can be described by (see [3])

J s ( x ) , y =Ax,y k = 1 N f s ( k , x ( k ) ) y(k)for yE,
(3.4)

where s[0,1] and

f s (k,t)=sf(k,t)+ 1 s 2 ( λ h + λ h + 1 )tfor (k,t)Z[1,N]×R.

Lemma 3.1 Assume that (f) holds. For any sequences { x n }E and { s n }[0,1], { x n } is bounded provided that

( 1 + x n ) J s n ( x n )0as n.
(3.5)

Moreover, for every s ˆ [0,1], J s ˆ satisfies (C).

Proof Assume, for a contradiction, that { x n } is unbounded. Then there exists a subsequence, which we still call { x n }, with KZ[1,N] being nonempty such that

lim n x n (k)=for kK

and either K=Z[1,N] or, for any fixed k K c Z[1,N]K, { x n (k)} is a bounded sequence.

Noticing that [[10], Lemma 3.7], with its proof being modified slightly, is applicable here, we know that

either  x n h x n 1or x n h + 1 x n 1as n.

Set

Γ 1 : = lim sup n k = 1 N { f s n ( k , x n ( k ) ) λ h x n ( k ) } x n h ( k ) , Γ 2 : = lim inf n k = 1 N { f s n ( k , x n ( k ) ) λ h + 1 x n ( k ) } x n h + 1 ( k ) .

Thus we have two cases to be considered.

Case 1. x n h / x n 1 as n. We have x n h as n and

lim n x n h + 1 x n =0, lim n x n v x n =0.
(3.6)

By (f)(i), there exist M>0 and ξ>0 such that t 2 (f(k,t)/t λ h )>ξ and t 2 ( λ h + 1 λ h )>ξ for |t|>M and kZ[1,N]. Then, for |t|>M, kZ[1,N] and s[0,1],

( f s ( k , t ) t λ h ) t 2 = s ( f ( k , t ) t λ h ) t 2 + 1 s 2 ( λ h + 1 λ h ) t 2 s ξ + 1 s 2 ξ ξ 2 .

Choose N 1 >0 such that | x n (k)|>M for kK and n> N 1 . It follows that

{ f s n ( k , x n ( k ) ) λ h x n ( k ) } x n h ( k ) = { f s n ( k , x n ( k ) ) x n ( k ) λ h } ( x n ( k ) ) 2 ( ( x n ( k ) z n ( k ) ) x n ( k ) ) { f s n ( k , x n ( k ) ) x n ( k ) λ h } ( x n ( k ) ) 2 ( ( | x n ( k ) | | z n ( k ) | ) x n ) ξ ( | x n ( k ) | | z n ( k ) | ) 2 x n for  k K  and  n > N 1 ,
(3.7)

where z n = x n h + 1 + x n v . Since E possesses an equivalent norm defined by x 1 k = 1 N |x(k)| for xE, there exists a positive constant C>0 such that x 1 Cx, xE. Thus, by (3.6) and (3.7),

Γ 1 lim sup n ξ 2 x n { k K | x n ( k ) | k K | z n ( k ) | } = lim sup n ξ 2 x n { k = 1 N | x n ( k ) | k = 1 N | z n ( k ) | } lim sup n ξ 2 x n ( C x n p z n ) = C ξ 2 ,

where the equality holds because | x n (k)|/ x n 0 as n for k K c in case K c .

Case 2. x n h + 1 / x n 1 as n. In this case, by using (f)(ii), we can show that Γ 2 <0 in the same way.

On the other hand, it follows from (3.5) that

x n J s n ( x n ) , x n x n 0as n,=h,h+1,

which implies that

J s n ( x n ) , x n 0as n,=h,h+1,

that is,

A x n , x n k = 1 N f s n ( k , x n ( k ) ) x n (k)0as n,=h,h+1.

Note that A x n , x n = λ x n , x n , =h,h+1, it follows that Γ 1 = Γ 2 =0. This contradiction proves the first conclusion.

By setting s n s ˆ [0,1] in the proven conclusion, we see that J s ˆ satisfies (C). The proof is complete. □

For xE, set x + (k)=max{0,x(k)}, kZ[1,N] and x + = ( x + ( 1 ) , , x + ( N ) ) T . The following lemma is derived from [[3], Lemma 2.1].

Lemma 3.2 [3]

If x is a solution of

Ax=f ( x + ) ,xE,

then xθ and hence it is also a solution of (3.1). Moreover, either x>θ or x=θ.

For x,yE, we say that xy (x>y) if x(k)y(k) (x(k)>y(k)) for kZ[1,N].

Lemma 3.3 Let ς j be the eigenvector corresponding to λ j , jZ[1,N], then ς 1 can be chosen to satisfy ς 1 >0. Moreover, for j2, neither ς j θ nor ς j θ.

Proof First we claim that ς 1 θ or ς 1 θ. Otherwise, by setting ς ¯ 1 = ( | ς 1 ( 1 ) | , , | ς 1 ( N ) | ) T , we have

k = 1 N | Δ ς ¯ 1 ( k ) | 2 < k = 1 N | Δ ς 1 ( k ) | 2 .
(3.8)

Since λ 1 = inf x = 1 Ax,x=A ς 1 , ς 1 / ς 1 2 , it follows from (3.8) that

λ 1 A ς ¯ 1 , ς ¯ 1 ς ¯ 1 = k = 1 N | Δ ς ¯ 1 ( k ) | 2 ς 1 2 < k = 1 N | Δ ς 1 ( k ) | 2 ς 1 2 = λ 1 .

This contradiction proves the above claim. Thus ς 1 can be assumed to satisfy ς 1 θ and then A ς 1 = λ 1 ς 1 + . It follows by Lemma 3.2 that ς 1 >θ and the first conclusion holds. Further, for k2, ς k and ς 1 are orthogonal to each other, which implies that neither ς j θ nor ς j θ. The proof is complete. □

Lemma 3.4 Let the function gC(Z[1,N]×R,R) be such that g(k,t)=0 for t<0. Assume that there exists hZ[2,N] such that

λ h lim inf t + g ( k , t ) t lim sup t + g ( k , t ) t λ h + 1 .
(3.9)

Then the functional

I(x)= 1 2 Ax,x k = 1 N G ( k , x ( k ) )

satisfies the (PS) condition, where G(k,t)= 0 t g(k,s)ds.

Proof Let { x n }E be such that

I ( x n )0as n.
(3.10)

We only need to prove that { x n } is bounded. In fact, if { x n } is unbounded, there exists a subsequence, still called { x n }, such that x n as n.

Let w n = x n / x n , then w n =1. There is a convergent subsequence of { w n }, call it { w n } again, such that w n wE as n. For every yE, we have I ( x n ),y/ x n 0 as n, that is,

A w n ,y k = 1 N g ( k , x n ( k ) ) x n y(k)0as n.
(3.11)

Set

K + = { k Z [ 1 , N ] | x n ( k ) +  as  n } .

We claim that K + , since otherwise (3.11) leads to A w n ,y0 (n) for yE, which leads to w=0, a contradiction. Thus we have by (3.9) that

λ h lim inf n g ( k , x n ( k ) ) x n ( k ) lim sup n g ( k , x n ( k ) ) x n ( k ) λ h + 1 for k K + ,

which implies that there exists a subsequence of { x n }, still called { x n }, and α k [ λ h , λ h + 1 ], k K + , such that

lim n g ( k , x n ( k ) ) x n ( k ) = α k for k K + .
(3.12)

If kZ[1,N] K + , then g(k, x n (k))/ x n 0 as n. Thus we can rewrite (3.11) as

A w n ,y k K + g ( k , x n ( k ) ) x n ( k ) x n ( k ) x n y(k)0as n.
(3.13)

Letting n in (3.13) and using (3.12), we get

Aw,y= k K + α k w(k)y(k)for yE.
(3.14)

Since w(k)0 for k K + , it follows from (3.14) that

Aw= k k + α k w + (k),
(3.15)

which, by Lemma 3.2, implies that w>0 and hence K + =Z[1,N]. Thus, (3.14) can be rewritten as

Aw,y= k = 1 N α k w(k)y(k)for yE.

Noticing that [[10], Lemma 3.4], with its proof being modified slightly, is applicable here, we know that w is an eigenvector corresponding to λ h or λ h + 1 . Since h2, it follows from Lemma 3.3 that wθ. This contradiction completes the proof. □

3.3 Critical group at infinity

Lemma 3.5 Let f satisfy (f). Then

C q (J,) δ q , h Z, C q (J,) δ q , N h Z.
(3.16)

Proof We claim that there exists M>0 such that

inf { ( 1 + x ) J s ( x ) : x > M , s [ 0 , 1 ] } >0,
(3.17)

otherwise there exist { x n }E and { s n }[0,1] such that x n and (1+ x n )J( x n )0 as n, which contradict Lemma 3.1. Moreover, it is easy to see that inf{ J s (x):s[0,1],xM}>. Thus, by Lemma 2.1, we have

C q (J,) C q ( J 0 ,).

On the other hand,

J 0 (x)= 1 2 Ax,x 1 4 ( λ h + λ h + 1 ) x 2 .

Note that x=θ is the unique critical point of J 0 with the Morse index μ:=dim( E E h )=h and nullity ν=0. Then, by (2.b) and (2.e),

C q ( J 0 ,) C q ( J 0 ,0) δ q , h Z.

Similarly, we have C q (J,) C q ( J 0 ,0) δ q , N h Z. The proof is complete. □

4 Proofs of main results

Now we prove the main results of this paper. First, by applying (3.16) and (2.a), we know that J has a critical point x satisfying

C h ( J , x ) 0.

Define α k = f (k, x (k)), kZ[1,N]. Then from (3.3) we know by calculation that ker J 1 ( x ) is the solution space of the system Bx=0, xE, where

B= ( 2 α 1 1 0 0 0 0 1 2 α 2 1 0 0 0 0 0 0 1 2 α N 1 1 0 0 0 0 1 2 α N ) N × N .

Thus ν 1 =dimker J 1 ( x )1 since B possesses non-degenerate (N1) order submatrixes. By (2.d)-(2.e), we further have

C q ( J , x ) δ q , h Z.
(4.1)

Proof of Theorem 1.1 First we give the proof for the case (i). By f (k,0)< λ 1 for kZ[1,N], we know that x=θ is a strict local minimizer of J. Thus, by (2.f), we have correspondingly

C q (J,θ) δ q , 0 Z.
(4.2)

Noticing that h2, by comparing (4.2) with (4.1), we have x θ.

For kZ[1,N], set f + (k,t)=f(k,t) for t0, f + (k,t)=0 for t<0. Let F + (k,t)= 0 t f + (k,s)ds. Then the critical points of

J + (x)= 1 2 Ax,x k = 1 N F + ( k , x ( k ) )

are exactly solutions of the problem

Ax= f + ( x + ) ,
(4.3)

where f + (x)= ( f + ( 1 , x ( 1 ) ) , , f + ( N , x ( N ) ) ) T , xE. By Lemma 3.4, we see that J + C 2 0 (E,R) satisfies the (PS) condition. From the definition of f + (k,) and the assumption f (k,0)< λ 1 , kZ[1,N], we know that there exists η>0 such that ( λ 1 t f + (k,t))t>0 for t(η,η){0}. For any fixed xE with 0<x<η, define a function ϕ(s)= J + (sx), s[0,1]. By the Lagrange mean value theorem, there exists ξ(0,1) such that

J + ( x ) = ϕ ( 1 ) ϕ ( 0 ) = ϕ ( ξ ) = A ( ξ x ) , x k = 1 N f + ( k , ξ x ( k ) ) x ( k ) k = 1 N { λ 1 ξ x ( k ) f + ( k , ξ x ( k ) ) } x ( k ) > 0 ,

which implies that there exist ρ>0 and τ>0 such that

J + (x)τ,xE with x=ρ.

In addition, let ς 1 be the eigenvector of A corresponding to λ 1 with ς 1 >0, then (f), with hZ[2,N1], implies that

J + (t ς 1 )as t+.

By Mountain Pass Theorem [17, 18], J + has a critical point x 1 θ with the critical group property for a mountain pass point [17], that is, C 1 ( J + , x 1 )0. Noticing that x 1 satisfies (4.3), we get by Lemma 3.2 that x 1 >θ and hence x 1 is also a mountain pass point of J, that is, C 1 (J, x 1 )0.

The same argument shows that J has a nontrivial critical point x 2 <θ with C 1 (J, x 2 )0. Noticing that h2, by comparing the critical groups, we see that x 1 , x 2 and x are three nontrivial critical points of J.

By the same argument as that for (4.1), we get C q (J, x i ) δ q , 1 Z, i=1,2. If x 1 , x 2 and x are all the nontrivial critical points of J, then K={θ, x 1 , x 2 , x } and then (2.c) reads

( 1 ) 0 ×1+ ( 1 ) 1 ×2+ ( 1 ) h ×1= ( 1 ) h ×1,

a contradiction. Thus we claim that there exist at least four nontrivial critical points of J.

In the case (ii), we consider the functional −J. By applying (3.16) and (2.a), we know that −J possesses a critical point x 1 satisfying

C N h ( J , x 1 ) 0.
(4.4)

Since f (k,0)> λ N for kZ[1,N], x=θ is a strict local minimizer of −J and

C q (J,θ) δ q , 0 Z.
(4.5)

Noticing that hN2, we know by comparing (4.5) with (4.4) that x 1 θ. The rest of the arguments are similar to that in case (i) and will be omitted. The proof is complete. □

Proof of Theorem 1.2 In view of (3.3) and the assumption λ m < f (k,0)< λ m + 1 , kZ[1,N], we see that x=θ is a non-degenerate critical point of J with the Morse index μ 0 =m. Thus

C q (J,0) δ q , m Z.
(4.6)

Noticing that hm, we know by comparing (4.6) with (4.1) that x θ.

We may assume that t 0 >0 in (f0). For kZ[1,N], set

f ˜ (k,t)= { 0 , t < 0 , f ( k , t ) , t [ 0 , t 0 ] , 0 , t > t 0 .

Define

J ˜ (x)= 1 2 Ax,x k = 1 N F ˜ ( k , x ( k ) ) ,xE,

where F ˜ (k,t)= 0 t f ˜ (k,s)ds. Since J ˜ (x)+ as x, there is a minimizer x 0 of J ˜ . Thus

A x 0 = f ˜ ( x 0 ),

where f ˜ (x)= ( f ˜ ( 1 , x ( 1 ) ) , f ˜ ( 2 , x ( 2 ) ) , , f ˜ ( N , x ( N ) ) ) T for xE. By the definition of f ˜ , the above equality can be rewritten as

A x 0 = f ˜ ( x 0 + ) .

From Lemma 3.4, we know that x 0 =θ or x 0 >θ. By assumption f (k,0)( λ m , λ m + 1 ) and m2, we know that θ is not a minimizer. Thus we have x 0 >θ. In the same way as the proof of Lemma 3.2, we can prove that x 0 (k)< t 0 for kZ[1,N]. Thus x 0 is a local minimizer of J, therefore

C q (J, x 0 ) δ q , 0 Z.
(4.7)

Define f ˆ (k,t)=f(k,t+ x 0 (k))f(k, x 0 (k)), (k,t)Z[1,N]×R and consider the functional

J ˆ (z)= 1 2 Az,z k = 1 N F ˆ ( k , z ( k ) ) ,zE,

where F ˆ (k,t)= 0 t f ˆ (k,s)ds. A simple calculation shows that if z is a positive critical point of J ˆ , then x 0 +z is a critical point of J, and, moreover, C q ( J ˆ ,z)= C q (J, x 0 +z).

Furthermore, define

f ˆ + (k,t)= { f ˆ ( k , t ) , t 0 , 0 , t < 0 , kZ[1,N]

and its energy functional

J ˆ + (z)= 1 2 Az,z k = 1 N F ˆ + ( k , z ( k ) ) ,zE,

where F ˆ + (k,t)= 0 t f ˆ + (k,s)ds. By (f), we see that f ˆ + satisfies

λ h lim inf t + f ˆ + ( k , t ) t lim inf t + f ˆ + ( k , t ) t λ h + 1 ,kZ[1,N].

It follows from Lemma 3.4 that J ˆ + satisfies the (PS) condition. If x 0 is not a strict local minimizer of J, then there exists infinitely many critical points near x 0 and the conclusion holds. Now we assume that x 0 is a strict local minimizer of J, then z=θ is a strict local minimizer of J ˆ + . In the same way as the proof of Theorem 1.1, we know that J ˆ + has a critical point z 1 , which is a mountain pass point of J ˆ + with C 1 ( J ˆ + , z 1 )0 and z 1 >θ. Thus z 1 is also a critical point of J ˆ with C 1 ( J ˆ , z 1 )0. Hence x 1 = x 0 + z 1 is a critical point of J with C 1 (J, x 1 )0.

In a similar way, we know that J has a critical point x 2 < x 0 with C 1 (J, x 2 )0. Finally, by comparing the critical groups and by using the condition m,h2 with mk, we see that x , x 0 , x 1 and x 2 are four nontrivial critical points of J in which x 0 and x 1 are positive. The proof is complete. □

The proof of the following lemma is similar to that of [[19], Theorem 3.1] and is omitted.

Lemma 4.1 [19]

Let f satisfy ( F 0 + ) (or ( F 0 )). Then J has a local linking at x=θ with respect to the decomposition E= H E + , where E := l m E l (or E := l < m E l respectively).

Proof of Theorem 1.3 In view of (3.3) and the assumption f (k,0)= λ m , kZ[1,N], we see that x=θ is a degenerate critical point of J with the Morse index μ 0 =m1 and nullity ν 0 =1. By Lemma 4.1 and (2.g), we have, corresponding to ( F 0 ) or ( F 0 + ) respectively,

C q (J,0) δ q , m 1 Zor C q (J,0) δ q , m Z.
(4.8)

which, compared with (4.1), implies that x 0 in both of cases (i) and (ii). The rest of the proof is similar to that of Theorem 1.2 and will be omitted. The proof is complete. □

References

  1. Agarwal RP Monographs and Textbook in Pure and Applied Mathematics 228. In Difference Equations and Inequalities. Dekker, New York; 2000.

    Google Scholar 

  2. Elaydi S: An Introduction to Difference Equations. Springer, New York; 2005.

    MATH  Google Scholar 

  3. Agarwal RP, Perera K, O’Regan D: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. 2004, 58: 69-73. 10.1016/j.na.2003.11.012

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonanno G, Candito P: Nonlinear difference equations investigated via critical point methods. Nonlinear Anal. 2009, 70: 3180-3186. 10.1016/j.na.2008.04.021

    Article  MATH  MathSciNet  Google Scholar 

  5. Galewski M, Smejda J: On variational methods for nonlinear difference equations. J. Comput. Appl. Math. 2010, 233: 2985-2993. 10.1016/j.cam.2009.11.044

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu JS, Wang SL, Zhang JM, Zhang FW: Nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups. Nonlinear Anal. TMA 2012, 75: 3809-3820. 10.1016/j.na.2012.02.003

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhu BS, Yu JS: Multiple positive solutions for resonant difference equations. Math. Comput. Model. 2009, 49: 1928-1936. 10.1016/j.mcm.2008.09.009

    Article  MATH  MathSciNet  Google Scholar 

  8. Mawhin J: Periodic solutions of second order nonlinear difference systems with ϕ -Laplacian: a variational approach. Nonlinear Anal. TMA 2012, 75: 4672-4687. 10.1016/j.na.2011.11.018

    Article  MATH  MathSciNet  Google Scholar 

  9. Tan FH, Guo ZM: Periodic solutions for second-order difference equations with resonance at infinity. J. Differ. Equ. Appl. 2012, 18: 149-161. 10.1080/10236191003730498

    Article  MATH  MathSciNet  Google Scholar 

  10. Zhang XS, Wang D: Multiple periodic solutions for difference equations with double resonance at infinity. Adv. Differ. Equ. 2011., 2011: Article ID 806458 10.1155/2011/806458

    Google Scholar 

  11. Liang ZP, Su JB: Multiple solutions for semilinear elliptic boundary value problems with double resonance. J. Math. Anal. Appl. 2009, 354: 147-158. 10.1016/j.jmaa.2008.12.053

    Article  MATH  MathSciNet  Google Scholar 

  12. Robinson S: Double resonance in semilinear elliptic boundary value problem over bounded and unbounded domains. Nonlinear Anal. TMA 1993, 21: 407-424. 10.1016/0362-546X(93)90125-C

    Article  MATH  Google Scholar 

  13. Su JB: Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues. Nonlinear Anal. 2002, 48: 881-895. 10.1016/S0362-546X(00)00221-2

    Article  MATH  MathSciNet  Google Scholar 

  14. Bartsch T, Li SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. TMA 1997, 28: 419-441. 10.1016/0362-546X(95)00167-T

    Article  MATH  MathSciNet  Google Scholar 

  15. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.

    Book  MATH  Google Scholar 

  16. Cerami G: Un criterio di esistenza per i punti critici su varietâ illimitate. Rend. - Ist. Lomb., Accad. Sci. Lett., a Sci. Mat. Fis. Chim. Geol. 1978, 112: 332-336.

    MATH  MathSciNet  Google Scholar 

  17. Chang KC: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston; 1993.

    Book  MATH  Google Scholar 

  18. Rabinowitz PH CBMS 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.

    Google Scholar 

  19. Li SJ, Liu JQ: Nontrivial critical point for asymptotically quadratic functions. J. Math. Anal. Appl. 1992, 165: 333-345. 10.1016/0022-247X(92)90044-E

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful for the referees’ careful reviewing and helpful comments. This work is supported by Beijing Municipal Commission of Education (KZ201310028031, KM2014).

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Zhang, X. Multiplicity results on discrete boundary value problems with double resonance via variational methods. Adv Differ Equ 2013, 309 (2013). https://doi.org/10.1186/1687-1847-2013-309

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