Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems

By using critical point theory, Lyapunov-Schmidt reduction method, and characterization of the Brouwer degree of critical points, sufficient conditions to guarantee the existence of five or six solutions together with their sign properties to discrete second-order two-point boundary value problem are obtained. An example is also given to demonstrate our main result.

Let , , and denote the sets of all natural numbers, integers, and real numbers, respectively. For , define , when . is the forward difference operator defined by .

Consider the following discrete second-order two-point boundary value problem (BVP for short):

where is a given integer.

By a solution to the BVP (1.1), we mean a real sequence satisfying (1.1). For with , we say that if there exists at least one such that . We say that is positive (and write ) if for all , and  : , and similarly, is negative () if for all , and  : . We say that is sign-changing if is neither positive nor negative. Under convenient assumptions, we will prove the existence of five or six solutions to (1.1), which include positive, negative, and sign-changing solutions.

Difference BVP has widely occurred as the mathematical models describing real-life situations in mathematical physics, finite elasticity, combinatorial analysis, and so forth; for example, see [1, 2]. And many scholars have investigated difference BVP independently mainly for two reasons. The first one is that the behavior of discrete systems is sometimes sharply different from the behavior of the corresponding continuous systems. For example, every solution of logistic equation is monotone, but its discrete analogue has chaotic solutions; see [3] for details. The second one is that there is a fundamental relationship between solutions to continuous systems and the corresponding discrete systems by employing discrete variable methods [1]. The classical results on difference BVP employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point theorems. We remark that, usually, the application of the fixed point theorems yields existence results only.

Recently, however, a few scholars have used critical point theory to deal with the existence of multiple solutions to difference BVP. For example, in 2004, Agarwal et al. [4] employed the mountain pass lemma to study (1.1) with and obtained the existence of multiple solutions. Very recently, Zheng and Zhang [5] obtained the existence of exactly three solutions to (1.1) by making use of three-critical-point theorem and analytic techniques. We also refer to [6–9] for more results on the difference BVP by using critical point theory. The application of critical point theory to difference BVP represents an important advance as it allows to prove multiplicity results as well.

Here, by using critical point theory again, as well as Lyapunov-Schmidt reduction method and degree theory, a sharp condition to guarantee the existence of five or six solutions together with their sign properties to (1.1) is obtained. And this paper offers, to the best of our knowledge, a new method to deal with the sign of solutions in the discrete case.

Here, we assume that and

Hence, grows asymptotically linear at infinity.

The solvability of (1.1) depends on the properties of both at zero and at infinity. If

where is one of the eigenvalues of the eigenvalue problem

then we say that (1.1) is resonant at infinity (or at zero); otherwise, we say that (1.1) is nonresonant at infinity (or at zero). On the eigenvalue problem (1.4), the following results hold (see [1] for details).

Proposition 1.1.

For the eigenvalue problem (1.4), the eigenvalues are

and the corresponding eigenfunctions with are .

Remark 1.2.

(i) The set of functions is orthogonal on with respect to the weight function ; that is,

Moreover, for each .

(ii) It is easy to see that is positive and changes sign for each ; that is,  :  and  :  for .

The main result of this paper is as follows.

Theorem 1.3.

If with , and , then (1.1) has at least five solutions. Moreover, one of the following cases occurs:

(i)   is even and (1.1) has two sign-changing solutions,

(ii)   is even and (1.1) has six solutions, three of which are of the same sign,

(iii)   is odd and (1.1) has two sigh-changing solutions,

(iv)   is odd and (1.1) has three solutions of the same sign.

Remark 1.4.

The assumption in Theorem 1.3 is sharp in the sense that when for , Theorem 1.4 of [5] gives sufficient conditions for (1.1) to have exactly three solutions with some restrictive conditions.

Example 1.5.

Consider the BVP

where is defined as follows:

It is easy to verify that , , and . So, all the conditions in Theorem 1.3 are satisfied with . And hence (1.7) has at least five solutions, among which two sign-changing solutions or three solutions of the same sign.

By the computation of critical groups, for , we have the following.

Corollary 1.6 (see Remark 3.7 below).

If , and , then (1.1) has at least one positive solution and one negative solution.

Let

Then, is a -dimensional Hilbert space with inner product

by which the norm can be induced by

Here, denotes the Euclidean norm in , and denotes the usual inner product in .

Define

Then, the functional is of class with

So, solutions to (1.1) are precisely the critical points of in .

As we have mentioned, we will use critical point theory, Lyapunov-Schmidt reduction method, and degree theory to prove our result. Let us collect some results that will be used below. One can refer to [10–12] for more details.

Let be a Hilbert space and . Denote

for . The following is the definition of the Palais-Smale (PS) compactness condition.

Definition 2.1.

The functional satisfies the (PS) condition if any sequence such that is bounded and as has a convergent subsequence.

In [13], Cerami introduced a weak version of the (PS) condition as follows.

Definition 2.2.

The functional satisfies the Cerami (C) condition if any sequence such that is bounded and , as has a convergent subsequence.

If satisfies the (PS) condition or the (C) condition, then satisfies the following deformation condition which is essential in critical point theory (cf. [14, 15]).

Definition 2.3.

The functional satisfies the () condition at the level if for any and any neighborhood of , there are and a continuous deformation such that

(i)   for all ,

(ii)   for all ,

(iii)    is non-increasing in for any ,

(iv)  .

satisfies the (D) condition if satisfies the () condition for all .

Let denote singular homology with coefficients in a field . If is a critical point of with critical level , then the critical groups of are defined by

Suppose that is strictly bounded from below by and that satisfies () for all . Then, the th critical group at infinity of is defined in [16] as

Due to the condition , these groups are not dependent on the choice of .

Assume that and satisfies the (D) condition. The Morse-type numbers of the pair are defined by , and the Betti numbers of the pair are defined by . By Morse theory [10, 11], the following relations hold:

It follows that for all . If , then for all . Thus, when for some must have a critical point with .

The critical groups of at an isolated critical point describe the local behavior of near , while the critical groups of at infinity describe the global property of . In most applications, unknown critical points will be found from (2.9) or (2.10) if we can compute both the critical groups at known critical points and the critical groups at infinity. Thus, the computation of the critical groups is very important. Now, we collect some useful results on computation of critical groups which will be employed in our discussion.

Proposition 2.4 (see [16]).

Let be a real Hilbert space and . Suppose that splits as such that is bounded from below on and for as . Then for .

Proposition 2.5 (see [17]).

Let be a separable Hilbert space with inner product and corresponding norm , , closed subspaces of such that . Assume that satisfies the (PS) condition and the critical values of are bounded from below. If there is a real number such that for all and , there holds

then there exists a -functional  :  such that

Moreover, if and , then .

Let denote the open ball in about 0 of the radius , and let denote its boundary.

Lemma 2.6 (Mountain Pass Lemma [10, 11]).

Let be a real Banach space and satisfying the (PS) condition. Suppose that and

(J1)  there are constants such that , and

(J2)  there is a such that .

Then, possesses a critical value . Moreover, can be characterized as

where

Definition 2.7 (Mountain pass point).

An isolated critical point of is called a mountain pass point if .

To compute the critical groups of a mountain pass point, we have the following result.

Proposition 2.8 (see [11]).

Let be a real Hilbert space. Suppose that has a mountain pass point and that is a Fredholm operator with finite Morse index satisfying

Then,

The following theorem gives a relation between the Leray-Schauder degree and the critical groups.

Theorem 2.9 (see [10, 11]).

Let be a real Hilbert space, and let be a function satisfying the (PS) condition. Assume that , where  :  is a completely continuous operator. If is an isolated critical point of , that is, there exists a neighborhood of , such that is the only critical point of in , then

where denotes the Leray-Schauder degree.

Finally, we state a global version of the Lyapunov-Schmidt reduction method.

Lemma 2.10 (see [18]).

Let be a real separable Hilbert space. Let and be closed subspaces of such that and . If there are such that for all ,

then the following results hold.

(i)  There exists a continuous function  :  such that

Moreover, is the unique member of such that

(ii)  The function  :  defined by is of class , and

(iii)  An element is a critical point of if and only if is a critical point of .

(iv)  Let and be the projection onto across . Let and be open bounded regions such that

If for , then

where denotes the Leray-Schauder degree.

(v)  If is a critical point of mountain pass type of , then is a critical point of mountain pass type of .

In this section, firstly, we obtain a positive solution and a negative solution with to (1.1) by using cutoff technique and the mountain pass lemma. Then, we give a precise computation of . And we remark that under the assumptions of Theorem 1.3, can be completely computed by using Propositions 2.4 and 2.5. Based on these results, four nontrivial solutions to (1.1) can be obtained by (2.9) or (2.10). However, it seems difficult to obtain the sign property of and through their depiction of critical groups. To conquer this difficulty, we compute the Brouwer degree of the sets of positive solutions and negative solutions to (1.1). Finally, the third nontrivial solution to (1.1) is obtained by Lyapunov-Schmidt reduction method, and its characterization of the local degree results in one or two more nontrivial solutions to (1.1) together with their sign property.

Let

and . The functionals are defined as

Remark 3.1.

From the definitions of and , it is easy to see that if is a critical point of (or ), then (or ).

Lemma 3.2.

The functionals satisfy the (PS) condition; that is, every sequence in such that is bounded, and as has a convergent subsequence.

Proof.

We only prove the case of . The case of is completely similar. Since is finite dimensional, it suffices to show that is bounded. Suppose that is unbounded. Passing to a subsequence, we may assume that and for each , either or is bounded.

Set . For a subsequence, converges to some with . Since for all , we have

Hence,

If , then

where , . If is bounded, then

Letting in (3.4), we have

which implies that satisfies

Because , we see that if is a solution to (3.8), then is positive. Since this contradicts , we conclude that is the only solution to (3.8). A contradiction to .

Lemma 3.3.

Under the conditions of Theorem 1.3, has a positive mountain pass-type critical point with ; has a negative mountain pass-type critical point with .

Proof.

We only prove the case of . Firstly, we will prove that satisfies all the conditions in Lemma 2.6. And hence, has at least one nonzero critical point . In fact, , and satisfies the (PS) condition by Lemma 3.2. Clearly, . Thus, we still have to show that satisfies (J1), (J2). To verify (J1), set , then for any , there exists , such that

So, by Taylor series expansion,

Take , then . If we set , then

Since for all , if , then for every and hence

where , . If we take

then . And hence, (J1) holds.

To verify (J2), note that implies that there exist and , such that

So, if we take with , then

So, if we take sufficiently large such that and for , then (J2) holds.

Now, by Lemma 2.6, has at least a nonzero critical point . And for all , we claim that . If not, set , then for all . By for all . Hence, .

In the following, we will compute the critical groups by using Proposition 2.8.

Assume that

and that there exists such that

This implies that satisfies

Hence, the eigenvalue problem

has an eigenvalue . Condition implies that 1 must be a simple eigenvalue; see [1]. So, . Since is finite dimensional, the Morse index of must be finite and must be a Fredholm operator. By Proposition 2.8, . Finally, choose the neighborhood of such that for all , then

The proof is complete.

Lemma 3.4.

By , one has

Proof.

By assumption, we have and for all ,

which implies that 0 is a local minimizer of both and . Hence, (3.21) holds.

Remark 3.5.

Under the conditions of Theorem 1.3, we have

We will use Propositions 2.4 and 2.5 to prove (3.23). Very similar to the proof of Lemma 3.2, we can prove that satisfies the (PS) condition. And it is easy to prove that satisfies (2.11). In fact, let

By , for all and , we have

Hence, if we set , then (2.11) holds.

Now, noticing that implies that there exist , and such that

Hence, we have

Then, (3.23) is proved by Propositions 2.4 and 2.5.

Remark 3.6.

Following the proof of Theorem 3.1 in [17], (3.23) implies that there must exist a critical point of satisfying

It is known that the critical groups are useful in distinguishing critical points. So far, we have obtained four critical points 0, , , and together with their characterization of critical groups. Assume that 0, , , and are the only critical points of . Then, the Morse inequality (2.10) becomes

This is impossible. Thus, must have at least one more critical point . Hence, (1.1) has at least five solutions. However, it seems difficult to obtain the sign property of and . To obtain more refined results, we seek the third nontrivial solution to (1.1) by Lyapunov-Schmidt reduction method and then its characterization of the local degree results in one or two more nontrivial solutions to (1.1) together with their sign property.

Remark 3.7.

The condition in Theorem 1.3 is necessary to obtain three or more nontrivial solutions to (1.1). In fact, if , then we have

Hence, may coincide with or which becomes an obstacle to seek other critical points by using Morse inequality. If , then

Hence, one cannot exclude the possibility of .

To compute the degree of the set of positive (or negative) solutions to (1.1), we need the following lemma.

Lemma 3.8.

There exists large enough, such that

Proof.

We only prove the case of . For any , define as

Let . The functional  :  is defined as

It is obvious that is of class and its critical points are precisely solutions to

Since , we see that if is a solution to (3.36), then is positive. Because this contradicts , we conclude that is the only critical point of .

We claim that if is a ball in containing zero, then . In fact, since and for . Hence, for , we have

where we have used the fact that is positive on . Then, for each and , we have

Hence, by invariance under homotopy of Brouwer degree, we have

where .

Now, let . We claim that for large enough and for all , the function has no zero on .

In fact, we have proved that for all and for all , we have

On the other hand, by the definition of , for all , there exists large enough such that for . Since , for , take , then

For , take , then

Hence, if we take , then for , we have , and for , we have . So, if we let

then for all . And for all , we have

So far, we have proved that for large enough, has no zero point on for each . Hence, by invariance under homotopy of Brouwer degree, we obtain

This completes the proof.

Remark 3.9.

By Theorem 2.9 and the above results, we have the following characterization of degree of critical points.

  If () is a neighborhood of () containing no other critical points, then

  Assume that is a ball centered at zero containing on other critical points, then

Hence, if is a bounded region containing the positive critical points and no other critical points, then by (3.33) we have

Similarly, we see that if is a bounded region containing the negative critical points and no other critical points, then

Now, we can give the proof of Theorem 1.3.

Proof of Theorem 1.3.

The functional satisfies (2.18) in Lemma 2.10 due to the fact that satisfies (2.11). Hence, by Lemma 2.10, there exists  :  such that

Moreover, is the unique member of such that

The function  :  defined by is of class . Because , (3.27) implies that as . Since , there must exist such that . Take , then by (iii) of Lemma 2.10. If is a neighborhood of containing no other critical points of , taking , then . Then, by part (iv) of Lemma 2.10, we have

Suppose that Is Even

Let be large enough so that if , then . Because and is of class , there exists such that for . Because is coercive, . Hence, if we set  : , then by (iv) of Lemma 2.10, we have

Suppose that is finite. Let , , and be disjoint open bounded regions in such that is the set of positive critical points of , and is the set of negative critical points of . So far, we have proved that

(i) If , then is sign changing. Let denote an open bounded region disjoint from such that . By the excision property of Brouwer degree, we have

Thus, by Kronecker existence property of Brouwer degree, we see that there must exist such that , which proves that (1.1) has at least five solutions. In this case, both and change sign.

(ii) Suppose now that . Without loss of generality, we may assume that . Let be a neighborhood of such that . By Lemma 3.3, there exists a critical point of mountain pass type such that if is a neighborhood of such that , then . Thus,

Thus, by Kronecker existence property of Brouwer degree, there exists such that . Finally,

Thus, there must exist such that . Thus, the set together with a critical point of in shows that (1.1) has five nontrivial solutions. Since and , is a sign-changing solution, and , , and have the same sign. This completes the proof of Theorem 1.3, when is even.

Suppose that Is Odd

(iii) Let , , and be as above. If , the proof follows very closely that of the case (i).

(iv) Suppose that , hence . Because , there exists such that if . So, if , then and is a local maximum of . Since we are assuming (1.1) to have only finitely many solutions, is a strictly local maximum of . Let be such that if . Since , is path connected. Thus, is not a critical point of mountain pass type. By Lemma 3.3, has a critical point of mountain pass type . By (v) of Lemma 2.10, , and hence . Let be neighborhoods of and , respectively, such that and . Thus,

Thus, by Kronecker existence property of Brouwer degree, there exists a third positive solution . So far, we have proved that (1.1) has at least four nontrivial solutions and that have the same sign. This proves Theorem 1.3.

Project supported by National Natural Science Foundation of China (no. 11026059) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. LYM09105).

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