- Open Access
Nontrivial solutions for discrete Kirchhoff-type problems with resonance via critical groups
© Yang and Liu; licensee Springer. 2013
- Received: 25 May 2013
- Accepted: 14 October 2013
- Published: 8 November 2013
In this paper, we study the existence of nontrivial solutions for discrete Kirchhoff-type problems with resonance at both zero and infinity by using variational methods and the computations of critical groups.
- discrete Kirchhoff-type problem
- variational method
- critical point
- critical group
for all .
and let be an eigenvector corresponding to the eigenvalue , then is an orthogonal basis of , where , , .
(see Theorem 2.1).
Remark 1.1 By referring to the notion of resonance of elliptic equations of Kirchhoff-type [1, 2], we call problem (1.1) resonant at zero if there exists some such that and resonant at infinity if there exists some eigenvalue μ of problem (1.4) such that .
and make the following assumptions:
(F0) There exists such that , , .
() , .
() There exist and such that , , .
Firstly, we consider the case that problem (1.1) is only resonant at zero.
Next, we investigate the case that problem (1.1) is resonant at both zero and infinity.
() and ;
(), and .
Thirdly, we study the case that problem (1.1) is only resonant at infinity.
(), and ;
(), and .
Finally, we deal with the case that problem (1.1) is non-resonant.
Now, we give some examples to illustrate our main results.
where is a smooth connection such that , it is easy to know that , and the condition (F0) holds, so problem (1.1) has at least a nontrivial solution by (i) of Theorem 1.1.
where is a smooth connection such that , it is easy to know that , , and the condition (F0) holds, so problem (1.1) has at least a nontrivial solution by (ii) of Theorem 1.1.
for all . They investigated the existence and multiplicity of nontrivial solutions for problem (1.6) by using various methods and techniques, such as minimax methods, bifurcation theory, critical groups, Morse theory and so on. However, to our knowledge, there are few results on the existence of nontrivial solutions of a discrete Kirchhoff-type resonance problem which is an extension of problem (1.6).
proposed by Kirchhoff  as an extension of the classical D’Alembert wave equation for free vibrations of elastic stings. After the famous article by Lions , this type of problems has been the subject of numerous studies. Due to the importance of equation (1.6), in recent years, many authors have studied the existence of solutions of equation (1.6) and the corresponding general elliptic equations with Dirichlet boundary value condition (see, e.g., [1, 2, 15–18] and the references therein). Obviously, problem (1.1) is the discrete form of equation (1.6). However, there are significant differences between (1.1) and (1.6) in some aspects such as properties of eigenvalues (see Theorem 2.1 and Proposition 3.2 in ), which justifies the necessity of research on problem (1.1).
In the current paper, we conclude the existence of eigenvalues for nonlinear eigenvalue problem (1.4) via the Lagrange multiplier rule. This appears to be first such result for eigenvalue problem (1.4). Furthermore, the existence of nontrivial solutions of discrete Kirchhoff-type problem (1.1) with resonance at both zero and infinity is also studied by employing the critical point theory, especially the local linking, Morse theory and the computations of critical groups.
The rest of this paper is organized as follows. In Section 2, we give the energy functional of problem (1.1) and study the eigenvalue of problem (1.4). In order to prove our main results, some facts about the critical groups are also recalled in this section. In Section 3, the proofs of main results are provided.
Hence the solutions of problem (1.1) are exactly the critical points of J in H and .
Now we consider nonlinear eigenvalue problem (1.4). Firstly, we introduce the Lagrange multiplier rule.
Lemma 2.1 
Let X, Y be real Banach spaces, , and continuously differentiable, and closed. Suppose also that . Then there exist Lagrange multipliers and , not all zero, such that . If , then .
Proof Firstly, we prove the existence of and .
Now, we claim that the number of eigenvalues of problem (1.4) is finite.
Hence the number of eigenvalues of problem (1.4) is finite due to the finiteness of . Besides, we can obtain by (2.5). This proof is completed. □
Definition 2.1 
is called the q th critical group of J at , where , denotes the q th singular relative homology group of the topological pair with coefficients in a field .
Definition 2.2 
Let H be a Hilbert space and , . Assume that is the supremum of the vector subspaces of H on which is negative definite. The Morse index of J at is defined as the dimension of . The nullity of J at is defined as the dimension of . is called a non-degenerate critical point of J if has a bounded inverse.
Then , if or .
Lemma 2.3 
If is a minimum point of J, then ;
If is a maximum point of J and , then ;
If is a non-degenerate critical point of J with the Morse index , then .
Now we give the proofs of Theorems 1.1-1.4.
Noting that , we can choose ε small enough such that for and .
- (i)By (1.3), and the continuity of F, we know that for any , there exists such that for all and . Together with (2.4), we have
- (ii)It follows from (1.3), and the continuity of F that we can find , such that for all and . By (2.1), (2.2) and (2.4), we can see that
By (3.2), (3.4) and , it is easy to see that . Therefore J has at least a nontrivial critical point, which completes the proof. □
- (ii)One concludes that there exists such that for all by (1.3), , () and the continuity of F. Hence, we infer from (2.1) and (2.4) that
From (3.2), (3.6) and , we get . So, J has at least a nontrivial critical point. The proof is completed. □
Equation (1.3), and () mean that (3.5) holds. Thus, by (3.5), (3.7) and , it is easily seen that , which implies that J has at least a nontrivial critical point.
It follows from (1.3), and () that (3.6) holds. Therefore by (3.6), (3.7) and , we have . This shows that J admits at least a nontrivial critical point. Then the conclusion holds. □
Equation (1.3) and show that (3.3) holds. Hence, by (3.3), (3.7) and , we have . This implies that J has at least a nontrivial critical point.
We can deduce (3.4) from (1.3) and . Therefore we can obtain from (3.4), (3.7) and . This implies that J has at least a nontrivial critical point. This proof is completed. □
We are grateful to the referees for their suggestions that helped improve the paper greatly. In addition, this work is supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-3).
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