# Nontrivial solutions for discrete Kirchhoff-type problems with resonance via critical groups

- Jinping Yang
^{1}and - Jinsheng Liu
^{1}Email author

**2013**:308

https://doi.org/10.1186/1687-1847-2013-308

© Yang and Liu; licensee Springer. 2013

**Received: **25 May 2013

**Accepted: **14 October 2013

**Published: **8 November 2013

## Abstract

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.

## Keywords

## 1 Introduction and main results

*f*near zero and near infinity. In this work, we consider the cases where

*f*satisfies the asymptotic conditions

for all $k\in \mathbb{Z}[1,N]$.

and let ${\xi}_{i}={({\xi}_{i}(1),{\xi}_{i}(2),\dots ,{\xi}_{i}(N))}^{T}$ be an eigenvector corresponding to the eigenvalue ${\lambda}_{i}$, then $\{{\xi}_{1},{\xi}_{2},\dots ,{\xi}_{N}\}$ is an orthogonal basis of ${\mathbb{R}}^{N}$, where ${\lambda}_{i}=4{sin}^{2}\frac{i\pi}{2(N+1)}$, ${\xi}_{i}(j)=sin\frac{ij\pi}{N+1}$, $i,j\in \mathbb{Z}[1,N]$.

(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 $m\in \mathbb{Z}[1,N]$ such that ${f}_{0}={\lambda}_{m}$ and resonant at infinity if there exists some eigenvalue *μ* of problem (1.4) such that ${f}_{\mathrm{\infty}}=\mu $.

and make the following assumptions:

(F_{0}) There exists ${\delta}_{1}>0$ such that $F(k,t)\ge \frac{a}{2}{\lambda}_{m}{t}^{2}+\frac{b}{4}m{\lambda}_{m}^{2}{t}^{4}$, $k\in \mathbb{Z}[1,N]$, $|t|\le {\delta}_{1}$.

(${\text{F}}_{\mathrm{\infty}}^{1}$) ${lim}_{|t|\to \mathrm{\infty}}(tf(k,t)-4F(k,t))=+\mathrm{\infty}$, $k\in \mathbb{Z}[1,N]$.

(${\text{F}}_{\mathrm{\infty}}^{2}$) There exist ${\delta}_{2}>0$ and $c\in (\frac{a}{2}{\lambda}_{N},+\mathrm{\infty})$ such that $F(k,t)\ge c{t}^{2}+\frac{b}{4}{f}_{\mathrm{\infty}}{t}^{4}$, $k\in \mathbb{Z}[1,N]$, $|t|\ge {\delta}_{2}$.

Firstly, we consider the case that problem (1.1) is only resonant at zero.

**Theorem 1.1**

*If*${f}_{0}={\lambda}_{m}$

*and*(F

_{0})

*hold*,

*then problem*(1.1)

*has at least one nontrivial solution in each of the following cases*:

- (i)
${f}_{\mathrm{\infty}}<{\mu}_{1}$;

- (ii)
${f}_{\mathrm{\infty}}>{\mu}_{2}$

*and*$m\ne N$.

Next, we investigate the case that problem (1.1) is resonant at both zero and infinity.

**Theorem 1.2**

*If*${f}_{0}={\lambda}_{m}$

*and*(F

_{0})

*hold*,

*then problem*(1.1)

*has at least one nontrivial solution in each of the following cases*:

- (i)
(${\text{F}}_{\mathrm{\infty}}^{1}$)

*and*${f}_{\mathrm{\infty}}={\mu}_{1}$; - (ii)
(${\text{F}}_{\mathrm{\infty}}^{2}$), ${f}_{\mathrm{\infty}}={\mu}_{2}$

*and*$m\ne N$.

Thirdly, we study the case that problem (1.1) is only resonant at infinity.

**Theorem 1.3**

*If*${f}_{0}\ne {\lambda}_{i}$, $i\in \mathbb{Z}[1,N]$

*hold*,

*then problem*(1.1)

*has at least one nontrivial solution in each of the following cases*:

- (i)
(${\text{F}}_{\mathrm{\infty}}^{1}$), ${f}_{\mathrm{\infty}}={\mu}_{1}$

*and*${\mu}_{0}\ne 0$; - (ii)
(${\text{F}}_{\mathrm{\infty}}^{2}$), ${f}_{\mathrm{\infty}}={\mu}_{2}$

*and*${\mu}_{0}\ne N$.

Finally, we deal with the case that problem (1.1) is non-resonant.

**Theorem 1.4**

*If*${f}_{0}\ne {\lambda}_{i}$, $i\in \mathbb{Z}[1,N]$

*hold*,

*then problem*(1.1)

*has at least one nontrivial solution in each of the following cases*:

- (i)
${f}_{\mathrm{\infty}}<{\mu}_{1}$

*and*${\mu}_{0}\ne 0$; - (ii)
${f}_{\mathrm{\infty}}>{\mu}_{2}$

*and*${\mu}_{0}\ne N$.

Now, we give some examples to illustrate our main results.

**Example 1.1**If

where ${f}_{1}$ is a smooth connection such that $f\in {C}^{1}(\mathbb{Z}[1,N]\times \mathbb{R},\mathbb{R})$, it is easy to know that ${f}_{0}={\lambda}_{1}$, ${f}_{\mathrm{\infty}}=\frac{1}{2}{\mu}_{1}<{\mu}_{1}$ and the condition (F_{0}) holds, so problem (1.1) has at least a nontrivial solution by (i) of Theorem 1.1.

**Example 1.2**If

where ${f}_{2}$ is a smooth connection such that $f\in {C}^{1}(\mathbb{Z}[1,N]\times \mathbb{R},\mathbb{R})$, it is easy to know that ${f}_{0}={\lambda}_{1}$, ${f}_{\mathrm{\infty}}=2{\mu}_{2}>{\mu}_{1}$, $N\ne 1$ and the condition (F_{0}) holds, so problem (1.1) has at least a nontrivial solution by (ii) of Theorem 1.1.

*f*satisfies the resonance condition

for all $k\in \mathbb{Z}[1,N]$. 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 [13] as an extension of the classical D’Alembert wave equation for free vibrations of elastic stings. After the famous article by Lions [14], 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 [1]), 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.

## 2 Preliminaries

*N*-dimensional real Hilbert space with the inner product $\u3008u,v\u3009={u}^{T}Av$ and the norm $\parallel u\parallel =\sqrt{\u3008u,u\u3009}$. For $1\le p<+\mathrm{\infty}$, we denote ${\parallel u\parallel}_{p}={({\sum}_{k=1}^{N}{|u(k)|}^{p})}^{\frac{1}{p}}$. Then, for all $u\in H$, we have

*J*is a functional with Fréchet derivatives given by

Hence the solutions of problem (1.1) are exactly the critical points of *J* in *H* and $J\in {C}^{2}(H,\mathbb{R})$.

Now we consider nonlinear eigenvalue problem (1.4). Firstly, we introduce the Lagrange multiplier rule.

**Lemma 2.1** [19]

*Let* *X*, *Y* *be real Banach spaces*, ${B}_{r}({x}_{0})\subset X$, $\phi :{B}_{r}({x}_{0})\to \mathbb{R}$ *and* $F:{B}_{r}({x}_{0})\to Y$ *continuously differentiable*, $F({x}_{0})=0$ *and* $R({F}^{\prime}({x}_{0}))$ *closed*. *Suppose also that* $\phi ({x}_{0})=min\{\phi (x):x\in {B}_{r}({x}_{0})\text{and}F(x)=0\}$. *Then there exist Lagrange multipliers* $\lambda \in \mathbb{R}$ *and* ${y}^{\ast}\in {Y}^{\ast}$, *not all zero*, *such that* $\lambda \phi ({x}_{0})+{({F}^{\prime}({x}_{0}))}^{\ast}{y}^{\ast}=0$. *If* $R({F}^{\prime}({x}_{0}))=Y$, *then* $\lambda \ne 0$.

**Theorem 2.1**

*Problem*(1.4)

*only has finitely many eigenvalues which are in*$[{\lambda}_{1}^{2},N{\lambda}_{N}^{2}]$,

*and moreover*,

*Proof* Firstly, we prove the existence of ${\mu}_{1}$ and ${\mu}_{2}$.

*u*is a nonzero solution of problem (1.4) corresponding to some eigenvalue

*μ*, then

*i.e.*,

*μ*is also an eigenvalue of problem (1.4), then there exists $u\ne 0$ on the sphere ${\parallel u\parallel}_{4}^{4}=1$ such that

Now, we claim that the number of eigenvalues of problem (1.4) is finite.

*u*is a solution of the equation ${\parallel u\parallel}^{2}Au=\mu {u}^{3}$ for some

*μ*, so is

*cu*. Thus we can suppose that ${\parallel u\parallel}^{2}=1$. Hence, we only need to consider the following equations:

Hence the number of eigenvalues of problem (1.4) is finite due to the finiteness of ${c}_{2},{c}_{3},\dots ,{c}_{N}$. Besides, we can obtain $u(2),u(3),\dots ,u(N)$ by (2.5). This proof is completed. □

In the following we recall some facts about the critical groups and Morse theory; see [20–22] for more details.

**Definition 2.1** [20]

*H*be a Hilbert space and $J\in {C}^{1}(H,\mathbb{R})$, let ${u}_{0}$ be an isolated critical point of

*J*with $J({u}_{0})=c$, and let

*U*be a neighbourhood of ${u}_{0}$, containing the unique critical point, the group

is called the *q* th critical group of *J* at ${u}_{0}$, where ${J}^{c}=\{u\in H|J(u)\le c\}$, ${H}_{q}(A,B)$ denotes the *q* th singular relative homology group of the topological pair $(A,B)$ with coefficients in a field $\mathbb{F}$.

**Definition 2.2** [20]

Let *H* be a Hilbert space and $J\in {C}^{2}(H,\mathbb{R})$, ${J}^{\prime}({u}_{0})=0$. Assume that ${H}^{-}$ is the supremum of the vector subspaces of *H* on which ${J}^{\u2033}({u}_{0})$ is negative definite. The Morse index of *J* at ${u}_{0}$ is defined as the dimension of ${H}^{-}$. The nullity of *J* at ${u}_{0}$ is defined as the dimension of $ker{J}^{\u2033}({u}_{0})$. ${u}_{0}$ is called a non-degenerate critical point of *J* if ${J}^{\u2033}({u}_{0})$ has a bounded inverse.

*Let*0

*be an isolated critical point of*$J\in {C}^{2}(H,\mathbb{R})$

*with the Morse index*${\mu}_{0}$

*and the nullity*${\nu}_{0}$.

*Assume that*

*J*

*has a local linking at*0

*with respect to*$H={H}^{-}\oplus {H}^{+}$, $l=dim{H}^{-}<\mathrm{\infty}$,

*i*.

*e*.,

*there exists*$\rho >0$

*such that*

*Then* ${C}_{q}(J,0)={\delta}_{q,l}\mathbb{F}$, *if* $l={\mu}_{0}$ *or* $l={\mu}_{0}+{\nu}_{0}$.

**Lemma 2.3** [20]

*Let*${u}_{0}$

*be an isolated critical point of*$J\in {C}^{2}(H,\mathbb{R})$,

*then the following statements are true*:

- (i)
*If*${u}_{0}$*is a minimum point of**J*,*then*${C}_{q}(J,{u}_{0})={\delta}_{q,0}\mathbb{F}$; - (ii)
*If*${u}_{0}$*is a maximum point of**J**and*$dimH=l<\mathrm{\infty}$,*then*${C}_{q}(J,{u}_{0})={\delta}_{q,l}\mathbb{F}$; - (iii)
*If*${u}_{0}$*is a non*-*degenerate critical point of**J**with the Morse index*${\mu}_{0}$,*then*${C}_{q}(J,{u}_{0})={\delta}_{q,{\mu}_{0}}\mathbb{F}$.

## 3 Proofs of main results

Now we give the proofs of Theorems 1.1-1.4.

*Proof of Theorem 1.1*Since (1.2), ${f}_{0}={\lambda}_{m}$ and (F

_{0}) hold, for any $\epsilon >0$, there exists $\rho >0$ such that $\frac{\rho}{\sqrt{{\lambda}_{1}}}={\delta}_{1}\le \delta $ and

Noting that ${\lambda}_{m}<{\lambda}_{m+1}$, we can choose *ε* small enough such that $J(u)\ge 0$ for $\parallel u\parallel \le \rho $ and $u\in {H}_{0}^{+}$.

*J*has a local linking at 0 with respect to $H={H}_{0}^{-}\oplus {H}_{0}^{+}$. Using Lemma 2.2, we can obtain

- (i)By (1.3), ${f}_{\mathrm{\infty}}<{\mu}_{1}$ and the continuity of
*F*, we know that for any $\epsilon >0$, there exists ${M}_{\epsilon}>0$ such that $F(k,t)\le \frac{1}{4}b({\mu}_{1}-\epsilon ){t}^{4}+{M}_{\epsilon}$ for all $t\in \mathbb{R}$ and $k\in \mathbb{Z}[1,N]$. Together with (2.4), we have$\begin{array}{rl}J(u)& =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}F(k,u(k))\\ \ge \frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}(\frac{1}{4}b({\mu}_{1}-\epsilon ){u}^{4}(k)+{M}_{\epsilon})\\ =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\frac{1}{4}b({\mu}_{1}-\epsilon ){\parallel u\parallel}_{4}^{4}-N{M}_{\epsilon}\\ \ge \frac{a}{2}{\parallel u\parallel}^{2}-N{M}_{\epsilon}.\end{array}$

*J*is coercive in

*H*. By the continuity of

*J*,

*J*must have a minimum point ${u}_{1}$. According to Lemma 2.3(i), we conclude that

*J*has at least a nontrivial critical point.

- (ii)It follows from (1.3), ${f}_{\mathrm{\infty}}>{\mu}_{2}$ and the continuity of
*F*that we can find $\epsilon >0$, ${M}_{\epsilon}>0$ such that $F(k,t)\ge \frac{b}{4}({\mu}_{2}+\epsilon ){t}^{4}-{M}_{\epsilon}$ for all $t\in \mathbb{R}$ and $k\in \mathbb{Z}[1,N]$. By (2.1), (2.2) and (2.4), we can see that$\begin{array}{rl}J(u)& =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}F(k,u(k))\\ \le \frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}(\frac{b}{4}({\mu}_{2}+\epsilon ){u}^{4}(k)-{M}_{\epsilon})\\ \le \frac{a}{2}{\parallel u\parallel}^{2}-\frac{b\epsilon}{4}{\parallel u\parallel}_{4}^{4}+N{M}_{\epsilon}\\ \le \frac{a}{2}{\parallel u\parallel}^{2}-\frac{b\epsilon}{4N{\lambda}_{N}^{2}}{\parallel u\parallel}^{4}+N{M}_{\epsilon},\end{array}$

*J*is inverse coercive in

*H*. Because

*J*is continuous,

*J*must have a maximum point ${u}_{2}$. By Lemma 2.3(ii), we can conclude that

By (3.2), (3.4) and $m\ne N$, it is easy to see that ${u}_{2}\ne 0$. Therefore *J* has at least a nontrivial critical point, which completes the proof. □

*Proof of Theorem 1.2*We have (3.2) from (1.2), ${f}_{0}={\lambda}_{m}$ and (F

_{0}).

- (i)Let $G(k,t)=F(k,t)-\frac{b}{4}{\mu}_{1}{t}^{4}$, then ${lim}_{|t|\to +\mathrm{\infty}}\frac{G(k,t)}{{|t|}^{4}}=0$ by (1.3) and ${f}_{\mathrm{\infty}}={\mu}_{1}$. Meanwhile, combining (${\text{F}}_{\mathrm{\infty}}^{1}$), we can get ${lim}_{|t|\to \mathrm{\infty}}G(k,t)=-\mathrm{\infty}$ [9, 18]. This means that ${lim}_{\parallel u\parallel \to \mathrm{\infty}}{\sum}_{k=1}^{N}G(k,u(k))=-\mathrm{\infty}$. Then$\begin{array}{rl}J(u)& =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}F(k,u(k))\\ \ge \frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\frac{b}{4}{\mu}_{1}{\parallel u\parallel}_{4}^{4}-\sum _{k=1}^{N}G(k,u(k))\\ \ge \frac{a}{2}{\parallel u\parallel}^{2}-\sum _{k=1}^{N}G(k,u(k)),\end{array}$

*J*is coercive. Since

*J*is continuous,

*J*must have a minimum point ${u}_{1}$. Thus we obtain

*J*has at least a nontrivial critical point.

- (ii)One concludes that there exists $M>0$ such that $F(k,t)\ge c{t}^{2}+\frac{b}{4}{\mu}_{2}{t}^{4}-M$ for all $t\in \mathbb{R}$ by (1.3), ${f}_{\mathrm{\infty}}={\mu}_{2}$, (${\text{F}}_{\mathrm{\infty}}^{2}$) and the continuity of
*F*. Hence, we infer from (2.1) and (2.4) that$\begin{array}{rl}J(u)& =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}F(k,u(k))\\ \le \frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\sum _{k=1}^{N}(\frac{b}{4}{\mu}_{2}{u}^{4}(k)+c{u}^{2}(k)-M)\\ =\frac{a}{2}{\parallel u\parallel}^{2}+\frac{b}{4}{\parallel u\parallel}^{4}-\frac{b}{4}{\mu}_{2}{\parallel u\parallel}_{4}^{4}-c{\parallel u\parallel}_{2}^{2}+NM\\ \le (\frac{a}{2}-\frac{c}{{\lambda}_{N}}){\parallel u\parallel}^{2}+NM.\end{array}$

*J*is inverse coercive in

*H*. Due to the continuity of

*J*,

*J*must have a maximum point ${u}_{2}$. Therefore we have

From (3.2), (3.6) and $m\ne N$, we get ${u}_{2}\ne 0$. So, *J* has at least a nontrivial critical point. The proof is completed. □

*Proof of Theorem 1.3*It follows from (1.2), ${f}_{0}\ne {\lambda}_{i}$, $i\in \mathbb{Z}[1,N]$, and ${J}^{\u2033}(0)=a{I}_{N}-{A}^{-1}{f}_{t}^{\prime}(k,0)$, $k\in \mathbb{Z}[1,N]$ that $u=0$ is a non-degenerate critical point of

*J*with the Morse index ${\mu}_{0}$. By Lemma 2.3(iii), we know that

- (i)
Equation (1.3), ${f}_{\mathrm{\infty}}={\mu}_{1}$ and (${\text{F}}_{\mathrm{\infty}}^{1}$) mean that (3.5) holds. Thus, by (3.5), (3.7) and ${\mu}_{0}\ne 0$, it is easily seen that ${u}_{1}\ne 0$, which implies that

*J*has at least a nontrivial critical point. - (ii)
It follows from (1.3), ${f}_{\mathrm{\infty}}={\mu}_{2}$ and (${\text{F}}_{\mathrm{\infty}}^{2}$) that (3.6) holds. Therefore by (3.6), (3.7) and ${\mu}_{0}\ne N$, we have ${u}_{2}\ne 0$. This shows that

*J*admits at least a nontrivial critical point. Then the conclusion holds. □

*Proof of Theorem 1.4*It is easy to verify from (1.2) and ${f}_{0}\ne {\lambda}_{i}$, $i\in \mathbb{Z}[1,N]$ that (3.7) holds.

- (i)
Equation (1.3) and ${f}_{\mathrm{\infty}}<{\mu}_{1}$ show that (3.3) holds. Hence, by (3.3), (3.7) and ${\mu}_{0}\ne 0$, we have ${u}_{1}\ne 0$. This implies that

*J*has at least a nontrivial critical point. - (ii)
We can deduce (3.4) from (1.3) and ${f}_{\mathrm{\infty}}>{\mu}_{2}$. Therefore we can obtain ${u}_{2}\ne 0$ from (3.4), (3.7) and ${\mu}_{0}\ne N$. This implies that

*J*has at least a nontrivial critical point. This proof is completed. □

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

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