# Multiplicity results for the Kirchhoff type equation via critical groups

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## Abstract

In this paper, we will compute critical groups at zero for the Kirchhoff type equation using the property that critical groups are invariant under homotopies preserving isolatedness of critical points. Using this results, we can get more nontrivial solutions when the functional of this equation is coercive.

## Keywords

Kirchhoff type equations Multiple solutions Morse theory## MSC

35J20 35B34 58E05## 1 Introduction

*∂*Ω, and we study the following Kirchhoff type equation:

- (\(f_{0}\))
- \(f \in\mathcal{C}^{1}(\overline{\Omega} \times\mathbb {R},\mathbb {R})\), \(f(x,0)=0\) and there is \(c>0\) such that$$ \bigl\vert f'(x,u) \bigr\vert \leq c\bigl(1+ \vert u \vert ^{\gamma-2}\bigr), \quad \text{for some } 2\leq\gamma< 2^{*}= \textstyle\begin{cases} +\infty, &N=1,2, \\ \frac{2N}{N-2},&N\geq3, \end{cases} $$

*f*is superlinear, the existence results of solutions can be found in [13, 18, 19, 26], and for the case where the nonlinearity is asymptotically linear, we refer to [4, 15, 24, 26] for details and further references. For example, by using the condition

*I*. Moreover, the authors in [19] assume

*δ*, \(C_{1}\), \(C_{2}\) are positive constants, and they show that the functional

*I*has a local linking at zero.

*f*is superlinear near zero but asymptotically 4-linear at infinity, in [18] one computes the relevant critical groups and obtains nontrivial solutions.

*I*and its applications to the existence and multiplicity results for equation (1.1) by Morse theory. Therefore, we recall the following notions (see [2, 14]). We assume that \(u_{0}\) is an isolated critical point of

*I*,

*U*is an isolated neighborhood of \(u_{0}\), and \(I(u_{0})=c\in\mathbb{R}\), the group

*I*at \(u_{0}\), where \(I^{c}=\{u\in H_{0}^{1}(\Omega): I(u)\leq c\}\), and \(H_{*}(\cdot,\cdot)\) are the singular relative homological groups with a coefficient group \({\mathbb{F}}\).

*f*the following non-resonance and resonance conditions:

- (\(f_{1}\))
- there exists \(\lambda \in\mathbb{R}\) such that$$\lim_{|u|\to0}\frac{f(x,u)}{au}=\lambda ,\quad \text{uniformly in } x \in \Omega ; $$
- (\(f_{2}\))
- there exists \(\alpha>0\) such thatwhere \(g(x,u)=f(x,u)-a\lambda _{1}u\).$$ug(x,u)\leq0, \quad \text{for } |u|\leq\alpha, x\in\Omega, $$

### Theorem 1.1

*Assume that*(\(f_{0}\))

*and*(\(f_{1}\))

*hold*.

*If*\(\lambda \in(\lambda _{k},\lambda _{k+1})\),

*then*\(u=0\)

*is an isolated critical point of*

*I*

*such that*

### Theorem 1.2

*Assume that*(\(f_{0}\)), (\(f_{1}\))

*and*(\(f_{2}\))

*hold*.

*If*\(\lambda =\lambda _{1}\),

*then*\(u=0\)

*is an isolated critical point of*

*I*

*such that*

### Remark 1

Note that, for the semilinear elliptic equation, i.e., \(b=0\), Theorem 1.1 can be found in [2], now we can generalize the same results to Eq. (1.1) with any \(b>0\). However, we cannot directly use the methods in [2], because there are many difficulties to get the critical group estimates for the functional *I*. For example, although we can get a space decomposition according to the eigenfunctions which is the basis of linking theorem by (\(f_{1}\)), the second derivative of *I* in each critical point is complex, so that we are not sure that the generalized Morse splitting lemma can be used. In spite of these difficulties, we can obtain critical groups estimates at zero by using the basic properties of critical groups (see [3]), that is, critical groups are invariant under homotopies preserving isolatedness of critical points.

### Remark 2

Obviously, (\(f_{2}\)) is known as one of the sign conditions in resonance problems. For the results of sign conditions with \(b=0\) we refer to [9, 11, 16, 17] for details and further references.

- (\(f_{3}\))
- there exist \(M>0\) and \(\beta<\frac{a\lambda _{1}}{2}\) such that$$F(x,u)-\frac{b}{4}\mu_{1}|u|^{4} \leq\beta u^{2},\quad \text{for } |u|\geq M, x\in \Omega , $$

### Theorem 1.3

*Assume that*\(N\leq3\), (\(f_{0}\)), (\(f_{1}\)) *and* (\(f_{3}\)) *hold*. *If*\(\lambda \in(\lambda _{k},\lambda _{k+1})\)*with*\(k\geq2\), *then Eq*. (1.1) *has at least three nontrivial solutions*.

### Remark 3

Using similar conditions, the paper [25] has studied the Kirchhoff type equations involving the nonlocal fractional *p*-Laplacian and can get at least two nontrivial solutions by the three-critical point theorem (see [12, Theorem 2.1]). Because of the exact calculations of the critical groups at zero, our theorem can get more nontrivial solutions. Then our result is new.

### Remark 4

*I*with \(b=0\) is coercive ([7]). For Eq. (1.1), because of the existence of Laplacian operator, we can also prove that the functional

*I*with \(b>0\) is coercive with (\(f_{3}\)). For other results of (1.1) we refer to [5, 7, 20, 21, 22, 23, 25] and references therein.

This paper is organized as follows. The proofs of Theorems 1.1–1.3 are given in Sects. 2–4, respectively. In the sequel, we use the letter *C* to denote a suitable positive constant whose value may change from line to line.

## 2 Proof of Theorem 1.1

*I*is a \(C^{2}\) functional with Fréchet derivatives

### Definition 2.1

*I*is said to satisfy the Palais–Smale (for short \((P.S)\)) condition.

### Proposition 2.2

([3])

*Assume*\(\tau\in[0,1]\),

*let*\(\Phi_{\tau}\in C^{1}(H_{0}^{1}(\Omega))\)

*and*

*If*\(U\subset H_{0}^{1}(\Omega)\)

*is a closed neighborhood of*\(u_{0}\)

*such that*

- (i)
\(\Phi_{\tau}\)

*satisfies the*\((P.S)\)*condition in**U**for all*\(\tau\in[0,1]\), - (ii)
\(K(\Phi_{\tau})\cap U=\{u_{0}\}\)

*for all*\(\tau\in[0,1]\), - (iii)
*the mapping*\(\tau\to\Phi_{\tau}\)*is continuous between*\([0,1]\)*and*\(C^{1}(U)\),

*then we have*

### Lemma 2.3

*Assume that* (\(f_{0}\)) *and* (\(f_{1}\)) *hold*. *If*\(\lambda \in(\lambda _{k},\lambda _{k+1})\)*then*\(u=0\)*is an isolated critical point of**I*.

### Proof

*I*. To see that \(u=0\) is isolated, we argue by contradiction: assume that there exists a sequence \((u_{n})\) in \(H_{0}^{1}(\Omega) \setminus \{0\}\) such that

*λ*is an eigenvalue of −Δ with

*v*as an associated eigenfunction, contrary to the assumption \(\lambda \in(\lambda _{k},\lambda _{k+1})\). The proof is completed. □

### Proof of Theorem 1.1

From Lemma 2.3, we know that \(u=0\) is an isolated critical point of *I*. Next, we will use Proposition 2.2 to compute the critical groups of zero.

*U*of 0 such that \(u=0\) is the only critical point of \(J_{s}\) in

*U*for all \(s\in[0,1]\).

Using the methods in the proof of Lemma 2.3, we deduce that \(v_{n}\to v\) in \(H_{0}^{1}(\Omega) \) and \(\|v\|=1\). Passing to the limit in (2.8) again we get a contradiction.

## 3 Proof of Theorem 1.2

Now, we give the proof of Theorem 1.2.

### Lemma 3.1

*Assume that* (\(f_{0}\)), (\(f_{1}\)) *and* (\(f_{2}\)) *hold*. *If*\(\lambda =\lambda _{1}\), *then*\(u=0\)*is an isolated critical point of**I*.

### Proof

*I*. □

*α*is defined in \((f_{2})\). We define a functional \(\Phi_{\tau}\in C^{1}(H_{0}^{1}(\Omega),\mathbb{R})\) by setting (see for example [6, Lemma 4.4])

### Proof of Theorem 1.2

By Lemma 3.1, we know that \(u=0\) is an isolated critical point of *I*.

*n*big enough,

*n*big enough, we get

*I*in Lemma 3.1. Then (3.2) is true.

## 4 Proof of Theorem 1.3

### Lemma 4.1

*If* (\(f_{0}\)) *and* (\(f_{3}\)) *hold*, *then**I**and*\(I_{\pm}\)*satisfy the*\((P.S)\)*condition*.

### Proof

*I*and \(I_{\pm}\) are coercive on \(H_{0}^{1}(\Omega)\). The following method is similar to [7]. For the functional

*I*, by contradiction, there is a sequence \(\{u_{n}\}\subset H_{0}^{1}(\Omega)\) such that

The case of \(I_{+}\) (\(I_{-}\)) is similar. □

Let \(e_{1}>0\) be the eigenfunction associated with \(\lambda _{1}\).

### Lemma 4.2

*If* (\(f_{1}\)) *with*\(k\geq2\)*holds*, *then there exists*\(t>0\)*such that*\(I_{\pm}(\pm te_{1})<0\).

### Proof

### Proof of Theorem 1.3

*I*has a positive critical point \(u_{1}\) and a negative critical point \(u_{2}\) such that

## 5 Conclusions

There are many difficulties if we want to obtain critical groups estimates for the Kirchhoff type equation; for example, we are not sure if the generalized Morse splitting lemma can be used. Then in this paper, by using the basic properties that critical groups are invariant under homotopies preserving the isolatedness of critical points, we can compute critical groups at zero when we impose on *f* the non-resonance and resonance conditions. Moreover, using these critical groups estimates our theorem can get more nontrivial solutions. The main results presented in this paper improve and generalize many results in [4, 19, 25].

## Notes

### Acknowledgements

The authors thank Professor Jiabao Su for many valuable discussions and suggestions.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Funding

This paper is supported by the NSFC (11771302, 11601353, 1174013), the fund of Beijing Education Committee (KM201710009012, 6943), the fund of North China University of Technology (XN018010, XN012).

### Competing interests

The authors declare that they have no competing interests.

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