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Multiplicity results for the Kirchhoff type equation via critical groups
 Zhenting Wang^{1},
 Mingzheng Sun^{1}Email authorView ORCID ID profile,
 Yutong Chen^{2} and
 Leiga Zhao^{3}
 Received: 4 September 2018
 Accepted: 3 December 2018
 Published: 6 December 2018
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
 (\(f_{0}\)):

\(f \in\mathcal{C}^{1}(\overline{\Omega} \times\mathbb {R},\mathbb {R})\), \(f(x,0)=0\) and there is \(c>0\) such that$$ \bigl\vert f'(x,u) \bigr\vert \leq c\bigl(1+ \vert u \vert ^{\gamma2}\bigr), \quad \text{for some } 2\leq\gamma< 2^{*}= \textstyle\begin{cases} +\infty, &N=1,2, \\ \frac{2N}{N2},&N\geq3, \end{cases} $$
 (\(f_{1}\)):

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
Theorem 1.2
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 pLaplacian and can get at least two nontrivial solutions by the threecritical 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
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
Definition 2.1
Proposition 2.2
([3])
 (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 \(C^{1}(U)\),
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
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.
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
Proof of Theorem 1.2
By Lemma 3.1, we know that \(u=0\) is an isolated critical point of I.
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
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
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 nonresonance 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].
Declarations
Acknowledgements
The authors thank Professor Jiabao Su for many valuable discussions and suggestions.
Availability of data and materials
Not applicable.
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).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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