- Research
- Open Access
Existence of nontrivial solutions for a class of nonlocal Kirchhoff type problems
- Zeng-Qi Ou^{1}Email author and
- Chun Li^{1}
- Received: 11 April 2018
- Accepted: 11 October 2018
- Published: 22 October 2018
Abstract
With the aid of the three-critical-point theorem due to Brezis and Nirenberg (see Brezis and Nirenberg in Commun. Pure Appl. Math. 44:939–963, 1991), two existence results of at least two nontrivial solutions for a class of nonlocal Kirchhoff type problems are obtained.
Keywords
- Kirchhoff type problem
- \((PS)\) condition
- Local linking
- Critical point
MSC
- 35D30
- 35J50
- 35J92
1 Introduction and main results
Theorem 1
- (i)
\(\mu_{1}(m_{\infty})>1\) or
- (ii)
\(\mu_{1}(m_{\infty})=1\) and (7) hold.
Theorem 2
Remark
If \(N=1, 2, 3\) and the nonlinearity f is 3-suplinear at infinity, Sun and Tang in [7] obtained a nontrivial solution for problem (1) by using the local linking theorem due to Li and Willem. In [8], when the nonlinearity F is some asymptotically 4-linear at infinity, Yang and Zhang proved the existence of at least two nontrivial solutions for problem (1) by means of the Morse theory and local linking. Since \(p=2^{*}\leq4\) (\(N\geq4\)), condition (9) implies that the nonlinearity f is 3-sublinear at infinity. Hence, our results are the complements for the ones of [7, 8].
2 Proof of the theorems
Recall that a sequence \(\{u_{n}\}\subset H_{0}^{1}(\Omega)\) is called a \((PS)_{c}\) sequence for any \(c\in R\) of the functional I on \(H_{0}^{1}(\Omega)\) if \(I(u_{n})\to c\) and \(I'(u_{n})\to0\) as \(n\to\infty\). The functional I is called to satisfy the \((PS)_{c}\) condition if any \((PS)_{c}\) sequence has a convergent subsequence. We will prove our theorems by using the following three-critical-point theorem related to local linking due to Brezis and Nirenberg (see Theorem 4 in [1]).
Theorem A
Proof of Theorem 1
(a) The functional I satisfies the local linking at zero with respect to \((V_{k}, V_{k}^{\bot})\), where \(V_{k}=\bigoplus_{i=1}^{k}\ker(-\Delta-\lambda_{i}(m_{0}))\) and \(V_{k}^{\bot}=\bigoplus _{i=k+1}^{+\infty}\ker(-\Delta-\lambda_{i}(m_{0}))\) such that \(H_{0}^{1}(\Omega )=V_{k}\oplus V_{k}^{\bot}\).
(b) The functional I satisfies the \((PS)\) condition. To the end, it suffices to say the functional I is coercive on \(H^{1}_{0}(\Omega)\), i.e., \(I(u)\to+\infty\) as \(\|u\|\to\infty\).
(c) From (b), we have that the functional I is bounded from below. From the fact that \(I(u)<0\) for any \(u\in V_{k}\) with \(0<\|u\|\leq r_{0}\), we have \(\inf_{u\in H^{1}_{0}(\Omega)}I(u)<0\). Moreover, \(I(0)=0\). Therefore, Theorem 1 is proved by Theorem A. □
Proof of Theorem 2
First of all, from (a) of the proof of Theorem 1, we have that the functional I satisfies the local linking at zero with respect to \((V_{k}, V_{k}^{\bot})\). And then, we know from (9) that \(f(x,t)\) is 3-sublinear at infinity, which implies that the functional I is coercive on \(H^{1}_{0}(\Omega)\) by a standard argument. We obtain that the functional I is bounded from below and satisfies the \((PS)\) condition for \(N=1,2,3\). In the following, we only prove that the functional I also satisfies the \((PS)\) condition for \(p=2^{*}\) (\(N\geq4\)), where \(f(x,t)\) is not only 3-sublinear at infinity, but also is asymptotically critical growth at infinity.
At last, similar to (c) of the proof of Theorem 1, Theorem 2 is proved. □
Declarations
Acknowledgements
The authors would like to thank the referees for their useful suggestions.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The work is supported by the National Natural Science Foundation of China (No. 11471267).
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both 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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