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Infinitely many solutions for a class of fourth-order partially sublinear elliptic problem
- Hua Gu^{1}Email authorView ORCID ID profile and
- Tianqing An^{1}
- Received: 17 July 2016
- Accepted: 7 December 2016
- Published: 3 January 2017
Abstract
In this paper, we study the existence of infinitely many solutions for a class of fourth-order partially sublinear elliptic problem with Navier boundary value condition by using an extension of Clark’s theorem.
Keywords
- multiple solution
- fourth-order elliptic problem
- critical point
- variational method
1 Introduction
The fourth-order elliptic equations can describe the static form change of beam or the motion of rigid body, so they are widely applied in physics and engineering. In the 1990s, Lazer and Mckenna (see [1, 2]) investigated the problem (1) as \(g(x,u)=d[(u+1)^{+}-1]\). In [1], they pointed out that this type of nonlinearity furnishes a model to study traveling waves in suspension bridges. In [2], they got \(2k-1\) solutions when \(N=1\) and \(d>\lambda_{k}(\lambda_{k}-c)\) (\(\lambda_{k}\) is the sequence of the eigenvalues of −△ in \(H_{0}^{1}(\varOmega )\)) by the global bifurcation method. In [3], Micheletti and Pistoia used a variational linking theorem to investigate the existence of two solutions for a more general nonlinearity \(g(\cdot,u)\). In 2001, Micheletti and Saccon (see [4]) obtained two results about the existence of two nontrivial solutions and four nontrivial solutions by a similar variational approach, depending on the position of a suitable parameter with respect to the eigenvalues of the linear part.
In recent years, many researchers have used variational approach to investigate the fourth-order elliptic equations. In [5], Pu et al. used the least action principle, the Ekeland variational principle and the mountain pass theorem to prove the existence and multiplicity of solutions of (1) when \(g(x,u)=a(x)|u|^{s-2}u+f(x,u)\) (\(a\in L^{\infty}(\varOmega )\), \(s\in(1,2)\) and \(f\in C(\overline{\varOmega }\times\mathbb{R},\mathbb{R})\)). In [6], Hu and Wang obtained the existence of nontrivial solutions to problem (1) under suitable assumptions of \(g(x,u)\) by a variant version of the mountain pass theorem. In [7], we have studied the existence of multiple solutions for problem (1) by using the variant fountain theorem without the condition \(c<\lambda_{1}\) (\(\lambda_{1}\) is the first eigenvalue of −△ in \(H_{0}^{1}(\varOmega )\)). In [8], Wang and Shen, under an improved Hardy-Rellich inequality, studied the existence of multiple and sign-changing solutions for a class of biharmonic equations in an unbounded domain by the minimax method and linking theorem. In 2015, Liu and Chen (see [9]) investigated the existence of ground-state solution and nonexistence of nontrivial solution for a similar biharmonic equation in [8] by using variational methods, also they explored the phenomenon of concentration of solutions. For other related results, see [10–15] and the references therein.
In critical point theory, Clark’s theorem [16] asserts the existence of a sequence of negative critical values tending to 0 for even coercive functionals. It is constantly and effectively applied to sublinear differential equations with symmetry. In 2001, Wang (see [17]) explored a variant of the Clark theorem given by Heinz in [18] to investigate a variety of nonlinear boundary value problems. Then, in 2015, Liu and Wang improved Clark’s theorem and gave an extension of Clark’s theorem in [19]. Their new results gave a more detailed structure of the set of critical points near the origin and are powerful in applications. In this paper, inspired by [17], we will use the new result of Liu and Wang in [19] to investigate the existence of infinitely many solutions for partially sublinear problems (1).
Our main result is the following theorem.
Theorem 1
Remark
The advantage of this theorem is that it not only obtained the existence of infinite solutions, but it also pointed out their positions.
2 Preliminaries
Lemma 1
The norm \(\|u\|\) is equivalent to the norm \(\|\triangle u\|_{2}\) in E.
Proof
This result can be found in [6] (Lemma 2.3), so we omit the proof here. □
To prove our main result, Theorem 1, we give the improved Clark theorem in [19].
Theorem 2
See [19], Theorem 1.1
- (i)
There exists a sequence of critical points \(\{ u_{k}\}\) satisfying \(\Phi(u_{k})<0\) for all k and \(\|u_{k}\| \rightarrow0\) as \(k\rightarrow\infty\).
- (ii)
There exists \(r>0\) such that for any \(0 < a< r\) there exists a critical point u such that \(\|u\|=a\) and \(\Phi(u)=0\).
In order to apply this theorem to prove our main result, we set our working space \(E=H^{2}(\varOmega )\cap H_{0}^{1}(\varOmega )\) as the space X in Theorem 2.
3 Proof of Theorem 1
Now we prove our main result, Theorem 1.
Proof of Theorem 1
Now the proof of conclusion of Theorem 1 is divided into three steps.
Step 1. Φ̂ satisfies the (PS) condition.
We assume the sequence \(\{u_{n}\}\subset E\) satisfies the requirement that \(\{\widehat{\Phi}(u_{n})\}\) is bounded and \(\widehat{\Phi}'(u_{n})\rightarrow0\) as \(n\rightarrow\infty\). Now we claim that the sequence \(\{u_{n}\}\) is bounded in E and possesses a strong convergent subsequence.
First, we claim that \(\{u_{n}\}\) is bounded in E. By (8), we set \(u_{n}=u_{n}^{-}+u_{n}^{0}+u_{n}^{+}\in E^{-}\oplus E^{0}\oplus E^{+}\).
Second, we claim that \(\{u_{n}\}\) possesses a strong convergent subsequence in E.
Step 2. Construction of space \(X^{k}\cap S_{\rho_{k}}\) and proof of \(\sup_{X^{k}\cap S_{\rho_{k}}}\Phi<0\).
Now we appeal to Theorem 2 to obtain infinitely many solutions \(u_{k}\) for (12) such that \(\|u_{k}\|\rightarrow0\) as \(k\rightarrow\infty\).
Step 3. Solutions of (12) are solutions of (1).
Since \(\|u_{k}\|\rightarrow0\), then \(\|u_{k}\|_{\infty}\rightarrow 0\). When k is large enough, there exists a constant \(\delta>0\) such that \(|u_{k}(x)|<\delta/2\). Then \(g(x,u_{k}(x))=\widehat{g}(x,u_{k}(x))\). Therefore, \(u_{k}\) are the solutions of (1) with k sufficiently large and \(\|u_{k}\|_{\infty}\rightarrow0\) as \(k\rightarrow\infty\). □
Declarations
Acknowledgements
The authors would like to thank the referee for his/her careful reading of the manuscript and the useful suggestions and comments. Supported by the Fundamental Research Funds for the Central Universities of China (No. 2016B08614).
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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