 Research
 Open Access
Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth
 Ruichang Pei^{1, 2}Email author and
 Jihui Zhang^{2}
 Received: 23 February 2015
 Accepted: 18 June 2015
 Published: 9 July 2015
Abstract
The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourthorder elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.
Keywords
 Morse theory
 Adamstype inequality
 subcritical polynomial growth
 subcritical exponential growth
1 Introduction
Fourthorder problems of this class with \(N>4\) have been studied by many authors. In [1], Lazer and Mckenna pointed out that this type of nonlinearity provides a model to study traveling waves in suspension bridges. Since then, more general nonlinear fourthorder elliptic boundary value problems have been studied. For problem (2), Lazer and Mckenna [2] proved the existence of \(2k1\) solutions when \(N=1\), and \(b>\lambda_{k}(\lambda_{k}c)\) by the global bifurcation method. In [3], Tarantello found a negative solution when \(b\geq \lambda_{1}(\lambda_{1}c)\) by a degree argument. For problem (1) when \(f(x,u)=bg(x,u)\), Micheletti and Pistoia [4] proved that there exist two or three solutions for a more general nonlinearity g by the variational method. Xu and Zhang [5] discussed the problem when f satisfies the local superlinearity and sublinearity. Zhang [6] proved the existence of solutions for a more general nonlinearity \(f(x,u)\) under some weaker assumptions. Zhang and Li [7] proved the existence of multiple nontrivial solutions by means of Morse theory and local linking. An and Liu [8] and Liu and Wang [9] also obtained the existence result for nontrivial solutions when f is asymptotically linear at positive infinity. In [10], Zhang and Wei obtained the existence of infinitely many solutions when the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1) is concerned, existence results of signchanging solutions were also obtained (see, e.g., [11, 12]).
 (SCP):there exist positive constants \(c_{1}\) and \(c_{2}\) and \(q_{0}\in (1, p^{*}1)\) such that$$\bigl\vert f(x,t)\bigr\vert \leq c_{1}+ c_{2}t^{q_{0}} \quad \mbox{for all } t\in\mathbb{R} \mbox{ and } x\in\Omega, $$
 (H_{1}):

\(f(x,t)t\geq0\) for all \(x\in\Omega\), \(t\in{\mathbb{R}}\);
 (H_{2}):

\(\lim_{t\rightarrow0}\frac{f(x,t)}{t}=f_{0}\) uniformly for \(x \in\Omega\), where \(f_{0}\) is a constant;
 (H_{3}):

\(\lim_{t\rightarrow\infty}\frac{f(x,t)}{t} =+\infty\) uniformly for \(x \in\Omega\);
 (H_{4}):

\(\frac{f(x,t)}{t}\) is nondecreasing in \(t\in \mathbb{R}\) for any \(x\in\Omega\).
Theorem 1.1
Let \(N>4\) and assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H_{1})(H_{4}). If \(f_{0}<\mu_{1}\), then problem (1) has at least three nontrivial solutions.
 (SCE):
f has subcritical (exponential) growth on Ω, i.e., \(\lim_{t\rightarrow\infty}\frac{f(x,t)}{\exp(\alpha t^{2})} =0 \) uniformly on \(x\in\Omega\) for all \(\alpha>0\).
When \(N=4\) and f has the subcritical (exponential) growth (SCE), our work is still to study problem (1) without the (AR) condition. Our result is as follows.
Theorem 1.2
Let \(N=4\) and assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H_{1})(H_{4}). If \(f_{0}<\mu_{1}\), then problem (1) has at least three nontrivial solutions.
2 Preliminaries and auxiliary lemmas
Definition 2.1
We have the following version of the mountain pass theorem (see [17]).
Proposition 2.1
Lemma 2.1
 (i)
there exist \(\rho, \alpha>0\) such that \(I_{+}(u)\geq\alpha\) for all \(u\in E \) with \(\ u \=\rho\),
 (ii)
\(I_{+}(t\varphi_{1} )\rightarrow\infty\) as \(t\rightarrow +\infty\).
Proof
Lemma 2.2
(see [16])
Lemma 2.3
 (i)
there exist \(\rho, \alpha>0\) such that \(I_{+}(u)\geq\alpha\) for all \(u\in E \) with \(\ u\=\rho\),
 (ii)
\(I_{+}(t\varphi_{1} )\rightarrow\infty\) as \(t\rightarrow +\infty\).
Proof
Lemma 2.4
Proof
This lemma is essentially due to [18]. We omit the proof. □
Lemma 2.5
Under the assumptions of Theorem 1.1, then \(I_{+}\) and I satisfy the \((C)_{c}\) condition.
Proof
Lemma 2.6
Under the assumptions of Theorem 1.2, \(I_{+}\) and I satisfy the \((C)_{c}\) condition.
Proof
3 Computation of the critical groups
It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equations. Let us recall some results which will be used later. We refer the readers to the book [20] for more information on Morse theory.
Proposition 3.1
Proof
Proposition 3.2
4 Proof of the main results
Proofs of Theorem 1.1 and Theorem 1.2
Declarations
Acknowledgements
This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C). The authors are very grateful for the reviewers’ valuable comments and suggestions in improving this paper.
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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