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Reversed S-shaped connected component for a fourth-order boundary value problem
- Jinxiang Wang^{1, 2} and
- Ruyun Ma^{1}Email author
https://doi.org/10.1186/s13662-017-1167-5
© The Author(s) 2017
- Received: 17 October 2016
- Accepted: 4 April 2017
- Published: 18 April 2017
Abstract
Keywords
- boundary value problem
- positive solutions
- principal eigenvalue
- bifurcation
MSC
- 34B10
- 34B18
1 Introduction
However, to the best of our knowledge, when parameter λ varies in \(\mathbb{R}^{+}\), there are few papers concerned with the global behavior of positive solutions of (1.1), see, for example, [18–20]. By using Rabinowitz’s or Dancer’s global bifurcation theorem, [18–20] investigated the global structure of the solutions set of (1.1), and accordingly, obtained the existence and multiplicity of positive solutions and nodal solutions. Notice that these results give no information on direction turns of the connected component.
- (H1)
\(h(x) \geq0\) in \([0,1]\) and \(h\not\equiv0\) in any subinterval of \([0,1]\);
- (H2)
there exist \(\alpha>0\), \(f_{0}>0\), and \(f_{1}>0\) such that \(\lim_{s\rightarrow0^{+}}\frac{f(s)-f_{0}s}{s^{1+\alpha}}=f_{1}\);
- (H3)
\(f_{\infty}:=\lim_{s\rightarrow\infty}\frac{f(s)}{s}=\infty\);
- (H4)
Arguing the shape of a component in the positive solutions set of problem (1.3), we have the following result.
Theorem 1.1
- (i)
(1.3) has at least one positive solution if \(0<\lambda <\lambda_{\ast}\);
- (ii)
(1.3) has at least two positive solutions if \(\lambda =\lambda_{\ast}\);
- (iii)
(1.3) has at least three positive solutions if \(\lambda_{\ast}<\lambda<\frac{\mu_{1}}{f_{0}}\);
- (iv)
(1.3) has at least two positive solutions if \(\frac {\mu_{1}}{f_{0}}\leq\lambda<\lambda^{\ast}\);
- (v)
(1.3) has at least one positive solution if \(\lambda =\lambda^{\ast}\);
- (vi)
(1.3) has no positive solution if \(\lambda>\lambda ^{\ast}\).
Remark 1.1
Indeed, condition (H2) pushes the direction of bifurcation to the left near \(u=0\), while conditions (H4) and (H3) guarantee that the bifurcation curve grows to the right at some point and grows to the left near \(\lambda=0\), respectively.
Remark 1.2
The rest of this paper is arranged as follows. In Section 2, we show global bifurcation phenomena from the trivial branch with the leftward direction. Section 3 is devoted to showing that there are at least two direction turns of the component and to completing the proof of Theorem 1.1.
2 Leftward bifurcation
In this section, we state some preliminary results and show global bifurcation phenomena from the trivial branch with the leftward direction.
- (i)
\(0\leq G(t,s)\leq G(s,s)\), \(\forall t,s\in[0,1]\);
- (ii)
\(G(t,s)\geq\frac{1}{4}G(s,s)\), \(\forall t\in[\frac {1}{4},\frac{3}{4}]\), \(s\in[0,1]\),
Lemma 2.1
see [19], Theorem 2.1
Lemma 2.2
Assume that (H1) and (H2) hold, then from \((\frac{\mu_{1}}{f_{0}},0)\) there emanates an unbounded subcontinuum \(\mathcal{C}\) of positive solutions of (1.3) in the set \(\mathbb{R}\times E\), where \(E=\{u\in C^{3}[0,1]\mid u(0)=u(1)=u''(0)=u''(1)=0\}\) with the norm \(\Vert u \Vert = \Vert u \Vert _{\infty}+ \Vert u' \Vert _{\infty}+ \Vert u'' \Vert _{\infty }+ \Vert u''' \Vert _{\infty}\).
Lemma 2.3
Assume that (H1) and (H2) hold. Let \(\{ (\lambda_{n},u_{n})\}\) be a sequence of positive solutions to (1.3) which satisfies \(\lambda_{n}\rightarrow\frac{\mu_{1}}{f_{0}}\) and \(\Vert u_{n} \Vert \rightarrow0\). Let ϕ be the first eigenfunction of (2.2) which satisfies \(\Vert \phi \Vert _{\infty}=1\). Then there exists a subsequence of \(\{u_{n}\}\), again denoted by \(\{u_{n}\}\), such that \(\frac{u_{n}}{ \Vert u_{n} \Vert _{\infty}}\) converges uniformly to ϕ on \([0,1]\).
Proof
Since \(\Vert v_{n}' \Vert _{\infty}\) is bounded, by the Ascoli-Arzela theorem, a subsequence of \(\{v_{n}\}\) uniformly converges to a limit \(v\in C[0,1]\) with \(\Vert v \Vert _{\infty}=1\), and we again denote by \(\{v_{n}\}\) the subsequence.
Lemma 2.4
Assume that (H1) and (H2) hold. Let \(\mathcal{C}\) be as in Lemma 2.2. Then there exists \(\delta>0\) such that \((\lambda,u)\in\mathcal{C}\) and \(\vert \lambda-\frac {\mu_{1}}{f_{0}} \vert + \Vert u \Vert \leq\delta\) imply \(\lambda<\frac{\mu_{1}}{f_{0}}\).
Proof
3 Direction turns of the component and the proof of Theorem 1.1
In this section, we show that there are at least two direction turns of the component under conditions (H4) and (H3), that is, the component is reversed S-shaped, and accordingly we finish the proof of Theorem 1.1.
Lemma 3.1
Assume that (H1) and (H4) hold. Let u be a solution of (1.3) with \(\Vert u \Vert _{\infty}=s_{0}\), then \(\lambda>\frac{\mu_{1}}{f_{0}}\).
Proof
Lemma 3.2
Assume that (H1), (H2), and (H3) hold. Let \(\mathcal{C}\) be as in Lemma 2.2. Then \(\sup\{\lambda\mid (\lambda ,u)\in\mathcal{C}\}<\infty\).
Proof
Lemma 3.3
Assume that (H1), (H2), and (H3) hold. Let \(\{(\lambda_{n},u_{n})\}\) be a sequence of positive solutions to (1.3), then \(\Vert u_{n} \Vert \rightarrow\infty\) implies \(\Vert u_{n} \Vert _{\infty}\rightarrow\infty\).
Proof
From Lemma 3.2, we conclude that \(\{\lambda _{n}\}\) is bounded. Assume on the contrary that \(\Vert u_{n} \Vert _{\infty}\) is bounded. By recalling (2.5) and (2.4), we have that \(\Vert u_{n}' \Vert _{\infty}\) and \(\Vert u_{n}'' \Vert _{\infty}\) are bounded too.
Lemma 3.4
Assume that (H1), (H2), and (H3) hold. Let \(\{(\lambda_{n},u_{n})\}\) be a sequence of positive solutions to (1.3), then \(\Vert u_{n} \Vert \rightarrow\infty\) implies \(\lambda_{n}\rightarrow0\).
Proof
Proof of Theorem 1.1
Let \(\mathcal{C}\) be as in Lemma 2.2. By Lemma 2.4, \(\mathcal{C}\) is bifurcating from \((\frac{\mu _{1}}{f_{0}},0)\) and goes leftward. Since \(\mathcal{C}\) is unbounded, there exists \(\{(\lambda_{n},u_{n})\}\) such that \((\lambda _{n},u_{n})\in\mathcal{C}\) and \(\vert \lambda_{n} \vert + \Vert u_{n} \Vert \rightarrow\infty\). By Lemmas 3.2 and 3.4, we have that \(\Vert u_{n} \Vert \rightarrow\infty\) and \(\lambda_{n}\rightarrow0\). Lemma 3.3 implies that \(\Vert u_{n} \Vert _{\infty}\rightarrow\infty\), then there exists \((\lambda_{0},u_{0})\in\mathcal{C}\) such that \(\Vert u_{0} \Vert _{\infty}=s_{0}\), and Lemma 3.1 shows that \(\lambda _{0}>\frac{\mu_{1}}{f_{0}}\).
- (i)
if \(\lambda\in(\underline{\lambda},\frac{\mu_{1}}{f_{0}})\), then there exist u and v such that \((\lambda,u),(\lambda,v)\in \mathcal{C}\) and \(\Vert u \Vert _{\infty}< \Vert v \Vert _{\infty}<s_{0}\);
- (ii)
if \(\lambda\in[\frac{\mu_{1}}{f_{0}},\overline{\lambda})\), then there exist u and v such that \((\lambda,u),(\lambda,v)\in \mathcal{C}\) and \(\Vert u \Vert _{\infty}< s_{0}< \Vert v \Vert _{\infty}\).
Define \(\lambda_{\ast}=\inf\{\underline{\lambda}: \underline{\lambda} \text{ satisfies (i)}\}\) and \(\lambda ^{\ast}=\sup\{\overline{\lambda}: \overline{\lambda} \text{ satisfies (ii)} \}\). Then (1.3) has a positive solution \(u_{\lambda_{\ast}}\) at \(\lambda=\lambda_{\ast}\) and \(u_{\lambda ^{\ast}}\) at \(\lambda=\lambda^{\ast}\), respectively. Clearly, the component curve turns to the right at \((\lambda_{\ast}, \Vert u_{\lambda_{\ast}} \Vert _{\infty})\) and to the left at \((\lambda^{\ast}, \Vert u_{\lambda^{\ast}} \Vert _{\infty})\) (see Figure 2). That is, \(\mathcal{C}\) is a reversed S-shaped component, this together with Lemma 3.4 completes the proof of Theorem 1.1. □
Declarations
Acknowledgements
This work was supported by the Natural Science Foundation of China (No. 11671322, No. 11626016).
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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