Sshaped connected component for the fourthorder boundary value problem
 Jinxiang Wang^{1, 2} and
 Ruyun Ma^{1}Email author
Received: 17 July 2016
Accepted: 18 October 2016
Published: 28 October 2016
Abstract
Keywords
boundary value problem positive solutions principal eigenvalue bifurcationMSC
34B10 34B181 Introduction
The positivity (or negativity) of solutions of beam equation (i.e., the deflection of the roadbed) is crucial for the whole system and properties of possibly nonstationary solutions. Even if we focus on onedimensional ODE problems, we would like to mention works of Grunau and Sweers [7, 8], where positive solutions of fourthorder PDEs subject to different types of boundary conditions are investigated. The existence and multiplicity of positive solutions of (1.1) or its particular case have been investigated via theory of fixed point index in a cone and LeraySchauder degree by many authors; see, for example, [9–17] and the references therein. Notice that these results give no information on the interesting problem as to what happens to the norms of positive solutions of (1.1) as λ varies in \(\mathbb{R} ^{+}\).
Recently, the global behavior of solution set of (1.1) has been studied by using Dancer’s or Rabinowitz’s global bifurcation theorem (see Ma [18], Ma, Gao, and Han [19], and Dai and Han [20]), and accordingly, the existence and multiplicity of positive solutions and nodal solutions have been obtained. However, the sublinear and superlinear conditions or asymptotic linear growth conditions imposed on the nonlinearities only deduce a relatively simple “shape of the component.”
Theorem A
[21], Theorem 1.1
 (F1):

there exist \(x_{1}, x_{2}\in [0,1]\) such that \(x_{1}< x_{2}\), \(a(x)>0\) on \((x_{1}, x_{2})\), and \(a(x) \leq 0\) on \([0,1]\backslash [x_{1}, x_{2}]\);
 (F2):

there exist \(\alpha >0, f_{0}>0\), and \(f_{1}>0\) such that \(\lim_{s\rightarrow 0^{+}}\frac{f(s)f_{0}s^{p1}}{s^{p1+ \alpha }}=f_{1}\);
 (F3):

\(f_{\infty }:=\lim_{s\rightarrow \infty }\frac{f(s)}{s ^{p1}}=0\);
 (F4):

there exists \(s_{0}>0\) such that
 (i)
(1.3) has at least one positive solution if \(\lambda =\lambda_{\ast }\);
 (ii)
(1.3) has at least two positive solutions if \(\lambda_{\ast }< \lambda \leq \frac{\mu_{1}}{f_{0}}\);
 (iii)
(1.3) has at least three positive solutions if \(\frac{\mu_{1}}{f _{0}}<\lambda <\lambda^{\ast }\);
 (iv)
(1.3) has at least two positive solutions if \(\lambda = \lambda^{\ast }\);
 (v)
(1.3) has at least one positive solution if \(\lambda >\lambda ^{\ast }\).
Of course, the natural question is what happens if we consider the fourthorder problem (1.1)? As we know, there are great differences between secondorder and fourthorder BVPs; for example, for secondorder BVPs, the existence of a wellordered pair of lower and upper solutions is sufficient to ensure the existence of a solution enclosed by them; see [22]. But this is not correct for fourthorder BVPs even for the simple boundary conditions, as the authors showed in [6]. On the other hand, the concavity and convexity of the solutions of secondorder BVPs can be deduced directly from the nonlinearity in the equation, but for fourthorder BVPs, this becomes complicated, especially when the nonlinearity changes the sign.
The purpose of this paper is to investigate the existence of an Sshaped component in the solution set of problem (1.1).
 (i)
\(\Vert u_{\lambda^{\ast }} \Vert _{\infty }<\Vert u_{\lambda_{\ast }} \Vert _{ \infty }\),
 (ii)
at \((\lambda^{\ast }, \Vert u_{\lambda^{\ast }} \Vert _{\infty })\), the component curve \(\widetilde{\mathcal{C}}\) turns to the left,
 (iii)
at \((\lambda_{\ast }, \Vert u_{\lambda_{\ast }} \Vert _{\infty })\), the component curve \(\widetilde{\mathcal{C}}\) turns to the right.
As we mentioned before, the change of sign of f brings a great difficulty to the solvability of (1.1), so in this paper, we only consider the nonnegative case. Evidently, if there exists an Sshaped component in the positive solutions set of problem (1.1), then we can accordingly deduce the existence and multiplicity of positive solutions for problem (1.1) and, especially, establish the existence of three distinct positive solutions for λ being in a certain interval.
 (H1):

\(f: [0,1]\times [0,\infty )\times (\infty,0] \rightarrow [0, \infty )\) is continuous, and \(f(x,s,p)>0\) for \(x\in [0,1]\) and \((s,p)\in ([0,\infty )\times (\infty,0])\backslash \{(0,0)\}\);
 (H2):

there exist constants \(a,b\in [0,\infty )\) with \(a+b>0\) and \(c>0,d>0\) such that$$\begin{aligned} \lim_{\sqrt{s^{2}+p^{2}}\rightarrow 0}\frac{f(x,s,p)(asbp)}{\sqrt{s ^{2}+p^{2}}^{1+c}}=d \quad \text{uniformly for } x \in [0,1]; \end{aligned}$$
 (H3):

$$\begin{aligned} \lim_{\sqrt{s^{2}+p^{2}}\rightarrow \infty }\frac{f(x,s,p)}{\sqrt{s ^{2}+p^{2}}}=0 \quad \text{uniformly for } x \in [0,1]; \end{aligned}$$
 (H4):

there exists \(s_{0}>0\) such thatwhere \(\lambda_{1}(a,b)>0\) is the generalized principal eigenvalue of the linear problem$$\begin{aligned} \min_{s\in [s_{0},4s_{0}]}\frac{f(x,s,p)}{s}\geq \frac{16\pi ^{4}}{\lambda_{1}(a,b)}\quad \text{uniformly for } x\in [0,1], p\in (\infty,0], \end{aligned}$$defined in Lemma 2.1.$$\begin{aligned} \left \{ \textstyle\begin{array}{l} u''''(x)=\lambda (aubu''), \quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0 \end{array}\displaystyle \right . \end{aligned}$$
Considering the shape of a component in the positive solution set of problem (1.1), we have the following result.
Theorem 1.1
 (i)
(1.1) has at least one positive solution if \(\lambda =\lambda _{\ast }\);
 (ii)
(1.1) has at least two positive solutions if \(\lambda_{\ast }< \lambda \leq \lambda_{1}(a,b)\);
 (iii)
(1.1) has at least three positive solutions if \(\lambda_{1}(a,b)< \lambda <\lambda^{\ast }\);
 (iv)
(1.1) has at least two positive solutions if \(\lambda = \lambda^{\ast }\);
 (v)
(1.1) has at least one positive solution if \(\lambda >\lambda ^{\ast }\).
Remark 1.1
The rest of this paper is arranged as follows. In Section 2, we show a global bifurcation phenomena from the trivial branch with rightward direction. Section 3 is devoted to show that there are at least two direction turns of the component and complete the proof of Theorem 1.1.
2 Rightward bifurcation
In this section, we state some preliminary results and show a global bifurcation phenomenon from the trivial branch with rightward direction.
Definition 2.1
Lemma 2.1
[18], Theorem 3.1
Remark 2.1
 (i)
\(0\leq G(t,t)G(s,s)\leq G(t,s)\leq G(s,s), \forall t,s \in (0,1)\);
 (ii)
\(G(t,s)\geq \frac{1}{4}G(s,s), \forall t\in [\frac{1}{4},\frac{3}{4}], s\in (0,1)\).
Using the Dancer bifurcation theorem and following the arguments in the proof of Theorem 4.1 in [18], we have the following:
Lemma 2.2
[18]
Lemma 2.3
Assume that (H1), (H2), and (H3) hold. Let \(\{(\lambda_{n},u_{n})\}\) be a sequence of positive solutions to (1.1) that satisfies \(\lambda_{n}\rightarrow \lambda_{1}(a,b)\) and \(\Vert u_{n} \Vert _{X}\rightarrow 0\). Then there exists a subsequence of \(\{u_{n}\}\), again denoted by \(\{u_{n}\}\), such that \(\frac{u_{n}}{\Vert u_{n} \Vert _{X}}\) converges uniformly to \(\frac{\sin \pi x}{\pi^{2}}\) on \([0,1]\).
Proof
Set \(v_{n}:=\frac{u_{n}}{\Vert u_{n} \Vert _{X}}\). Then \(\Vert v_{n} \Vert _{X}=1\), and (2.3) implies that \(\Vert v_{n} \Vert _{\infty }\) and \(\Vert v_{n}'' \Vert _{\infty }\) are bounded. By the AscoliArzelà theorem a subsequence of \(v_{n}\) uniformly converges to a limit v, and we again denote by \(v_{n}\) the subsequence.
Remark 2.2
From the proof of Lemma 2.3 we have that \(\frac{u _{n}''}{\Vert u_{n} \Vert _{X}}\) converges uniformly to \(\sin \pi x\) on \([0,1]\).
Lemma 2.4
Assume that (H1), (H2), and (H3) hold. Let \(\mathcal{C}\) be as in Lemma 2.2. Then there exists \(\delta >0\) such that \((\lambda,u)\in \mathcal{C}\) and \(\vert \lambda \lambda_{1}(a,b) \vert + \Vert u \Vert _{X}\leq \delta \) imply \(\lambda >\lambda_{1}(a,b)\).
Proof
3 Direction turns of component and proof of Theorem 1.1
In this section, we show that there are at least two direction turns of the component under conditions (H3) and (H4), that is, the component is Sshaped, and accordingly, we finish the proof of Theorem 1.1.
Lemma 3.1
Assume that (H1) and (H4) hold. Let u be a positive solution of (1.1) with \(\Vert u \Vert _{\infty }=4s_{0}\). Then \(\lambda < \lambda_{1}(a,b)\).
Proof
Lemma 3.2
Assume that (H1), (H2), and (H3) hold. Then, \(\mathcal{C}\) joins \((\lambda_{1}(a,b),0)\) to \((\infty,\infty )\) in \([0,\infty )\times P\).
Proof
We divide the proof into two steps.
Step 1. We show that \(\sup \{\lambda \mid (\lambda,u)\in\mathcal{C}\}=\infty \).
Step 2. We show that \(\sup \{\Vert u \Vert _{X}\mid (\lambda,u)\in\mathcal{C}\}=\infty \).
Proof of Theorem 1.1.
 (i)
if \(\lambda \in (\lambda_{1}(a,b),\overline{\lambda}]\), then there exist u and v such that \((\lambda,u),(\lambda,v)\in \mathcal{C}\) and \(\Vert u \Vert _{\infty }<\Vert v \Vert _{\infty }<4s_{0}\);
 (ii)
if \(\lambda \in (\underline{\lambda},\lambda_{1}(a,b)]\), then there exist u and v such that \((\lambda,u),(\lambda,v)\in \mathcal{C}\) and \(\Vert u \Vert _{\infty }<4s_{0}<\Vert v \Vert _{\infty }\).
Define \(\lambda^{\ast }=\sup\{\overline{\lambda}: \overline{ \lambda} \text{ satisfies } \mbox{(i)} \}\) and \(\lambda_{\ast }= \inf\{\underline{\lambda}: \underline{\lambda} \text{ satisfies } \mbox{(ii)} \}\). Then (1.1) has positive solutions \(u_{\lambda_{\ast }}\) at \(\lambda =\lambda_{\ast }\) and \(u_{ \lambda^{\ast }}\) at \(\lambda =\lambda^{\ast }\), respectively. Clearly, the component curve \(\widetilde{\mathcal{C}}\) turns to the left at \((\lambda^{\ast }, \Vert u_{\lambda^{\ast }} \Vert _{\infty })\) and to the right at \((\lambda_{\ast }, \Vert u_{\lambda_{\ast }} \Vert _{\infty })\), that is, \(\mathcal{C}\) is an Sshaped component. This, together with Lemma 3.2, completes the proof of Theorem 1.1. □
Declarations
Acknowledgements
This work was supported by the NSFC (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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