 Research
 Open Access
Global behavior of positive solutions for some semipositone fourthorder problems
 Dongliang Yan^{1} and
 Ruyun Ma^{1}Email author
https://doi.org/10.1186/s1366201819044
© The Author(s) 2018
 Received: 2 July 2018
 Accepted: 22 November 2018
 Published: 3 December 2018
Abstract
Keywords
 34B18
 34B16
 34B25
 47H11
MSC
 Positive solutions
 Topological degree
 Connected set
 Bifurcation
1 Introduction
 \((f_{1})\) :

\(f(x, 0) < 0\) \(\forall x \in (0,1)\).
The existence of positive solutions of secondorder positone problems have been extensively studied via the Leray–Schauder degree theory, fixed point theorem on a cone, and the method of lower and upper solutions; see [1–3] and the references therein.
 1.
Spectrum theory for singular secondorder linear eigenvalue problems has been established via Prüfer transform in [13]. However, the spectrum structure of singular fourthorder linear eigenvalue problems is not established so far.
 2.The uniqueness of solutions of secondorder problemshas been obtained in [14]. However, the uniqueness of solution of$$ \textstyle\begin{cases} u''=\lambda u^{q}, &x\in (a,b), \\ u>0,&x\in (a,b), \\ u(a)=u(b)=0 \end{cases} $$is not obtained so far.$$ \textstyle\begin{cases} w''''=b\vert w \vert ^{\alpha }, \quad x\in (0,1), \\ w(0)=w(1)=w''(0)=w''(1)=0 \end{cases} $$
 3.
It is well known that, for a secondorder differential equation with periodic, Neumann, or Dirichlet boundary conditions, the existence of a wellordered pair of lower and upper solutions \(\alpha \leq \beta \) is sufficient to ensure the existence of a solution in the sector enclosed by them. However, this result it is not true for fourthorder differential equations; see Remark 3.1 in [15].
Motivated by Ambrosetti [4], we investigate the global behavior of positive solutions of the fourthorder boundary value problem (1.1). Depending on the behavior of \(f = f(x, s)\) as \(s\rightarrow +\infty \), we handle both asymptotically linear, superlinear, and sublinear problems. All results are obtained by showing that there exists a global branch of solutions of (1.1) “emanating from infinity” and proving that for λ near the bifurcation value, solutions of large norms are indeed positive to which bifurcation theory or topological methods apply in a classical fashion. Since there are a lot of differences between second and fourthorder cases, we have to overcome several new difficulties in the proof of our main results.
We deal in Sect. 2 with asymptotically linear problems. In Sect. 3, we discuss superlinear problems, and we show that (1.1) possesses positive solutions for \(0 < \lambda < \lambda^{*}\). Similar arguments can be used in the sublinear case, discussed in Sect. 4, to show that (1.1) has positive solutions provided that λ is large enough.
2 Asymptotically linear problems
For Lebesgue spaces, we use standard notation. We work in \(X=C[0,1]\). The usual norm in such spaces is denoted by \(\Vert u \Vert =\max_{t \in [0,1]}\vert u(t) \vert \), and we set \(B_{r}=\{u\in X: \Vert u \Vert \leq r\}\). The first eigenvalue of \(u''''\) with boundary conditions \(u(0)=u(1)=u''(0)=u''(1)=0\) is denoted by \(\lambda_{1}\); \(\phi_{1}\) is the corresponding eigenfunction such that \(\phi_{1}>0\) in \((0,1)\). We also set \(\mathbb{R^{+}}=[0, \infty )\).
Hereafter we will use the same symbol to denote both the function and the associated Nemitski operator.
We say that \(\lambda_{\infty }\) is a bifurcation from infinity for (2.1) if there exist \(\mu_{n}\rightarrow \lambda_{\infty }\) and \(u_{n} \in X\) such that \(u_{n}\mu_{n}Kf(u_{n})=0\) and \(\Vert u_{n} \Vert \rightarrow \infty \).
In some situations, like the specific ones we will discuss later, an appropriate rescaling allows us to find bifurcation from infinity by means of the Leray–Schauder topological degree, denoted by \(\deg (\cdot , \cdot ,\cdot )\). Recall that \(K: X\rightarrow X\) is (continuous and) compact, and hence it makes sense to consider the topological degree of \(I\lambda Kf\), where I is the identity map.
 \((f_{2})\) :

there is \(m>0\) such that$$ \lim_{u\rightarrow +\infty }\frac{f(x,u)}{u}=m. $$
Theorem 2.1
 (i)
\(a>0\) (possibly +∞) in \([0,1]\), and \(\lambda \in ( \lambda_{\infty }\epsilon ,\lambda_{\infty })\); or
 (ii)
\(A <0\) (possibly −∞) in \([0,1]\), and \(\lambda \in ( \lambda_{\infty },\lambda_{\infty }+\epsilon )\).
Clearly, any \(u>0\) such that \(\varPhi (\lambda ,u)=0\) is a positive solution of (1.1).
Lemma 2.1
 (i)
if \(a>0\), then we can also take \(\varLambda =[\lambda_{\infty }, \lambda ]\) for all \(\lambda > \lambda_{\infty }\), and
 (ii)
if \(A<0\), then we can also take \(\varLambda =[0,\lambda_{\infty }]\).
Proof
We prove statement (ii) similarly to (i). Taking \(\mu_{n}\uparrow \lambda_{\infty }\), it follows that \(w\geq 0\) satisfies (2.2), and hence there exists \(\beta >0\) such that \(w=\beta \phi_{1}\). Then we have \(u_{n}=\Vert u_{n} \Vert w_{n}\rightarrow +\infty \) and \(F(u_{n})=f(u_{n})\) for n large.
Lemma 2.2
Proof
Taking into account that \(F(x, u)\simeq m\vert u \vert \) as \(\vert u \vert \rightarrow \infty \), we can repeat the arguments of Lemma 3.3 of [16] with some minor changes. □
Lemma 2.3
\(\lambda_{\infty }\) is a bifurcation from infinity for (2.1). More precisely, there exists an unbounded closed connected set \(\varSigma_{\infty }\subset \varSigma \) that bifurcates from infinity. Moreover, \(\varSigma_{\infty }\) bifurcates to the left (to the right), provided that \(a>0\) (respectively, \(A<0\)).
Proof of Theorem 2.1
Example 2.1
Notice that \(\lambda_{1}=\pi^{4}\) and \(\lambda_{\infty }=\frac{\pi ^{4}}{10}\). Thus by Theorem 2.1 there exists \(\epsilon >0\) such that (2.6) has positive solutions, provided that \(\lambda \in ( \lambda_{\infty }\epsilon , \lambda_{\infty })\). Moreover, Lemma 2.3 guarantees that there exists an unbounded closed connected set of positive solutions \(\varSigma_{\infty }\subset \varSigma \) that bifurcates from infinity and bifurcates to the left of \(\lambda_{\infty }\).
3 Superlinear problems
 \((f_{3})\) :

there is \(b\in C([0,1]),b>0\), such that \(\lim_{u\rightarrow \infty }u^{p}f(x,u) =b\) uniformly in \(x\in [0,1]\) with \(p>1\).
Lemma 3.1
([6])
 (1)
If \(\Vert Fx \Vert \leq \Vert x \Vert \) for \(x\in \partial \varOmega_{p}\), then \(i(F,\varOmega_{p},\varOmega )=1\).
 (2)
If \(\Vert Fx \Vert \geq \Vert x \Vert \) for \(x\in \partial \varOmega_{p}\), then \(i(F,\varOmega_{p},\varOmega )=0\).
Our main result is the following:
Theorem 3.1
Let \(f \in C([0,1]\times \mathbb{R^{+}},\mathbb{R})\) satisfy \((f_{1})\) and \((f_{3})\). Then there exists \(\lambda_{*}>0\) such that (1.1) has positive solutions for all \(0<\lambda \leq \lambda_{*}\). More precisely, there exists a connected set of positive solutions of (1.1) bifurcating from infinity at \(\lambda_{\infty }=0\).
Proof
Lemma 3.2
 (i)
\(\deg (S(\gamma , \cdot ), \varOmega_{R}\setminus \bar{\varOmega }_{r}, 0)=1\) \(\forall 0 \leq \gamma \leq \gamma_{0}\);
 (ii)
if \(S(\gamma ,w)=0\), \(\gamma \in [0,\gamma_{0}]\), \(r \leq \Vert w \Vert \leq R\), then \(w>0\) in \((0,1)\).
Proof
Proof of Theorem 3.1 completed
4 Sublinear problems
 \((f_{4})\) :

\(\exists b\in C([0,1]), b>0\), such that \(\lim_{u\rightarrow \infty }u^{q}f(x,u)= b\) uniformly in \(x\in [0,1]\) with \(0\leq q<1\).
Lemma 4.1
 (i)
\(\deg (S(\gamma , \cdot ), O_{R}\setminus \bar{O}_{r}, 0)=1\) \(\forall 0 \leq \gamma \leq \gamma_{0}\);
 (ii)
if \(S(\gamma ,w)=0\), \(\gamma \in [0,\gamma_{0}]\), \(r \leq \Vert w \Vert \leq R\), then \(w>0\) in \((0,1)\).
Proof
Theorem 4.1
Let \(f\in C([0,1]\times \mathbb{R^{+}}, \mathbb{R})\) satisfy \((f_{1})\) and \((f_{4})\). Then there is \(\lambda^{*}>0\) such that (1.1) has positive solutions for all \(\lambda \geq \lambda^{*}\). More precisely, there exists a connected set of positive solutions of (1.1) bifurcating from infinity for \(\lambda_{\infty }=+\infty \).
Proof of Theorem 4.1
Declarations
Acknowledgements
The authors are very grateful to an anonymous referee for very valuable suggestions.
Availability of data and materials
Data sharing not applicable to this paper as no datasets were generated.
Funding
This work was supported by National Natural Science Foundation of China (No. 11671322).
Authors’ contributions
The authors claim that the research was realized in collaboration with the same responsibility. Both authors read and approved the last version of the manuscript.
Competing interests
Both authors of this paper declare that they have no competing interests.
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Authors’ Affiliations
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