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Global behavior of positive solutions for some semipositone fourth-order problems

Advances in Difference Equations20182018:443

https://doi.org/10.1186/s13662-018-1904-4

  • Received: 2 July 2018
  • Accepted: 22 November 2018
  • Published:

Abstract

In this paper, we study the global behavior of positive solutions of fourth-order boundary value problems
$$ \textstyle\begin{cases} u''''=\lambda f(x,u), \quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0, \end{cases} $$
where \(f: [0,1]\times \mathbb{R^{+}} \to \mathbb{R}\) is a continuous function with \(f(x,0)<0\) in \((0, 1)\), and \(\lambda >0\). The proof of our main results are based upon bifurcation techniques.

Keywords

  • 34B18
  • 34B16
  • 34B25
  • 47H11

MSC

  • Positive solutions
  • Topological degree
  • Connected set
  • Bifurcation

1 Introduction

In this paper, we study the global behavior of positive solutions of fourth-order boundary value problems
$$ \textstyle\begin{cases} u''''=\lambda f(x,u),\quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0, \end{cases} $$
(1.1)
where \(\lambda > 0\) and \(f:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R}\). If \(f(x, 0)\geq 0\), then (1.1) is called a positone problem.
On the contrary, here we deal with the so-called semipositone problem when f is such that
\((f_{1})\)

\(f(x, 0) < 0\) \(\forall x \in (0,1)\).

The existence of positive solutions of second-order 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 [13] and the references therein.

Ambrosetti [4] studied the existence of positive solutions for semipositone elliptic problems via bifurcation theory. Recently, Hai and Shivaji [5] obtained the existence of positive solutions for second-order semipositone problems
$$ \textstyle\begin{cases} -u''=\lambda h(t)f(u), & t\in(0,1), \\ u(0)=0,&u'(1)+c (u(1) )u(1)=0 \end{cases} $$
via a Krasnosel’skii fixed-point-type theorem in a Banach space.
The existence and multiplicity of positive solutions of fourth-order positone problems have been studied by several authors; see [611] along this line. However, there are few results for fourth-order semipositone problems; see [12]. Ma [12] used the fixed point theorem in cones to show that the problem
$$ \textstyle\begin{cases} u''''=\lambda\tilde{f} (x,u(x),u'(x) ), \quad t\in (0,1), \\ u(0)=u'(0)=u''(1)=u'''(1)=0 \end{cases} $$
has a positive solution if \(\lambda >0\) is small enough, where \(\tilde{f}(x, u, p)\geq -M\) for some positive constant M, and
$$ \lim_{p\rightarrow \infty }\frac{f(x,u,p)}{p}=\infty . $$
There is a big difference in the study of fourth- and second-order problems. For example:
  1. 1.

    Spectrum theory for singular second-order linear eigenvalue problems has been established via Prüfer transform in [13]. However, the spectrum structure of singular fourth-order linear eigenvalue problems is not established so far.

     
  2. 2.
    The uniqueness of solutions of second-order problems
    $$ \textstyle\begin{cases} -u''=\lambda u^{q}, &x\in (a,b), \\ u>0,&x\in (a,b), \\ u(a)=u(b)=0 \end{cases} $$
    has been obtained in [14]. However, the uniqueness of solution of
    $$ \textstyle\begin{cases} w''''=b\vert w \vert ^{\alpha }, \quad x\in (0,1), \\ w(0)=w(1)=w''(0)=w''(1)=0 \end{cases} $$
    is not obtained so far.
     
  3. 3.

    It is well known that, for a second-order differential equation with periodic, Neumann, or Dirichlet boundary conditions, the existence of a well-ordered 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 fourth-order differential equations; see Remark 3.1 in [15].

     

Motivated by Ambrosetti [4], we investigate the global behavior of positive solutions of the fourth-order 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 fourth-order 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 \(\mathbb{R^{+}}=[0, \infty )\).

We define \(K: X\rightarrow X\) by
$$ Ku(t):= \int_{0}^{1} \int_{0}^{1}G(t,s)G(s,\tau )f \bigl(\tau ,u(\tau ) \bigr)\,d\tau \,ds $$
and
$$ G(t,s)= \textstyle\begin{cases} t(1-s),&0 \leq t \leq s \leq 1, \\ s(1-t),&0 \leq s \leq t \leq 1. \end{cases} $$
We write \(u=Kv\) if
$$ \textstyle\begin{cases} u''''=v,\quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0. \end{cases} $$
With this notation, problem (1.1) is equivalent to
$$ u-\lambda Kf(u)=0,\quad u\in X. $$
(2.1)

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.

We suppose that \(f\in C([0,1]\times \mathbb{R^{+}}, \mathbb{R})\) satisfies \((f_{1})\) and
\((f_{2})\)
there is \(m>0\) such that
$$ \lim_{u\rightarrow +\infty }\frac{f(x,u)}{u}=m. $$
Let \(\lambda_{\infty }=\frac{\lambda_{1}}{m}\) and define
$$ a(x)=\liminf_{u\rightarrow +\infty } \bigl(f(x,u)-mu \bigr),\qquad A(x)= \limsup _{u\rightarrow +\infty } \bigl(f(x,u)-mu \bigr). $$

Theorem 2.1

Suppose that f satisfies \((f_{1})\) and \((f_{2})\). Then there exists \(\epsilon >0\) such that (1.1) has positive solutions, provided that either
  1. (i)

    \(a>0\) (possibly +∞) in \([0,1]\), and \(\lambda \in ( \lambda_{\infty }-\epsilon ,\lambda_{\infty })\); or

     
  2. (ii)

    \(A <0\) (possibly −∞) in \([0,1]\), and \(\lambda \in ( \lambda_{\infty },\lambda_{\infty }+\epsilon )\).

     
The proof of Theorem 2.1 will be carried out in several steps. First of all, we extend \(f(x,\cdot )\) to the whole \(\mathbb{R}\) by setting
$$ F(x, u)=f \bigl(x,\vert u \vert \bigr). $$
For \(u \in X\),
$$ \varPhi (\lambda ,u):=u-\lambda KF(u). $$

Clearly, any \(u>0\) such that \(\varPhi (\lambda ,u)=0\) is a positive solution of (1.1).

Lemma 2.1

For every compact interval \(\varLambda \subset \mathbb{R^{+}}\backslash \{\lambda_{\infty }\}\), there exists \(r>0\) such that
$$ \varPhi (\lambda ,u)\neq 0\quad \forall \Vert u \Vert \geq r. $$
Moreover,
  1. (i)

    if \(a>0\), then we can also take \(\varLambda =[\lambda_{\infty }, \lambda ]\) for all \(\lambda > \lambda_{\infty }\), and

     
  2. (ii)

    if \(A<0\), then we can also take \(\varLambda =[0,\lambda_{\infty }]\).

     

Proof

Suppose on the contrary that there exists a sequence \(\{(\mu _{n}, u_{n})\}\) satisfying
$$ \mu _{n}\in \varLambda ; \qquad \Vert u_{n} \Vert \geq n \quad \text{for } n\in \mathbb{N}; \qquad u_{n}=\mu _{n}KF(u_{n}). $$
Obviously, \(\Vert u_{n} \Vert \geq n\) implies that \(u_{n}(x)\not \equiv 0\). We may assume that \(\mu _{n}\rightarrow \mu \) for some \(\mu \neq \lambda _{\infty }\).
Setting \(w_{n}=u_{n}\Vert u_{n} \Vert ^{-1}\), we find
$$ w_{n}=\mu_{n}\Vert u_{n} \Vert ^{-1}KF(u_{n}). $$
Since \(w_{n}\) is bounded in X, after taking a subsequence if necessary, we have that \(w_{n}\rightarrow w\) in X, where w is such that \(\Vert w \Vert =1\) and satisfies
$$ \textstyle\begin{cases} w''''=\mu m\vert w \vert , \quad x\in (0,1), \\ w(0)=w(1)=w''(0)=w''(1). \end{cases} $$
By the maximum principle it follows that \(w\geq 0\). Since \(\Vert w \Vert =1\), we infer that \(\mu m=\lambda_{1}\), namely \(\mu =\lambda_{\infty }\), a contradiction that proves the first statement.
We will give a short sketch of (i). Taking \(\mu_{n}\downarrow \lambda_{\infty }\), it follows that \(w\geq 0\) satisfies
$$ \textstyle\begin{cases} w''''=\lambda_{1}w,\quad x\in (0,1), \\ w(0)=w(1)=w''(0)=w''(1)=0, \end{cases} $$
(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.
From \(\varPhi (\lambda_{n},u_{n})=0\) it follows that
$$ \lambda_{1} \int_{0}^{1}u_{n}\phi_{1}\,dx= \mu_{n} \int_{0}^{1} \bigl(f(u_{n})-mu _{n} \bigr)\phi_{1}\,dx+\mu_{n}m \int_{0}^{1}u_{n}\phi_{1}\,dx. $$
(2.3)
Since \(\mu_{n}>\lambda_{\infty }\) and \(\int_{0}^{1}u_{n}\phi_{1}\,dx>0\) for n large, we infer that \(\int_{0}^{1}(f(u_{n})-mu_{n})\phi_{1}\,dx<0\) for n large, and the Fatou lemma yields
$$\begin{aligned} 0&\geq \liminf\int_{0}^{1} \bigl(f(u_{n})-mu_{n} \bigr)\phi_{1}\,dx \\ &\geq \int_{0} ^{1}a\phi_{1}\,dx, \end{aligned}$$
a contradiction if \(a>0\).

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.

From \(\varPhi (\lambda_{n},u_{n})=0\) we have (2.3); since \(\mu_{n}< \lambda_{\infty }\) and \(\int_{0}^{1}u_{n}\phi_{1}\,dx>0\) for n large, we infer that \(\int_{0}^{1}(f(u_{n})-mu_{n})\phi_{1}\,dx>0\) for n large, and the Fatou lemma yields
$$ 0\leq \liminf \int_{0}^{1} \bigl(f(u_{n})-mu_{n} \bigr)\phi_{1}\,dx\leq \int_{0} ^{1}A\phi_{1}\,dx, $$
a contradiction if \(A<0\). □

Lemma 2.2

If \(\lambda >\lambda_{\infty }\), then there exists \(r>0\) such that
$$ \varPhi (\lambda ,u)\neq t\phi_{1}\quad \forall t\geq 0,\Vert u \Vert \geq r. $$

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. □

For \(u\neq 0\), we set \(z=u\Vert u \Vert ^{-2}\). Letting
$$\begin{aligned} \varPsi (\lambda ,z)&=\Vert u \Vert ^{2}\varPhi (\lambda ,u) =z-\lambda \Vert z \Vert ^{2}KF \biggl(\frac{z}{ \Vert z \Vert ^{2}} \biggr), \end{aligned}$$
we have that \(\lambda_{\infty }\) is a bifurcation from infinity for (2.1) if and only if it is a bifurcation from the trivial solution \(z=0\) for \(\varPsi =0\). From Lemma 2.1 by homotopy it follows that
$$\begin{aligned} \deg \bigl(\varPsi (\lambda ,\cdot ),B_{1/r},0 \bigr)&=\deg \bigl(\varPsi (0, \cdot ),B_{1/r},0 \bigr) \\ &= \deg (I,B_{1/r},0)=1\quad \forall \lambda < \lambda_{\infty }. \end{aligned}$$
(2.4)
Similarly, by Lemma 2.2 we infer that, for all \(\tau \in [0,1]\) and \(\lambda >\lambda_{\infty }\),
$$\begin{aligned} \deg \bigl(\varPsi (\lambda ,\cdot ),B_{1/r},0 \bigr)&=\deg \bigl(\varPsi ( \lambda ,\cdot )- \tau \phi_{1},B_{1/r},0 \bigr) \\ &=\deg \bigl(\varPsi (\lambda ,\cdot )-\phi_{1},B_{1/r},0 \bigr)=0\quad \forall \lambda < \lambda_{\infty }. \end{aligned}$$
(2.5)
Let us set
$$ \varSigma = \bigl\{ (\lambda ,u)\in \mathbb{R^{+}}\times X: u\neq 0,\varPhi ( \lambda ,u)=0 \bigr\} . $$
From (2.4) and (2.5) and the preceding discussion we deduce the following:

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

By the previous lemmas it suffices to show that if \(\mu_{n}\rightarrow \lambda_{\infty }\) and \(\Vert u_{n} \Vert \rightarrow \infty \), then \(u_{n}>0\) in \([0,1]\) for n large. Setting
$$ w_{n}=u_{n}\Vert u_{n} \Vert ^{-1} $$
and using the preceding arguments, we find that, up to subsequence, \(w_{n}\rightarrow w\) in X and \(w=\beta \phi_{1}\), \(\beta >0\). Then it follows that
$$ u_{n}>0 $$
in \((0,1)\) for n large. □

Example 2.1

Let us consider the fourth-order semipositone boundary value problem
$$ \textstyle\begin{cases} x''''(t)=\lambda f(t,x),\quad t\in (0,1), \\ x(0)=x(1)=x''(0)=x''(1)=0, \end{cases} $$
(2.6)
where \(\lambda >0\) and \(f(t,x)=10x+t\ln (1+x)-t\).
Obviously,
$$\begin{aligned}& f(t, 0)< 0,\quad t\in (0,1); \\& \lim_{x\to \infty }\frac{f(t,x)}{x}=10=:m; \\& a(t)=\liminf_{x\rightarrow +\infty } \bigl(f(t,x)-mx \bigr)=\liminf _{x\rightarrow +\infty } \bigl(t\ln (1+x)-t \bigr)>0, \quad t\in (0,1). \end{aligned}$$

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

We study the existence of positive solutions of problem (1.1) when \(f(x,\cdot )\) is superlinear. Precisely, we suppose that \(f\in C([0,1] \times \mathbb{R^{+}},\mathbb{R})\) satisfies \((f_{1})\) and
\((f_{3})\)

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])

Let X be a Banach space, and let \(\varOmega \subset X\) be a cone in X. For \(p>0\), define \(\varOmega_{p}=\{x\in\varOmega \mid \vert x \vert < p\}\). Assume that \(F:\varOmega_{p}\rightarrow \varOmega \) is completely continuous such that
$$ Fx\neq x, \quad x\in \partial \varOmega_{p}= \bigl\{ x\in \varOmega \mid \vert x \vert =p \bigr\} . $$
  1. (1)

    If \(\Vert Fx \Vert \leq \Vert x \Vert \) for \(x\in \partial \varOmega_{p}\), then \(i(F,\varOmega_{p},\varOmega )=1\).

     
  2. (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

As before, we set
$$ F(x,u)=f \bigl(x,\vert u \vert \bigr) $$
and let
$$ G(x,u)=F(x,u)-b\vert u \vert ^{p}. $$
For the remainder of the proof, we omit the dependence with respect to \(x\in [0,1]\).
To prove that \(\lambda_{\infty }=0\) is a bifurcation from infinity for
$$ u-\lambda KF(u)=0, $$
(3.1)
we use the rescaling \(w=\gamma u, \lambda =\gamma^{p-1}, \gamma >0\). A direct calculation shows that \((\lambda ,u)\), \(\lambda >0\), is a solution of (3.1) if and only if
$$ w-K\tilde{F}(\gamma ,w)=0, $$
(3.2)
where
$$ \tilde{F}(\gamma ,w):=b\vert w \vert ^{p}+\gamma^{p}G \bigl(\gamma^{-1}w \bigr). $$
(3.3)
We can extend to \(\gamma =0\) by setting
$$ \tilde{F}(0,w)=b\vert w \vert ^{p}, $$
and by \((f_{3})\) such an extension is continuous. We set
$$ S(\gamma ,w)=w-K\tilde{F}(\gamma ,w),\quad \gamma \in \mathbb{R^{+}}. $$
Let us point out explicitly that \(S(\gamma ,\cdot )=I-K\) with compact K. For \(\gamma =0\), solutions of \(S_{0}(w):=S(0,w)=0\) are nothing but solutions of
$$ \textstyle\begin{cases} w''''=b\vert w \vert ^{p},\quad x\in (0,1), \\ w(0)=w(1)=w''(0)=w''(1)=0. \end{cases} $$
(3.4)
We claim that there exist two constants \(R>r>0\) such that
$$\begin{aligned}& S_{0}(w)\neq 0 \quad \forall \Vert w \Vert \geq R, \end{aligned}$$
(3.5)
$$\begin{aligned}& S_{0}(w)\neq 0 \quad \forall \Vert w \Vert \leq r. \end{aligned}$$
(3.6)
Assume on the contrary that (3.5) is not true. Then there exists a sequence \(\{w_{n}\}\) of solutions of (3.4) satisfying
$$ \Vert w_{n} \Vert \to \infty , n\rightarrow \infty . $$
(3.7)
In fact, we have from (3.4) that
$$ \textstyle\begin{cases} w_{n}''''= (b\vert w_{n} \vert ^{p-1} )w_{n}, \quad x\in (0,1), \\ w_{n}(0)=w_{n}(1)=w_{n}''(0)=w_{n}''(1)=0, \end{cases} $$
since
$$ \lim_{n\rightarrow \infty } \bigl(b\vert w_{n} \vert ^{p-1} \bigr)=\infty \quad \text{uniformly in } x\in [1/4,3/4], $$
which means that \(w_{n}\) must change its sign in \([1/4,3/4]\). However, this is a contradiction. Therefore (3.5) is valid.
Assume on the contrary that (3.6) is not true. Then there exists a sequence \({w_{n}}\) of solutions of (3.4) satisfying
$$ \Vert w_{n} \Vert >0 \quad \forall {n\in \mathbb{N}}; \qquad \Vert w_{n} \Vert \rightarrow 0, \quad n\rightarrow \infty . $$
(3.8)
Let \(v_{n}:= w_{n}/\Vert w_{n} \Vert \). From (3.4) we have
$$ \textstyle\begin{cases} v_{n}''''=(b\vert w_{n} \vert ^{p-1})v_{n},\quad x\in (0,1), \\ v_{n}(0)=v_{n}(1)=v_{n}''(0)=v_{n}''=0. \end{cases} $$
(3.9)
By the standard argument, after taking a subsequence and relabeling if necessary, it follows that
$$ \lim_{n\rightarrow \infty } \bigl(b\vert w_{n} \vert ^{p-1} \bigr)=0\quad \text{uniformly in } x\in [0,1], $$
and there exists \(v_{*}\in X\) with \(\Vert v_{*} \Vert =1\) such that
$$ v_{n}\rightarrow v_{*}, \quad n\rightarrow \infty , $$
and
$$ \textstyle\begin{cases} v_{*}''''=0,\quad x\in (0,1), \\ v_{*}(0)=v_{*}(1)=v_{*}''(0)=v_{*}''(1)=0, \end{cases} $$
which implies that \(v_{*}=0\). However, this is a contradiction, Therefore (3.6) is valid.
Now, from (3.5) and (3.6), we deduce
$$ S_{0}(w)\neq 0\quad \forall w\in \partial \varOmega_{R},\qquad S_{0}(w) \neq 0 \quad \forall w\in \partial \varOmega_{r}. $$
This implies
$$ S_{0}(w)\neq 0\quad \forall w\in \partial (\bar{\varOmega }_{R}\setminus \varOmega_{r}). $$
Thus the degree \(\deg (S_{0}, \varOmega_{R}\setminus \varOmega_{r},0)\) is well defined.
Next, we show that
$$ \deg (S_{0}, \varOmega_{R}\setminus \bar{\varOmega }_{r},0)=-1. $$
To this end, let us define
$$ \varOmega = \bigl\{ u\in C[0,1]:u(t)\geq 0 \text{ for } t\in [0,1] \bigr\} $$
and
$$ \varOmega_{\rho }= \bigl\{ u\in \varOmega : \bigl\Vert u(t) \bigr\Vert < \rho \bigr\} . $$
Using Lemma 3.1 and an argument similar to that in the proof of [6], Theorem 3, we deduce
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{r},\varOmega \bigr)=1, \qquad \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{R},\varOmega \bigr)=0. $$
(3.10)
By the excision and the additivity properties of the degree it follows that
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{R}\setminus \bar{\varOmega }_{r}, \varOmega \bigr)+\mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{r},\varOmega \bigr)= \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{R},\varOmega \bigr), $$
(3.11)
and accordingly,
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{R}\setminus \bar{\varOmega }_{r}, \varOmega \bigr)=\mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{R},\varOmega \bigr) - \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),\varOmega_{r},\varOmega \bigr)=-1, $$
(3.12)
that is,
$$ \deg (S_{0},\varOmega_{R}\setminus \bar{\varOmega }_{r},0)=-1. $$

Lemma 3.2

There exists \(\gamma >0\) such that
  1. (i)

    \(\deg (S(\gamma , \cdot ), \varOmega_{R}\setminus \bar{\varOmega }_{r}, 0)=-1\) \(\forall 0 \leq \gamma \leq \gamma_{0}\);

     
  2. (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

Clearly, (i) follows if we show that
$$ S(\gamma ,w)\neq 0, \quad 0 \leq \gamma \leq \gamma_{0}, \Vert w\Vert \in \{r,R\}. $$
Otherwise, there exists a sequence \((\gamma_{n}, w_{n})\) with \(\gamma_{n}\rightarrow 0\), \(\Vert w_{n} \Vert \in \{r,R\}\), and \(w_{n}=K \tilde{F}(\gamma_{n}, w_{n})\). Since K is compact, then, up to a subsequence, \(w_{n}\rightarrow w\), and
$$ S_{0}(w)=0,\quad \Vert w \Vert \in \{r,R\}, $$
a contradiction with (3.5) and (3.6).
Thus, by (3.7) and homotopy we get that
$$ \deg \bigl(S(\gamma ,\cdot ), \varOmega_{R}\setminus \varOmega_{r},0 \bigr)=-1. $$
To prove (ii), we argue again by contradiction. As in the preceding argument, we find a sequence \(w_{n}\in X\) with \(\{x\in [0,1]: w_{n}(x) \leq 0\}\neq \varnothing \) such that \(w_{n}\rightarrow w\), \(\Vert w \Vert \in [r,R]\), and \(S_{0}(w)=0\); namely, w solves (3.4). By the maximum principle, \(w>0\) on (0,1) and X. Moreover, without relabeling, \(w_{n}\rightarrow w\) in X. Therefore
$$ w_{n}>0,\quad x\in (0,1), $$
for n large, a contradiction. □

Proof of Theorem 3.1 completed

By Lemma 3.2 problem (3.2) has a positive solution \(w_{\gamma }\) for all \(0\leq \gamma \leq \gamma_{0}\). As remarked before, for \(\gamma >0\), the rescaling \(\lambda =\gamma^{p-1}, u=w/\gamma \) gives a solution \((\lambda , u_{\lambda })\) of (3.1) for all \(0<\lambda <\lambda_{*}:=\gamma_{0}^{p-1}\). Since \(w_{\gamma }>0\), \((\lambda , u_{\lambda })\) is a positive solution of (1.1). Finally, \(\Vert w_{\gamma } \Vert \geq r\) for all \(\gamma \in [0,\gamma_{0}]\) implies that
$$ \Vert u_{\lambda } \Vert =\Vert w \Vert _{\gamma }/\gamma \rightarrow \infty \quad \text{as }\gamma \rightarrow 0. $$
This completes the proof. □

4 Sublinear problems

In this final section, we deal with sublinear f, namely with \(f\in C([0,1]\times \mathbb{R^{+}},\mathbb{R})\) that satisfy \((f_{1})\) and
\((f_{4})\)

\(\exists b\in C([0,1]), b>0\), such that \(\lim_{u\rightarrow \infty }u^{-q}f(x,u)= b\) uniformly in \(x\in [0,1]\) with \(0\leq q<1\).

We will show that in this case positive solutions of (1.1) branch off from ∞ for \(\lambda_{\infty }=+\infty \). First, some preliminaries are in order. It is convenient to work on X. Following the same procedure as for the superlinear case, we employ the rescaling \(w=\gamma u,\lambda =\gamma^{q-1}\) and use the same notation with q instead of p. As before, \((\lambda , u)\) solves (3.1) if and only if \((\gamma ,w)\) satisfies (3.2). Note that now, since \(0\leq q<1\), we have that
$$ \lambda \rightarrow +\infty \quad \Leftrightarrow \quad \gamma \rightarrow 0. $$
(4.1)
For future reference, note that by Lemma 3.1
$$ \textstyle\begin{cases} u''''=bw^{q},\quad x\in (0,1), \\ u(0)=u(1)=u''(0)=u''(1)=0 \end{cases} $$
(4.2)
has a unique positive solution \(w_{0}\).
We claim that there exist two constants \(R>r>0\) such that
$$\begin{aligned}& S_{0}(w)\neq 0 \quad \forall \Vert w \Vert \leq R; \end{aligned}$$
(4.3)
$$\begin{aligned}& S_{0}(w)\neq 0 \quad \forall \Vert w \Vert \geq r; \end{aligned}$$
(4.4)
$$\begin{aligned}& \deg (S_{0}, O_{R}\setminus O_{r},0)=1. \end{aligned}$$
(4.5)
Assume on the contrary that (4.3) is not true. Then there exists a sequence \({w_{n}}\) of solutions of (4.4) satisfying
$$ \Vert w_{n} \Vert \rightarrow 0, \quad n\rightarrow \infty , $$
(4.6)
then \(w_{n}\equiv 0\) in \([0,1]\) for n large.
Let \(v_{n}:= w_{n}/\Vert w_{n} \Vert \). From (3.4) we have
$$ \textstyle\begin{cases} v_{n}''''=(b\vert w_{n} \vert ^{q-1})v_{n},\quad x\in (0,1),\\ v_{n}(0)=v_{n}(1)=v_{n}''(0)=v_{n}''(1)=0. \end{cases} $$
(4.7)
By the standard argument, after taking a subsequence and relabeling if necessary, it follows that
$$ \lim_{n\rightarrow \infty } \bigl(b\vert w_{n} \vert ^{q-1} \bigr)=0\quad \text{uniformly in } x\in [0,1], $$
and there exists \(v_{*}\in X\) with \(\Vert v_{*} \Vert =1\) such that
$$ v_{n}\rightarrow v_{*}, \quad n\rightarrow \infty , $$
and
$$ \textstyle\begin{cases} v_{*}''''=0,\quad x\in (0,1), \\ v_{*}(0)=v_{*}(1)=v_{*}''(0)=v_{*}''(1)=0, \end{cases} $$
which implies that \(v_{*}=0\). However, this is a contradiction, Therefore (4.3) is valid.
Assume on the contrary that (4.4) is not true. Then there exists a sequence \(\{w_{n}\}\) of solutions of (4.4) satisfying
$$ \Vert w_{n} \Vert \to \infty , \quad n\rightarrow \infty . $$
(4.8)
In fact, we have from (3.4) that
$$ \textstyle\begin{cases} w_{n}''''= (b\vert w_{n} \vert ^{q-1} )w_{n}, \quad x\in (0,1), \\ w_{n}(0)=w_{n}(1)=w_{n}''(0)=w_{n}''(1)=0, \end{cases} $$
since
$$ \lim_{n\rightarrow \infty } \bigl(b\vert w_{n} \vert ^{q-1} \bigr)=\infty \quad \text{uniformly in } x\in [1/4,3/4], $$
which shows that \(w_{n}\) must change its sign in \([1/4,3/4]\). However, this is a contradiction. Therefore (4.4) is valid.
Now, from (4.3) and (4.4) we deduce
$$ S_{0}(w)\neq 0 \quad \forall w\in \partial O_{R},\qquad S_{0}(w)\neq 0 \quad \forall w\in \partial O_{r}. $$
This implies that
$$ S_{0}(w)\neq 0\quad \forall w\in \partial (\bar{O}_{R} \setminus O _{r}). $$
Thus, the degree \(\deg (S_{0}, O_{R}\setminus \bar{O}_{r},0)\) is well defined.
Next, we show that
$$ \deg (S_{0}, O_{R}\setminus \bar{O}_{r},0)=1. $$
To this end, let us define
$$ O= \bigl\{ u\in C[0,1]:u(t)\geq 0 \text{ for } t\in [0,1] \bigr\} $$
and
$$ O_{\rho }= \bigl\{ u\in O:\Vert u \Vert < \rho \bigr\} . $$
Using Lemma 3.1 and an argument similar to that in the proof of [6], Theorem 3, we deduce
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),O_{r},O \bigr)=0,\qquad \mathrm{i} \bigl(K \tilde{F}(0,\cdot ),O_{R},O \bigr)=1. $$
By the excision and the additivity properties of the degree it follows that
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),O_{R}\setminus \bar{O}_{r},O \bigr)+ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),O_{r},O \bigr)=\mathrm{i} \bigl(K\tilde{F}(0, \cdot ),O_{R},O \bigr) , $$
and accordingly,
$$ \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),O_{R}\setminus \bar{O}_{r},O \bigr)= \mathrm{i} \bigl(K\tilde{F}(0,\cdot ),O_{R},O \bigr) -\mathrm{i} \bigl(K\tilde{F}(0, \cdot ),O_{r},O \bigr)=1, $$
that is,
$$ \deg (S_{0},O_{R}\setminus \bar{O}_{r},0)=1. $$

Lemma 4.1

There exists \(\gamma >0\) such that
  1. (i)

    \(\deg (S(\gamma , \cdot ), O_{R}\setminus \bar{O}_{r}, 0)=1\) \(\forall 0 \leq \gamma \leq \gamma_{0}\);

     
  2. (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

Clearly, (i) follows if we show that
$$ S(\gamma ,w)\neq 0, \quad 0 \leq \gamma \leq \gamma_{0}, \Vert w \Vert \in \{r,R\}. $$
Otherwise, there exists a sequence \((\gamma_{n}, w_{n})\) with \(\gamma_{n}\rightarrow 0\), \(\Vert w_{n} \Vert \in \{r,R\}\), and \(w_{n}=K \tilde{F}(\gamma_{n}, w_{n})\). Since K is compact, then, up to a subsequence, \(w_{n}\rightarrow w\), and
$$ S_{0}(w)=0, \quad \Vert w \Vert \in \{r,R\}, $$
a contradiction with (4.3) and (4.4).
To prove (ii), we argue again by contradiction. As in the preceding argument, we find a sequence \(w_{n}\in X\) with \(\{x\in [0,1]: w_{n}(x) \leq 0\}\neq \varnothing \) such that \(w_{n}\rightarrow w,\Vert w \Vert \in [r,R]\), and \(S_{0}(w)=0\); namely, w solves (3.2). By the maximum principle, \(w>0\) on (0,1) and X. Moreover, without relabeling, \(w_{n}\rightarrow w\) in X. Therefore
$$ w_{n}>0, \quad x\in (0,1), $$
for n large, a contradiction. □

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

By Lemma 4.1 problem (3.2) has a positive solution \(w_{\gamma }\) for all \(0\leq \gamma \leq \gamma_{0}\). As remarked before, for \(\gamma >0\), the rescaling
$$ \lambda =\gamma^{q-1},\quad u=w/\gamma $$
gives a solution \((\lambda , u_{\lambda })\) of (3.1) for all \(\lambda \geq \lambda^{*}:=\gamma_{0}^{q-1}\). Since \(w_{\gamma }>0\), \((\lambda , u_{\lambda })\) is a positive solution of (1.1). Finally, \(\Vert w_{\gamma } \Vert \geq r\) for all \(\gamma \in [0,\gamma_{0}]\) implies that
$$ \Vert u_{\lambda } \Vert =\Vert w \Vert _{\gamma }/\gamma\rightarrow \infty \quad \text{as } \gamma \rightarrow 0. $$
This completes the proof. □

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.

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

(1)
Department of Mathematics, Northwest Normal University, Lanzhou, P.R. China

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