Theory and Modern Applications

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

## 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.

## 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 [1,2,3] and the references therein.

Ambrosetti  studied the existence of positive solutions for semipositone elliptic problems via bifurcation theory. Recently, Hai and Shivaji  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 [6,7,8,9,10,11] along this line. However, there are few results for fourth-order semipositone problems; see . Ma  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 . 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 . 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 .

Motivated by Ambrosetti , 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.

## 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  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 $$\lambda_{\infty }$$.

## 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

()

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 , 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. □

## 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 , 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. □

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### Acknowledgements

The authors are very grateful to an anonymous referee for very valuable suggestions.

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## Funding

This work was supported by National Natural Science Foundation of China (No. 11671322).

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