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Global behavior of positive solutions for some semipositone fourthorder problems
Advances in Difference Equations volume 2018, Article number: 443 (2018)
Abstract
In this paper, we study the global behavior of positive solutions of fourthorder boundary value problems
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.
1 Introduction
In this paper, we study the global behavior of positive solutions of fourthorder boundary value problems
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 socalled semipositone problem when f is such that
 \((f_{1})\) :

\(f(x, 0) < 0\) \(\forall x \in (0,1)\).
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 [1,2,3] 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 secondorder semipositone problems
via a Krasnosel’skii fixedpointtype theorem in a Banach space.
The existence and multiplicity of positive solutions of fourthorder positone problems have been studied by several authors; see [6,7,8,9,10,11] along this line. However, there are few results for fourthorder semipositone problems; see [12]. Ma [12] used the fixed point theorem in cones to show that the problem
has a positive solution if \(\lambda >0\) is small enough, where \(\tilde{f}(x, u, p)\geq M\) for some positive constant M, and
There is a big difference in the study of fourth and secondorder problems. For example:

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 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.
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 \(\mathbb{R^{+}}=[0, \infty )\).
We define \(K: X\rightarrow X\) by
and
We write \(u=Kv\) if
With this notation, problem (1.1) is equivalent to
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
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

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

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

(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
Suppose on the contrary that there exists a sequence \(\{(\mu _{n}, u_{n})\}\) satisfying
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
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
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
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
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
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
a contradiction if \(A<0\). □
Lemma 2.2
If \(\lambda >\lambda_{\infty }\), then there exists \(r>0\) such that
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
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
Similarly, by Lemma 2.2 we infer that, for all \(\tau \in [0,1]\) and \(\lambda >\lambda_{\infty }\),
Let us set
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
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
in \((0,1)\) for n large. □
Example 2.1
Let us consider the fourthorder semipositone boundary value problem
where \(\lambda >0\) and \(f(t,x)=10x+t\ln (1+x)t\).
Obviously,
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

(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
As before, we set
and let
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
we use the rescaling \(w=\gamma u, \lambda =\gamma^{p1}, \gamma >0\). A direct calculation shows that \((\lambda ,u)\), \(\lambda >0\), is a solution of (3.1) if and only if
where
We can extend F̃ to \(\gamma =0\) by setting
and by \((f_{3})\) such an extension is continuous. We set
Let us point out explicitly that \(S(\gamma ,\cdot )=IK\) with compact K. For \(\gamma =0\), solutions of \(S_{0}(w):=S(0,w)=0\) are nothing but solutions of
We claim that there exist two constants \(R>r>0\) such that
Assume on the contrary that (3.5) is not true. Then there exists a sequence \(\{w_{n}\}\) of solutions of (3.4) satisfying
In fact, we have from (3.4) that
since
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
Let \(v_{n}:= w_{n}/\Vert w_{n} \Vert \). From (3.4) we have
By the standard argument, after taking a subsequence and relabeling if necessary, it follows that
and there exists \(v_{*}\in X\) with \(\Vert v_{*} \Vert =1\) such that
and
which implies that \(v_{*}=0\). However, this is a contradiction, Therefore (3.6) is valid.
Now, from (3.5) and (3.6), we deduce
This implies
Thus the degree \(\deg (S_{0}, \varOmega_{R}\setminus \varOmega_{r},0)\) is well defined.
Next, we show that
To this end, let us define
and
Using Lemma 3.1 and an argument similar to that in the proof of [6], Theorem 3, we deduce
By the excision and the additivity properties of the degree it follows that
and accordingly,
that is,
Lemma 3.2
There exists \(\gamma >0\) such that

(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
Clearly, (i) follows if we show that
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
a contradiction with (3.5) and (3.6).
Thus, by (3.7) and homotopy we get that
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
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 \(\lambda =\gamma^{p1}, u=w/\gamma \) gives a solution \((\lambda , u_{\lambda })\) of (3.1) for all \(0<\lambda <\lambda_{*}:=\gamma_{0}^{p1}\). 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
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 \(w=\gamma u,\lambda =\gamma^{q1}\) 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
For future reference, note that by Lemma 3.1
has a unique positive solution \(w_{0}\).
We claim that there exist two constants \(R>r>0\) such that
Assume on the contrary that (4.3) is not true. Then there exists a sequence \({w_{n}}\) of solutions of (4.4) satisfying
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
By the standard argument, after taking a subsequence and relabeling if necessary, it follows that
and there exists \(v_{*}\in X\) with \(\Vert v_{*} \Vert =1\) such that
and
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
In fact, we have from (3.4) that
since
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
This implies that
Thus, the degree \(\deg (S_{0}, O_{R}\setminus \bar{O}_{r},0)\) is well defined.
Next, we show that
To this end, let us define
and
Using Lemma 3.1 and an argument similar to that in the proof of [6], Theorem 3, we deduce
By the excision and the additivity properties of the degree it follows that
and accordingly,
that is,
Lemma 4.1
There exists \(\gamma >0\) such that

(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
Clearly, (i) follows if we show that
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
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
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
gives a solution \((\lambda , u_{\lambda })\) of (3.1) for all \(\lambda \geq \lambda^{*}:=\gamma_{0}^{q1}\). 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
This completes the proof. □
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Yan, D., Ma, R. Global behavior of positive solutions for some semipositone fourthorder problems. Adv Differ Equ 2018, 443 (2018). https://doi.org/10.1186/s1366201819044
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DOI: https://doi.org/10.1186/s1366201819044