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Nontrivial solutions to boundary value problem with general singular differential operator
Boundary Value Problems volume 2015, Article number: 70 (2015)
Abstract
In this paper, a class of boundary value problems with general singular differential operator is investigated. The nonlinear term in the boundary value problem is signchanging and may be unbounded from below. By means of the topological degree of a completely continuous field, the existence of nontrivial solutions is obtained. Finally, an example is given to illustrate the application of our main result.
Introduction
In [1], Han and Wu considered the singular boundary value problem
under the following assumptions:

(H1)
There exist three constants \(b>0\), \(c>0\) and \(\alpha\in (0,1)\) such that
$$f(u)\geqbcu^{\alpha}, \quad u\in\mathbb{R}. $$ 
(H2)
\(h\in C((0,1), [0,+\infty))\), \(h(t)\not\equiv0\) in \((0,1)\) and \(\int_{0}^{1}t(1t)h(t)\,\mathrm{d}t<+\infty\).

(H3)
\(f:\mathbb{R}\rightarrow \mathbb{R}\) is continuous.
Denote by \(\tilde{\lambda}_{1}\) the first eigenvalue of the eigenvalue problem
By computing the LeraySchauder degree, they established the following result.
Theorem A
Assume that (H1)(H3) hold. If
then the singular boundary value problem (1.1) has at least one nontrivial solution.
This paper is mainly focused on the existence of nontrivial solutions to singular boundary value problem with general differential operator such as
We will make the following assumptions:

(A1)
\(a\in C(0,1)\cap L^{1}(0,1)\); \(b\in C((0, 1),(\infty,0))\) and \(\int_{0}^{1}s(1s)b(s)\,\mathrm{d}s<+\infty\).

(A2)
\(h:(0, 1)\rightarrow[0, \infty)\) is continuous and does not vanish identically on any subinterval of \((0,1)\); furthermore, \(\int_{0}^{1} s(1s)h(s)\,\mathrm{d}s<+\infty\).

(A3)
\(f: [0, 1]\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous and there exist two nonnegative functions \(c, d\in C[0,1]\), \(d\not\equiv0\) and \(B\in C(\mathbb{R}, [0,+\infty))\) with
$$\begin{aligned}& \lim_{u\to+\infty}\frac{B(u)}{u}=0 \end{aligned}$$(1.4)such that
$$\begin{aligned}& f(t, u)\geqc(t)d(t) B(u), \quad t\in[0,1], u\in\mathbb{R}. \end{aligned}$$(1.5) 
(A4)
\(\liminf_{u\to+\infty}\frac{f(t,u)}{u}>\lambda _{1}\) uniformly on \(t\in[0,1]\).

(A5)
\(\limsup_{u\to0}  \frac{f(t,u)}{u} <\lambda_{1}\) uniformly on \(t\in[0,1]\).
Here \(\lambda_{1}\) is the first eigenvalue of the following linear eigenvalue problem corresponding to the boundary value problem (1.3):
We state our main theorem as follows.
Theorem 1.1
Let (A1)(A5) hold, then the singular boundary value problem (1.3) has at least one nontrivial solution.
Remark 1.1
(A1) and (A2) show that the functions a, b and h are all allowed to be singular at \(t=0,1\), which means that the differential operator in the equation of problem (1.3) is more general than the classic SturmLiouville differential operator where \(a, b\in C[0,1]\) are required. It is worth mentioning that the solvability of singular boundary value problems with general differential operators were also investigated in [2], where the results on the existence of solutions do not involve the eigenvalue of the corresponding linear eigenvalue problem, therefore the conditions laid on the functions a, b in [2] are more general than the conditions in this paper.
Remark 1.2
The existence of the first eigenvalue \(\lambda_{1}\) and the properties of its corresponding positive eigenfunction \(\varphi_{1} \) in (1.6) (see Lemma 2.4 in Section 2) play a very important role in the proof of our main theorem. The presence of \(a(t)u'+b(t)u\), however, brings difficulties when it comes to proving that \(\varphi_{1}\in C^{1}[0,1]\). We find a way to overcome this problem. For the special case of (1.6), i.e., the eigenvalue problem (1.2), the complete results on eigenvalues and eigenfunctions were obtained by Asakawa [3].
Remark 1.3
Choose \(B(u)=u^{\alpha}\), \(\alpha\in(0,1)\) and the functions c, d to be constants on \([0,1]\). Then (A3) can be reduced to (H1). Therefore, the main theorem in this paper generalizes Theorem A in [1].
We note that the results on positive solutions or nontrivial solutions for other kinds of boundary value problems have been considered in many publications such as [3–10] and the references therein. The existence of nontrivial solutions involving the relation between the principal eigenvalue and the growth of the nonlinearity has been investigated in [11].
The rest of the paper is arranged as follows. In Section 2, we state some notations and prove some preliminary results. In Section 3, we prove our main result. In Section 4, an example is given to illustrate the application of our main result.
Notations and preliminary results
In this section, we recall some notations, abstract theorems and auxiliary results, which are important for proving our main result.
We denote by \(C[0,1]\) the Banach space with the norm \(\u\=\max_{t\in[0,1]}u(t)\). Let
be a positive cone in \(C[0,1]\). Denote by \(B_{r}=\{u \in C[0, 1] \mid \u\< r\}\) (\(r>0\)) the open ball of radius r. We denote by \(AC[0,1]\) the space of all absolute continuous functions on \([0,1]\). Let
Lemma 2.1
[4]
Suppose that (A1) holds. Then

(i)
the initial value problem
$$ \begin{aligned} &u''+a(t)u'+b(t)u=0 , \quad t\in(0,1), \\ &u(0)=0, \qquad u'(0)=1 \end{aligned} $$(2.1)has a unique solution \(\alpha\in AC[0,1]\cap C^{1}[0,1)\) and \(\alpha '\in AC_{\mathrm{loc}}[0,1)\);

(ii)
the initial value problem
$$ \begin{aligned} &u''+a(t)u'+b(t)u=0 , \quad t\in(0,1), \\ &u(1)=0, \qquad u'(1)=1 \end{aligned} $$(2.2)has a unique solution \(\beta\in AC[0,1]\cap C^{1}(0,1]\) and \(\beta'\in AC_{\mathrm{loc}}(0,1]\);

(iii)
α is nondecreasing on \([0,1]\), β is nonincreasing on \([0,1]\).
We remark here that from the proof of Lemma 2.2 and Lemma 2.3 in [4], the existence and uniqueness of α and β for the initial value problem (2.1) and (2.2) have no relation with the sign of the functions a and b. The sign condition laid on b in (A1) is meant to obtain the monotonicity of α and β.
Let
where \(\rho=\alpha'(\frac{1}{2})\beta(\frac{1}{2})\alpha(\frac{1}{2})\beta'(\frac{1}{2})\) is a positive constant.
We give the following remark. Although its proof is trivial, the consequences of the result are of major importance.
Remark 2.1
It follows from the above lemma that there exist positive constants \(c_{1}\), \(c_{2}\), \(\rho_{1}\) and \(\rho_{2}\) such that
Furthermore, for \(t,s\in[0,1]\),
From Lemma 2.6 in [4], the singular boundary value problem (1.3) can be converted into the equivalent integral equation
where \(q(s)=\exp(\int_{\frac{1}{2}}^{s}a(\tau)\,\mathrm{d}\tau)\). Define \(\tilde {h}(t)=q(t)h(t)\) for ease of notation. Combining (A2) and the definition of q, we see that \(\tilde{h}\) satisfies the following:
 (\(\tilde{\mathrm{A}}2\)):

\(\tilde{h}:(0, 1)\rightarrow[0, \infty)\) is continuous and does not vanish identically on any subinterval of \((0,1)\); furthermore, \(0<\int_{0}^{1} s(1s)\tilde{h}(s)\,\mathrm{d}s<+\infty\).
For any \(u\in C[0,1]\), let
From Lemma 2.9 in [4], then \(A: C[0,1]\rightarrow C[0,1]\) and \(T: C[0,1]\rightarrow C[0,1]\) are completely continuous operators, respectively.
Lemma 2.2
Suppose that (A1) and (A2) hold. Let \(\sigma\in(0, 1)\), \(r\in C(0,1)\) be continuous and \(\int_{0}^{1}s(1s) r(s) <+\infty\). If ω is a solution of
such that \(\omega\in C[0, 1]\cap C^{1}(0,1)\) and \(\omega'\in AC_{\mathrm{loc}}(0, 1)\), then \(\omega\equiv0\).
Proof
Let ω be a solution of initial value problem (2.6). Multiplying both sides of the equation of (2.6) by q and integrating it from σ to t, we get
Integrating the above equation from σ to t again, we have
Let
If \(t\in[\sigma, 1)\), then it follows from (2.7) that
Since \(\int_{\sigma}^{1}(1s)b(s)+r(s)\,\mathrm{d}s<+\infty\), we get \(\omega (t)\equiv0\) for \(t\in[\sigma,1]\) by Gronwall’s inequality. Using the same argument, we can easily get \(\omega(t)\equiv0\) for \(t\in[0, \sigma ]\). This completes the proof. □
We now turn to the eigenvalue problem (1.6). Before stating the results on the eigenvalue and eigenfunction, we give the following lemma, which is a direct result of the KreinRutman theorem.
Lemma 2.3
Suppose that \(T: C[0, 1]\rightarrow C[0,1]\) is a completely continuous linear operator and \(T(P)\subset P\). If there exist \(\phi\in C[0, 1]\backslash(P)\) and a constant \(c>0\) such that \(cT\phi\geq\phi\), then the spectral radius \(r(T )\neq0\) and T has a positive eigenfunction corresponding to its first eigenvalue \(\lambda _{1}=(r(T))^{1}\).
We are now in a position to give the results on eigenvalue problem (1.6).
Lemma 2.4
Suppose that (A1) and (A2) are satisfied. Then T has a principle eigenvalue \(\lambda_{1}=(r(T))^{1}\) and a positive eigenfunction \(\varphi_{1}\in P\) corresponding to \(\lambda_{1}\). Furthermore,

(i)
\(\varphi'_{1}(0)>0\), \(\varphi'_{1}(1)<0\) and there exist positive constants \(\nu_{1}\), \(\nu_{2}\) such that \(\nu_{1}t(1t)\leq\varphi_{1}(t)\leq\nu _{2}t(1t)\), \(t\in[0,1]\);

(ii)
there exist \(\delta_{1}, \delta_{2}>0\) such that \(\delta_{1} G(t,s)\leq\varphi_{1}(s)\leq\delta_{2} s(1s)\), \(t,s\in[0,1]\).
Proof
It is obvious that there is \(t_{1}\in(0,1)\) such that \(G(t_{1}, t_{1})\tilde {h}(t_{1})>0\). Thus there exists \([a_{1}, b_{1}]\in(0, 1)\) such that \(t_{1}\in(a_{1}, b_{1})\) and \(G(t,s)\tilde {h}(s)>0\) for all \(t,s\in[a_{1}, b_{1}]\). Take \(\psi\in C[0, 1]\) such that \(\psi(t)\geq0\), \(t\in[0,1]\), \(\psi(t_{1})>0\) and \(\psi(t)= 0\), \(t\notin[a_{1}, b_{1}]\). Then, for \(t\in[a_{1}, b_{1}]\),
So there exists a constant \(c>0\) such that \(c(T \psi)(t)\geq\psi(t)\), \(t\in[0,1]\). From Lemma 2.3, we know that the spectral radius \(r(T) \neq0\) and T has a positive eigenfunction \(\varphi_{1}\) corresponding to its first eigenvalue \(\lambda_{1}=(r(T))^{1}\).
If (A1) and (A2) hold, then the conclusions of Lemma 2.1 are still valid for the initial value problems
and
We denote the unique solutions of the above two initial problems by ξ and ζ, respectively. Then Lemma 2.1 shows that \(\xi\in AC[0,1]\cap C^{1}[0,1)\), \(\xi'\in AC_{\mathrm{loc}}[0,1)\) and \(\zeta\in AC[0,1]\cap C^{1}(0,1]\), \(\zeta'\in AC_{\mathrm{loc}}(0,1]\).
(i) From the definition of eigenvalue, we know that \(\varphi_{1}\) satisfies
Since \(\varphi_{1}(0)=\varphi_{1}(1)=0\), there exists \(\tau\in(0,1)\) such that \(\varphi_{1}(\tau)=\\varphi_{1}\>0\) and \(\varphi_{1}'(\tau)=0\). Let
Then \(\chi\in C^{1}(0,1)\) and for \(t\in(0,1)\), it is easy to compute that
For \(0< t<\tau\), integrating \(\chi'\) from t to τ and letting \(t\to0\), we have
By Lemma 2.3 in [4] and the fact that \(\xi\in C^{1}[0,1)\), \(\xi (0)=0\), we have
Combining (2.10), (2.11), (2.12) and
we get
Since \(q(\tau)\neq0\) and \(\varphi_{1}(\tau)\neq0\), we have \(\xi'(\tau )=0\). Then \(\xi(\tau)\neq0\) due to Lemma 2.2. Let us define the function ϕ by
it is easy to check that ϕ is the solution of the following problem:
Then Lemma 2.2 yields \(\phi\equiv0\), that is, \(\varphi_{1}=[\varphi_{1}(\tau )/\xi(\tau)]\xi\). In particular, \(\varphi_{1}'(0)=\varphi_{1}(\tau)/\xi(\tau)\neq0\); furthermore, \(\varphi_{1}'(0)>0\) by positivity of \(\varphi_{1}\) on \((0,1)\). In the same manner, we can see that \(\varphi_{1}=[\varphi_{1}(\tau)/\zeta (\tau)]\zeta\) and \(\varphi_{1}'(1)=\varphi_{1}(\tau)/\zeta(\tau)< 0\).
Define
Then \(\Theta(\cdot)\) is continuous and \(\Theta(t)> 0\) on \([0,1]\), so there exist positive constants \(\nu_{1}\) and \(\nu_{2}\) such that \(\nu_{1}\leq \Theta(t)\leq\nu_{2}\) for \(t\in[0,1]\), i.e.,
(ii) From Lemma 2.1(iii), (∗) and (∗∗), for \(t,s\in[0,1]\), we have
Define
then
This completes the proof. □
Lemma 2.5
Suppose that (A1) and (A2) are satisfied. Let
where \(\varphi_{1}\) and \(\delta_{1}\) are defined by Lemma 2.4. Then \(P_{1}\) is a cone in \(C[0,1]\) and \(T(P)\subset P_{1}\).
Proof
It follows from (\(\tilde{\mathrm{A}}2\)) and (∗∗) that
for any \(u\in C[0,1]\), then \(P_{1}\) is well defined. Furthermore, for any \(u\in P\), from Lemma 2.4, we have
then \(\int_{0}^{1}\varphi_{1}(t)\tilde{h}(t)(Tu)(t)\,\mathrm{d}t\geq\lambda _{1}^{1}\delta_{1}\Tu\\), i.e., \(T(P)\subset P_{1}\).
This completes the proof. □
Proof of the main result
Proof of Theorem 1.1
From (A4), for \(\varepsilon>0\), there exists \(L>0\) such that
Combining (A3) and (3.1), we can see that there exists a nonnegative function \(c_{1}\in C[0,1]\) such that
In the following we shall prove
provided that R is large enough.
In fact, if (3.3) is not true, then there exist \(u_{0}\in C[0,1]\), \(\ u_{0}\=R\) and \(\mu_{0}\geq0\) such that
For \(t\in[0,1]\) and for any \(u\in C[0,1]\), we set
Then from (\(\tilde{\mathrm{A}}2\)) and (∗∗), functional g is well defined. Thus we have
Recall that
It follows from Lemma 2.5 and (3.2) that \(u_{0}+\eta_{1}+\eta_{2}\in P_{1}\).
From Lemma 2.4 and (3.5)(3.7), we have
By the same manner, we get
By (3.2), (3.7) and (3.8), we have
Then, from the above inequality, (3.8), (3.9) and Lemma 2.5, we have
We divide the proof into two cases as follows.
Case 1. \(B(\cdot)\) is bounded on \(\mathbb{R}\).
In this case, there exists \(M_{2}>0\), for any \(u\in\mathbb{R}\), \(B(u)\leq M_{2}\). Thus,
Therefore, from (3.10), (3.11) and the expression of \(\eta_{1}\), we get
provided that we take
where
Case 2. \(B(\cdot)\) is unbounded on \(\mathbb{R}\).
From (1.4), for \(\rho>0\), there exists a positive constant \(M_{1}\) such that
In this case, there exists a positive constant \(M_{3}>M_{1}\) such that
Since \(\u_{0}\=R\), then for \(s\in[0,1]\), \(u_{0}(s)\leq R< M_{3}\). By (3.12) and (3.13), we have
Therefore, from (3.10), (3.13) and (3.14), we have
provided that we take
where
Thus, no matter which case happens, if we choose \(R>\max\{R_{1}, R_{2}\}\), we have
On the other hand, from (3.4), (\(\tilde{\mathrm{A}}2\)) and the fact that \(\varphi_{1}(t)>0\) for \(t\in(0,1)\) and \(\mu_{0}\geq 0\), we have
which contradicts with (3.15). Therefore, (3.3) is true and we have
From (A5), for \(0<\varepsilon_{1}<\lambda_{1}\), there exists \(0< r<R\) such that
In the following we will prove that
If (3.18) is not true, then there exist \(u_{1}\in\partial B_{r}\) and \(\mu _{1}\in(0,1]\) such that \(u_{1}= \mu_{1} Au_{1}\), and then by (3.17) we have
which implies \(g(u_{1}(t))\leq0\).
On the other hand, from \(\u_{1}\=r>0\) and \(\varphi_{1}(t)>0\) for \(t\in (0,1)\), we have
which is a contradiction. Thus (3.18) holds. According to the invariance property of the LeraySchauder degree, we have
By (3.16), (3.19) and the additivity of the LeraySchauder degree, we obtain
Therefore, A has at least one fixed point on \(B_{R}\backslash\bar {B}_{r}\), i.e., problem (1.3) has at least one nontrivial solution. This completes the proof. □
Example
In this section, an example is given to illustrate the application of our main result (Theorem 1.1). Consider the following boundary value problem:
where
Obviously, conditions (A1)(A3) of Theorem 1.1 are satisfied. For the linear eigenvalue problem corresponding to (4.1), we compute that
Moreover,
which implies that (A4) and (A5) are satisfied. So, all of conditions in Theorem 1.1 are fulfilled. Therefore, the boundary value problem (4.1) has at least one nontrivial solution according to Theorem 1.1.
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Acknowledgements
The authors would like to thank the anonymous referees for their careful reading of the manuscript and for the constructive comments. This work is supported by the Natural Science Foundation of Gansu Province, China (1308RJZA113), the National Natural Science Foundation of China (11362008), by Youth Science Foundation of Lanzhou Jiaotong University (2012019) and the Fundamental Research Funds for the Universities of Gansu Province (212084).
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Li, J., Yan, D. Nontrivial solutions to boundary value problem with general singular differential operator. Bound Value Probl 2015, 70 (2015). https://doi.org/10.1186/s1366101503317
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MSC
 34B16
Keywords
 singularities
 eigenvalue
 topological degree
 nontrivial solutions