Nontrivial solutions to boundary value problem with general singular differential operator
 Jiemei Li^{1} and
 Dongming Yan^{2}Email author
https://doi.org/10.1186/s1366101503317
© Li and Yan; licensee Springer. 2015
Received: 7 October 2014
Accepted: 13 April 2015
Published: 25 April 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.
Keywords
MSC
1 Introduction
 (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.
Theorem A
 (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 \(B\in C(\mathbb{R}, [0,+\infty))\) withsuch that$$\begin{aligned}& \lim_{u\to+\infty}\frac{B(u)}{u}=0 \end{aligned}$$(1.4)$$\begin{aligned}& f(t, u)\geqc(t)d(t) B(u), \quad t\in[0,1], u\in\mathbb{R}. \end{aligned}$$(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 \(t\in[0,1]\).
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.
2 Notations and preliminary results
In this section, we recall some notations, abstract theorems and auxiliary results, which are important for proving our main result.
Lemma 2.1
[4]
 (i)the initial value problemhas a unique solution \(\alpha\in AC[0,1]\cap C^{1}[0,1)\) and \(\alpha '\in AC_{\mathrm{loc}}[0,1)\);$$ \begin{aligned} &u''+a(t)u'+b(t)u=0 , \quad t\in(0,1), \\ &u(0)=0, \qquad u'(0)=1 \end{aligned} $$(2.1)
 (ii)the initial value problemhas a unique solution \(\beta\in AC[0,1]\cap C^{1}(0,1]\) and \(\beta'\in AC_{\mathrm{loc}}(0,1]\);$$ \begin{aligned} &u''+a(t)u'+b(t)u=0 , \quad t\in(0,1), \\ &u(1)=0, \qquad u'(1)=1 \end{aligned} $$(2.2)
 (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.
We give the following remark. Although its proof is trivial, the consequences of the result are of major importance.
Remark 2.1
 (\(\tilde{\mathrm{A}}2\)):

\(\tilde{h}:(0, 1)\rightarrow[0, \infty)\) is continuous and does not vanish identically on any subinterval of \((0,1)\); furthermore, \(0<\int_{0}^{1} s(1s)\tilde{h}(s)\,\mathrm{d}s<+\infty\).
Lemma 2.2
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
 (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
This completes the proof. □
Lemma 2.5
Proof
This completes the proof. □
3 Proof of the main result
Proof of Theorem 1.1
We divide the proof into two cases as follows.
Case 1. \(B(\cdot)\) is bounded on \(\mathbb{R}\).
Case 2. \(B(\cdot)\) is unbounded on \(\mathbb{R}\).
4 Example
Declarations
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).
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
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