Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 24, 2006, 135–142

© W.Zhou, S. Cai

Abstract. This paper is concerned with the positive solutions for a singular second order ordinary differential equation. Under appropriate conditions, by the classical method of elliptic regularization, we prove the existence of position solutions.

1. Introduction

This paper is concerned with the existence of positive solutions for a singular second order ordinary differential equation

with one of the following boundary conditions

where $\lambda >0,m\ge \frac{1}{2},f\left(t\right)\in C^1\left[0,1\right]$ and $f\left(t\right)>0$ on $\left[0,1\right]$.

It is well known that boundary value problems (BVPs) for singular second order ordinary differential equations arise in the field of gas dynamics, flow mechanics, theory of boundary layer, and so on. In recent years, singular ordinary differential equation with dependence on the first order derivative have been studied extensively, see for example [1–8] and references therein where some general existence results were obtained. We point out that the case considered here is not in their considerations since it does not satisfy some sufficient conditions of those papers. Our considerations were motivated by [9] in which the authors studied the following singular differential equation

with the boundary conditions: $\varphi \left(1\right)={\varphi }^{\prime }\left(0\right)=0$. By ordinary differential equation theories, they obtained a decreasing positive solution. In the present paper, we consider (1) and will use the classical method of elliptic regularization to obtain positive solutions to BVP (1), (2) and BVP (1), (3). However, it is easy to see from the boundary conditions (2) (or (3)) that any positive solution to BVP (1), (2) (or BVP (1), (3)) must not be decreasing. Thus the existence results obtained here are not a simple extension of [9]

We say $\varphi \in C^2\left(0,1\right)\cap C\left[0,1\right]$ is a solution to BVP (1),(2) if it is positive in (0,1) and satisfies (1) and (2). Similarly, we say $\varphi \in C^2\left(0,1\right)\cap C^1\left[0,1\right]$ is a solution to BVP (1),(3) if it is positive in (0,1) and satisfies (1) and (3).

The main purpose of this paper is to prove the following theorems.

Theorem 1  Let $m\ge \frac{1}{2},\lambda >0,f\left(t\right)\in C^1\left[0,1\right]$ and $0<{min}_{\left[0,1\right]}f\le {max}_{\left[0,1\right]}f<1$. Then BVP (1),(2) admits at least a solution.

Theorem 2  Let $m\ge \frac{1}{2},f\left(t\right)\in C^1\left[0,1\right],0<{min}_{\left[0,1\right]}f\le {max}_{\left[0,1\right]}f<1$ and $\lambda >\frac{1}{2}{\left[1-{\left({max}_{\left[0,1\right]}f\right)}^2\right]}^{1∕2-m}$. Then BVP (1),(3) admits at least a solution.

2. Proofs of Theorems

We will use the classical method of elliptic regularization to prove Theorem 1. For this, we consider the following regularized problem:

where $\varepsilon \in \left(0,1\right)$, $I_{\varepsilon }\left(s\right)$ and ${sgn}_{\varepsilon }\left(s\right)$ are defined as follows:

Clearly, $I_{\varepsilon }\left(s\right),{sgn}_{\varepsilon }\left(s\right)\in C^1\left(\mathbb{ℝ}\right),$ and $I_{\varepsilon }\left(s\right)\ge \varepsilon ∕2,1\ge \mid {sgn}_{\varepsilon }\left(s\right)\mid ,{sgn}_{\varepsilon }\left(s\right)\cdot sgn\left(s\right)\ge 0$ in $\mathbb{ℝ}$.

For any $m\ge \frac{1}{2}$, it follows from Theorem 4.1 of Chapter 7 in [10] that for any fixed $\varepsilon \in \left(0,1\right)$, the above regularized problem admits a classical solution ${\varphi }_{\varepsilon }\in C^2\left(0,1\right)\cap C^1\left[0,1\right]$. By the maximal principle, it is easy to see that ${\varphi }_{\varepsilon }\left(t\right)\ge \varepsilon \text{on}\left[0,1\right]$. Thus ${\varphi }_{\varepsilon }$ satisfies

Note that (4) is equivalent to

Lemma 1  Under the assumptions of Theorem 1, for all $\varepsilon \in \left(0,1\right)$ there holds

where $\Lambda \stackrel{\Delta }{=}{max}_{\left[0,1\right]}f<1$.

Proof.  Noticing ${\varphi }_{\varepsilon }\left(1\right)={\varphi }_{\varepsilon }\left(0\right)=\varepsilon$ and ${\varphi }_{\varepsilon }\left(t\right)\ge \varepsilon$ for all $t\in \left[0,1\right]$, we have

On the other hand, it follows from $\left(5\right)$ that

i.e.

Thus the function ${\int }_0^{{\varphi }_{\varepsilon }^{\prime }\left(t\right)}\frac{1}{\sqrt{1+s^2}}ds+\Lambda t$ is non-decreasing on $\left[0,1\right]$, therefore

and hence

From this and using the inequality

we obtain

Noticing $\Lambda <1$, we obtain

This completes the proof of Lemma 1.

Denote ${\Lambda }_m$ by ${\Lambda }_m={\left(1-{\Lambda }^2\right)}^{1∕2-m}$, where $\Lambda$ is the same as that of Lemma 1. From (4) and Lemma 1 we obtain obtain

To obtain the uniform bounds of ${\varphi }_{\varepsilon }$, the following comparison theorem will be proved to be very useful.

Proposition 2  Let ${\varphi }_i\in C^2\left(0,1\right)\cap C\left[0,1\right]$ and ${\varphi }_i>0\text{on}\left[0,1\right]\left(i=1,2\right)$. If ${\varphi }_2\ge {\varphi }_1$ for $t=0,1$, and

where $\varrho$ and $\theta$ are positive constants, then

Proof.  From (8) and (9), we have

and

Combining the above inequalities, we obtain

where $w:\left[0,1\right]\to \mathbb{ℝ}$ is defined by

Clearly, $w\in C^2\left(0,1\right)\cap C\left[0,1\right]$.

To prove the proposition, we argue by contradiction and assume that there exists a point $t_0$ of $\left(0,1\right)$ such that ${\varphi }_2\left(t_0\right)-{\varphi }_1\left(t_0\right)<0$. From the assumption, it is easy to see that $w$ reaches a minimum at some point $t_{\ast }$ of $\left(0,1\right)$ such that

Combining $\left(12\right)$ with (10), we have

This implies ${\varphi }_2\left(t_{\ast }\right)\ge {\varphi }_1\left(t_{\ast }\right)$. However, from (11) we find that ${\varphi }_2\left(t_{\ast }\right)<{\varphi }_1\left(t_{\ast }\right)$, a contradiction. Thus the proof of Proposition 2 is completed.

Lemma 3  Under the assumptions of Theorem 1, for all $\varepsilon \in \left(0,1\right)$ there exists a positive constant $C$ independent of $\varepsilon$ such that

Proof. Let $w_{\varepsilon }=C{\left[t\left(1-t\right)+{\varepsilon }^{1∕2}\right]}^2$, where $C\in \left(0,1\right]$ will be determined later. By Proposition 2 and noticing (6), it suffices to show that

for some sufficiently small positive constant $C$ independent of $\varepsilon$. Simple calculation shows that

and

Choosing a positive constant $C$ such that

we find that (13) holds. Thus the proof of Lemma 3 is completed.

From (6), (7), Lemma 1 and Lemma 3, we derive that for any $\delta \in \left(0,1∕2\right)$ there exists a positive constant $C_{\delta }$ independent of $\varepsilon$ such that

From (4), we have

Differentiating the above equation with respect to $t$ we get

By (14), Lemma 1 and Lemma 3, we derive that for any $\delta \in \left(0,1∕2\right)$, there exists a positive constant $C_{\delta }$ independent of $\varepsilon$ such that

From this and Lemma 1 and using Alzel$á$-Ascoli theorem and diagonal sequential process, we see that there exists a subsequence $\left\{{\varphi }_{{\varepsilon }_n}\right\}$ of $\left\{{\varphi }_{\varepsilon }\right\}$ and a function $\varphi \in C^2\left(0,1\right)\cap C\left[0,1\right]$ such that, $as{\varepsilon }_n\to 0$,

Combining these with (4) (or (5)) and the boundary conditions satisfied by ${\varphi }_{{\varepsilon }_n}$, we find that $\varphi$ satisfies (1) and (2). By Lemma 3, we have

therefore $\varphi >0$ in (0,1), and thus $\varphi$ is a solution to BVP (1), (2). This completes the proof of Theorem 1.

Proof of Theorem 2.  From Theorem 1, we see that for any $\lambda >0$, BVP (1),(2) admits a solution $\varphi$ which can be approximated by ${\varphi }_{{\varepsilon }_n}$ satisfying (4) (or (5)) with $\varepsilon ={\varepsilon }_n$. Hence it suffices to show $\varphi$ satisfies ${\varphi }^{\prime }\left(1\right)={\varphi }^{\prime }\left(0\right)=0$ for $\lambda >\frac{1}{2}{\left[1-{\left({max}_{\left[0,1\right]}f\right)}^2\right]}^{1∕2-m}=\frac{1}{2}{\left(1-{\Lambda }^2\right)}^{1∕2-m}=\frac{1}{2}{\Lambda }_m$. We claim that if $\lambda >\frac{1}{2}{\Lambda }_m$, then there exist positive constants $C$ independent of ${\varepsilon }_n$ such that

We first show (16). Let $v_{{\varepsilon }_n}=C{\left(1+{\varepsilon }_n^{1∕2}-t\right)}^2$, where $C\ge 1$ will be determined later. A calculation shows that

Choosing a positive constant $C$ such that

and noticing $\lambda >\frac{1}{2}{\Lambda }_m$, we find that

and then, by Proposition 2 and noticing (7), we obtain (16). Similarly we can prove the claim (17). Letting ${\varepsilon }_n\to 0$ in (16) and (17) to yield

Combining this with (15) we immediately obtain ${\varphi }^{\prime }\left(1\right)={\varphi }^{\prime }\left(0\right)=0$. Thus the proof of Theorem 2 is completed.

References

Department of Mathematics, Jilin University, Changchun 130012, P.R.China

E-mail address: pdezhou@126.com

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P. R. China

Received August 26, 2006