Exact Number of Positive Solutions for a Class of Two-Point Boundary Value Problems
This paper considers the existence of positive solutions of the boundary value problems and , where and is a positive parameter. Using a time-map approach, we obtain the exact number of positive solutions in different cases.
1. Introduction and Main Results
The study of multiplicity results to boundary value problem where is a positive parameter, is very interesting because of its applications. As we know, when , the boundary value problem where and are fixed given numbers and is a parameter, comes from the elliptic problem with , which was raised by Ambrosetti et al. in .
Cheng  investigated the following two-point boundary value problem: where is a positive parameter and and got the exact number of positive solutions.
Now, in this paper we consider the more general case where is a positive parameter, , and .
Define , where satisfies
For and , let be given byThe main results of this paper are as follows.
Theorem 1. If , (5) has exactly one positive solution for any .
Theorem 2. If , (5) has exactly one positive solution for and none for .
Theorem 3. If , (5) has exactly one positive solution for and none for .
Theorem 4. If , (5) has exactly one positive solution for and none for .
Theorem 5. Assume that . Define where and . One has the following.(1)If , (5) has exactly one positive solution for and none for .(2)If , there exists such that (5) has exactly two positive solutions for , exactly one for or , and none for .
We assume throughout this section that and Denote that , where is given by (6). For and , let Define a function as It is clear that . Now we claim that for and if and only if .
Let and ; then . Immediately, we get that . In order to judge the sign of , we just judge the sign of . Since , , and , we can get the following two results.(1)For , the function has a stable point . When , we have , and when , we have .(2)For , we have on . Combining and with the above two results, we obtain the monotony of .(3)For , the function has a stable point on . When , we have and when , we have .(4)For , we have that on .(5)For or , we have that on .Then from , we infer that for .
We consider the integration . It is clear that is a flaw point. Since on , we consider the integration .
Using Lagrange theorem, we obtain that where are constants.
Proof. (1) Assume that and is a positive solution of (5). Let satisfy . It follows from that
This implies that and if . And combine to obtain
Then, we have that
It follows that
From (19) and we have that . With (17) and (18) we obtain the result (1) of this theorem.
(2) Since and is a positive solution of the boundary value problem we have that is a positive solution of (5).
Lemma 7. is differentiable on , and where
Lemma 8. Consider the following: where is given by (7).
Proof. From and the Lebesgue theorem, we have that On the other hand, from , we can obtain that This completes the proof of Lemma 8.
Lemma 9. Consider the following:
Proof. From we have thatFrom Lemma 7 and the Lebesgue theorem, we can obtain the results of this lemma.
3. The Proof of Theorem 5
Lemma 11. For and ,
Condition implies that
With and (44), we have that for .
It follows that Combining we have the results of this lemma.
Lemma 12. For ,
Lemma 13. has continuous derivatives up to second order on as follows: where .
Lemma 14. There exist and such that
Proof. From (41) and , we have that
And since , let satisfy
From , we have that and
On the other hand, from (10), (38), and Lemma 11, we have that
for , .
This means that for , . It follows from Lemma 13 and (58) that Now, from (59)–(62) we have (55), where
Lemma 15. If , then for . If , then there exists , such that for and for .
Proof. It follows from Lemma 14 that if and then By (40), (42), and (51) we can obtain that With (10) and (41), (66) implies that If , then from Lemma 12 and (64)–(67) we have the results of this lemma, immediately. If , then by Lemmas 12 and 14 we have that for near , and so for near . Thus, it follows from (64)–(67) that for .
S.-H. Wang and D.-M. Long, “An exact multiplicity theorem involving concave-convex nonlinearities and its application to stationary solutions of a singular diffusion problem,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 44, pp. 469–486, 2001.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet