International Journal of Differential Equations
Volume 2015 (2015), Article ID 138629, 11 pages
http://dx.doi.org/10.1155/2015/138629

Research Article

Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

Tetsutaro Shibata

Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

Received 22 October 2014; Accepted 15 December 2014

Academic Editor: Nasser-Eddine Tatar

Copyright © 2015 Tetsutaro Shibata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the nonlinear eigenvalue problem ,  ,  ,  , where is a cubic-like nonlinear term and is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions on , there exist exactly three bifurcation branches (), and these curves are parameterized by the maximum norm of the solution corresponding to . In this paper, we establish the precise global structures for (), which can be applied to the inverse bifurcation problems. The precise local structures for () are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solution , which corresponds to , as .

1. Introduction

We consider the following nonlinear eigenvalue problem:

where is a cubic-like nonlinear term and is a parameter. We assume the following conditions (A.1)–(A.3), which have been introduced in [1].

(A.1) is a -function on and has three positive roots at , and

(A.2)There exists a constant such that

(A.3)Let satisfy . Then

The typical example of which satisfies (A.1)–(A.3) is with and the area of the negative hump of is nearly equal to that of the positive hump. For example, if we choose appropriately, then satisfies (A.1)–(A.3).

Nonlinear elliptic eigenvalue problems have been studied by many authors. We refer to [2–8] and the references therein. Among other things, (1)–(3) have been investigated by many authors. We refer to [1, 9–12] and the references therein. In particular, the following basic properties of the structure of bifurcation diagram for (1)–(3) have been proved in [1, Theorem 3.1].

Theorem 1 (see [1, Theorem 3.1]). Assume (A.1)–(A.3). Then there exists a critical such that (1)–(3) have exactly one positive solution for , exactly two positive solutions for , and exactly three positive solutions for . Furthermore, all solutions lie on two smooth solution curves, and is parameterized by as . One of the curves, referred to as the lower curve , satisfies and increases in , and

where is a solution of (1)–(3) corresponding to . (Note that is equivalent to .) The upper curve, which consists of two branches (), is a parabola-like curve with exactly one turn to the right at . Furthermore,

where is a solution of (1)–(3) corresponding to () and is a constant which satisfies

(See Figure 1 for the bifurcation diagram.)

Figure 1

As we see from (9), (11), and (13), one of the most interesting facts in Theorem 1 is the difference between the asymptotic shapes of () and for sufficiently large (see Figures 2 and 3). This difference comes from the fact that () is stable and is unstable. This drives us to the question whether these facts give effect on the asymptotic behavior of the bifurcation branches or not.

Figure 2

Figure 3

The purpose of this paper is to establish the precise asymptotic formulas for () as , respectively, to clarify how the difference of the asymptotic shapes of solutions corresponding to these three curves gives effect to the asymptotic formulas for () and . Furthermore, we establish the asymptotic widths and of the spike layer of solution as .

Finally, we establish the precise asymptotic formulas for as and as (). By using the asymptotic formulas above, we propose new inverse bifurcation problems.

Now we state our main result. Let .

Theorem 2. Assume that . Let satisfy . Then as ,

Theorem 3. Assume that . Let satisfy .
(i) As ,

(ii) Let satisfy . Then as (namely, ),

(iii) Let be an arbitrary fixed constant. Furthermore, let satisfy . Then as (namely, ),

where

Theorem 4. Assume that . Let satisfy . Then as ,

Now, we establish the local asymptotic behavior of and () near and , respectively, where satisfy

Theorem 5. (i) As ,

where () are bounded functions of determined explicitly.
(ii) Assume that is analytic near . Then as ,

where () are bounded functions of determined explicitly.

Theorem 6. Let . Then as ,

where () are bounded functions determined explicitly.

Finally, we apply our results above to the inverse bifurcation problems.

Theorem 7. Let . Assume that satisfy (A.1)–(A.3), where are unknown constants. Let () be the bifurcation branches corresponding to . Further, let be the number defined in (14) for . Finally, for and , let

(i) Suppose that

Then, , , and .
(ii) Suppose that

Then, , , and .
(iii) Suppose that (28) and hold. Then, , , and .

Theorem 7(i) implies that if we assume that the unknown nonlinear term is a cubic, then the precise information about the local asymptotic behavior along with the rough global asymptotics of one branch determines the unknown . Theorem 7(ii) implies that the unknown is determined by only the asymptotic behavior of two branches.

Our methods to prove Theorems 2–4 are based on the precise calculation of the time map. We prove Theorems 5 and 6 by the method developed in [13]. By using Theorems 2, 4, and 5, we prove Theorem 7.

2. Proof of Theorem 2

In this section, let and let . In what follows, denotes various positive constants independent of . We know that if satisfies (1)–(3), then

We parameterize the solution pair by using the norm of the solution such as (). By (1), for ,

This implies that, for ,

By this, (31), and putting , for , we obtain

This implies that

By this and (32), for , we obtain

By this, for ,

By mean value theorem, for , we have

where . We know that , since, by (A.2), for . Let defined in Theorem 2. We easily see that as . Furthermore, by Taylor expansion, as ,

By (38) and (39), we obtain

We write (). Then,

Then,

where

by (40). We put

We note that . Then by (41) and (43),

We next calculate . We have

By (40)–(44), (46), (47), and Taylor expansion, we obtain

This implies (15). Thus, the proof is complete.

3. Proof of Theorem 3

In this section, let with . We write defined in the statement in Theorem 3. By (38), we have

Lemma 8. As ,

Proof. We have

We first calculate . For , by (14) and Taylor expansion, we have

By (51) and (52), we obtain

where

Then we obtain

We next calculate . For , by (A.2) and (14), we have

By this, we obtain

This along with (55) implies (50). Thus the proof is complete.

Lemma 9. As ,

Proof. We have

We first calculate . By (14) and Taylor expansion, for , we obtain

By this and (55), we obtain

We next calculate . Let an arbitrary be fixed. Then,

We calculate . For , we obtain

By this,

We next calculate . For ,

By this, we obtain

Finally, we calculate . For , by (14), we obtain

By this, we obtain

By (62), (64), (66), and (68), we obtain

By this, (59), and (61), we obtain (58). Thus, the proof is complete.

Lemma 10. As ,

Proof. By (A.3) and (14), we know that . Then for , we have

By this, for , we obtain

Thus, the proof is complete.

Proof of Theorem 3(i). Theorem 3(i) follows directly from Lemmas 8–10. Thus, the proof is complete.

Proof of Theorem 3(ii). By (37) and (49), we have

By this, Lemmas 8–10, and direct calculation, we obtain Theorem 3(ii). Thus, the proof is complete.

Proof of Theorem 3(iii). By (37) and (49), we have

We have only to calculate . We fix a constant sufficiently small. Then we have

By (A.2), for , we have

Further, since , we have

By this and (76), we obtain

We next calculate . For , we obtain

By (76) and (79), we obtain

By Lemma 10, (74), (75), (78), and (80), we obtain Theorem 3(iii). Thus, the proof is complete.

4. Proof of Theorem 4

Let and let in this section. The proof of Theorem 4 is similar to that of Theorem 2. Let defined in the statement of Theorem 4. By (38), we have

By the same argument as that to obtain in (41), if is close to , we obtain

Now, we calculate . We have

We calculate . Since , by (4) and (5), for , we have . By this, we obtain

We calculate . Since , by (4), for , we have

This implies that . Finally, we calculate . Let . By (6), we have for . By this and Taylor expansion, we obtain

where . By this, we obtain

By (81), (82), (83), (87), and the same calculation as that in (48), we obtain Theorem 4. Thus, the proof is complete.

5. Proofs of Theorems 5 and 6

We first characterize . Let be the curve consisting of and . We know that is determined by . By this and (38), we obtain

By this, we see that satisfies the following equation:

Proof of Theorem 6. We study the asymptotic behavior of () as . We put and consider the case where . For , we put

Then,

We show that, for and sufficiently small,

Let an arbitrary be fixed. First, we consider the case . By (A.2), we have

We put . By mean value theorem,

where . By this and (93), we obtain (92). Next, let . Since (cf. Figure 1), we have

Furthermore,

By this and (95), we obtain (92). For , we know that

where for . By this, (38), and (91), for , we obtain

By this, we obtain Theorem 6. Thus, the proof is complete.

Proof of Theorem 5. We write . For and , we put

By Taylor expansion,

By (99) and (100), for and , we obtain

By this, (97), and (98), we obtain

where

Thus, we obtain Theorem 5(i). Theorem 5(ii) is obtained by Theorem 5(i) and Taylor expansion of near . Thus, the proof is complete.

6. Proof of Theorem 7

Proof of Theorem 7(i). Let with (). By (26) and Theorem 2, we obtain . This implies that

By direct calculation,

Let . By using (38), (105), and the calculation in Section 5, we obtain

This implies that

By this, (27), (28), and Theorem 2, we obtain

By (104), (108), (109), and direct calculation, we easily obtain , , and . Thus, the proof is complete.

Proof of Theorem 7(ii). By Theorem 2 and (28), we obtain and . By this, we obtain (104) and

By this and (104), we obtain . By this and (104), we have

By (29) and (107), we obtain (108). By (108) and (111), we obtain

By this, we obtain . By this and (111), we obtain and . Thus, the proof is complete.

Proof of Theorem 7(iii). Let . We put

For , by (14) and (111), we obtain

We study the shape of for . Since by (5), we have . By this and direct calculation, we see that and it decreases for and attains its minimum at , and . Then increases for and attains its maximum at , and . Then, decreases for and as . Therefore, there exists such that . We assume that and we derive a contradiction. In this case, by (114), we obtain and . This implies that . This is a contradiction, since . If we assume that , we obtain the same contradiction as above. Consequently, we obtain . By this and (111), we obtain and . Thus, the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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