Research Article | Open Access
Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems
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 .
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 .(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.)
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.
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.
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
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(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.
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