Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity

∞ α uλ of the solution uλ associated with λ. In the field of bifurcation theory, the study of global structures of bifurcation curves is one of the main subjects of research, and it is important to investigate the influence of the oscillatory term on the global structure of bifurcation curve. Due to the effect of u sin , it is reasonable to expect that an oscillatory term appears in the second term of the asymptotic formula for ( ) λ α as → ∞ α (cf. [1]). Contrary to expectation, we show that the asymptotic formula for ( ) λ α as → ∞ α does not contain oscillatory terms by the third term of ( ) λ α . This result implies that the oscillatory term has almost no influence on the global structure of ( ) λ α . The result is, therefore, unexpected, new and novel, since such phenomenon as this is not known so far. For the proof, the involved time map method and stationary phase method are used.

− ″( ) = ( ( + ( )) + ( )) ( ) > ∈ ≔ (− ) (± ) = u t λ u t u t u t t I u log 1 sin , 0, 1, 1 , 1 0, where > λ 0 is a parameter. It is known that λ is a continuous function of > α 0, written as ( ) λ α , where α is the maximum norm = ‖ ‖ ∞ α u λ of the solution u λ associated with λ. In the field of bifurcation theory, the study of global structures of bifurcation curves is one of the main subjects of research, and it is important to investigate the influence of the oscillatory term on the global structure of bifurcation curve. Due to the effect of u sin , it is reasonable to expect that an oscillatory term appears in the second term of the asymptotic formula for ( ) λ α as → ∞ α (cf. [1]). Contrary to expectation, we show that the asymptotic formula for ( ) λ α as → ∞ α does not contain oscillatory terms by the third term of ( ) λ α . This result implies that the oscillatory term has almost no influence on the global structure of ( ) λ α . The result is, therefore, unexpected, new and novel, since such phenomenon as this is not known so far. For the proof, the involved time map method and stationary phase method are used.

Introduction
We study the global structure of the oscillatory perturbed bifurcation problem which comes from the stationary logarithmic Schrödinger equation where ( ) f u is continuous for ∈ [ ∞) u 0, and ( ) > f u 0 for > u 0. Then we know from [2] that, for a given > α 0, there exists a unique classical solution pair ( ) λ u , α of (1)-(3) satisfying = ‖ ‖ ∞ α u α for any given > α 0. Moreover, we know from [2, p. 2] that λ is a continuous function of > α 0, we write as = ( ) λ λ α for > α 0. The study of global and local structures of bifurcation diagrams has a long history. Many topics considered there have the background in mathematical biology, engineering, and have been investigated intensively by many authors. To take an example, for the bifurcation problems related to logistic equations, we refer to [3][4][5][6][7] and references therein. For the readers who are interested in various types of global behavior of bifurcation curves, we refer to [8]. In particular, global shapes of bifurcation diagrams has been considered by many researchers. However, the precise asymptotic behavior of ( ) λ α as → ∞ α is not elucidated so much yet. In this paper, we study an interesting nonlinear eigenvalue problem in which both oscillatory term and logarithmic term are included to show that, surprisingly, the oscillatory term hardly gives effect to the asymptotic behavior of ( ) λ α as → ∞ α . To clarify our intention, we introduce the global behavior of bifurcation curves when ( ) = + f u u u sin and ( ) = ( + ) f u u log 1 . As for oscillatory bifurcation problems, the typical model problem ( ) = + f u u u sin in (1) with (2)-(3) was first considered in [9]. Besides, the asymptotic formula for ( ) λ α is known (see [10,11]).
where C 1 is a constant specified in Section 3 and R 3 is the remainder term satisfying the following optimal estimate. Namely, there exists a constant < By Theorems 1.1 and 1.2, we expect that if ( ) = ( + ) + f u u u log 1 sin , then the asymptotic behavior of ( ) λ α is composed of the mixture of (5) and (6). Now we state our main result.
It should be mentioned that (7) is more precise than (5), and certainly, (7) contains the oscillatory terms. However, we note that (7) can be written as (5). Therefore, from a view point of asymptotic expansion formula for ( ) λ α , (5) and (7) are formally the same. In other words, the perturbed oscillatory term gives almost no influence on the global structure of ( ) λ α . As far as the author knows, such phenomenon has not been obtained yet. Therefore, the result here is novel and gives the original contribution to the development of the study of global behavior of bifurcation curves. It should be emphasized that, by the precise and very involved calculations to obtain (7), we reach the new aspect of bifurcation theory.
Here, { } b n and { ( )} d α n ( = … n 1, 2, ) are expected to be constants and oscillatory functions, which are determined by induction, respectively. If (9) is valid, then the oscillatory term does not appear in any nth term of the asymptotic expansion of ( ) λ α and it is a remarkable and significant result in the field of bifurcation theory. However, if the readers look at Section 3, then they understand immediately that it seems almost impossible to determine whether the conjecture (9) is valid or not, since the calculation is quite long and complicated to obtain even the third term of (5). One of the foreseeable extensions in this direction would be to present a computer-assisted analysis of (8) and (9).
The proof of Theorem 1.3 depends on the very involved time map method and stationary phase method. The essential point is that if the terms of time map (22) in Section 2 do not contain x sin , then we are able to calculate them by Taylor expansion and by a direct calculation. However, since they contain x sin , we need to calculate them by the involved stationary phase method. It seems to be new to apply stationary phase method to the calculation of the asymptotic behavior of bifurcation curve in such a complicated situation. The calculation is performed by the precise integration by parts again and again. It was possible to get all the Lemmas, especially in Section 3 by the very involved calculations.
In what follows, we denote by C the various positive constants independent of α.
We apply the standard time-map argument to (1) (cf. [1]). By (1), we have By this, (11) and putting = t 0, we obtain This along with (12) implies that for − ≤ ≤ t 1 0, By this and (13), we obtain log 1 cos cos . where By this, (15) and Taylor expansion, we obtain We see from (22) that the second term of ( ) λ α in Theorem 1.3 follows from Lemma 2.1.
The proof of Lemma 2.1 is a combination of Lemmas 2.4 and 2.5. By (18), we have , the most important point of Lemma 2.1 is to obtain the asymptotic behavior of L 1 . To do this, we apply the stationary phase method to L 1 .
Then as → ∞ μ , In particular, Here, we use l'Hôpital's rule to obtain 3 The third term of ( ) λ α in Theorem 1.3 By (22), to obtain the third term of ( ) λ α , we calculate Proof. We put = s θ sin in (38). By integration by parts and (45), we have By the same argument with l'Hôpital's rule as (33), we see that = J 0 10 . By using integration by parts, we obtain By (47)-(51), we obtain (46). Thus, the proof is complete. □ Lemma 3.2. As → ∞ α , Proof. We put = s θ sin . By (39), (45) and integration by parts, we have By l'Hopital's rule, we see that  . It was shown in [19] that for ≫ α 1, 1 sin sin 1 sin sin .
By the same argument as that in (58), we see that | | ≤ J C

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. By (17), we have By (58)-(60), we obtain that | | ≤ J C By this and (61), we obtain By direct calculation, we see that ( ) h x satisfies the conditions in Remark 2.3. For completeness, the proof will be given in Appendix. By (65) and l'Hôpital's rule, we have