Spectral asymptotics for inverse nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem −u ′′ (t) + f(u(t),u ′ (t)) = �u(t), u(t) > 0, t ∈ I := (−1/2,1/2), u(±1/2) = 0, where f(x,y) = |x| p−1 x − |y| m , p > 1,1 ≤ m 0 is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in R+ ×L q (I) (1 ≤ q < ∞) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue � = �q(�) as � :=

The purpose of this paper is to study precisely how the damping term |u ′ (t)| m gives effect on the global behavior of the bifurcation branch λ q (α) from a viewpoint of inverse eigenvalue problems.To this end, we establish the precise asymptotic formulas for λ q (α) as α → ∞.
To explain the background and the motivation of our problem here, we recall the known and related facts of our problems.The equation (1.1)-(1.3)without damping term (e.g. ) is well known as the diffusive equation of population dynamics (the solution is denoted by u 0,λ ), and has been studied extensively by many authors by local and global L ∞ bifurcation theory.We refer to [2] and [7][8][9][10].In particular, as a basic asymptotic behavior of u 0,λ as λ → ∞, the following formula is well known.
We also mention that relationship between nonlinear term and the property of eigenvalues has been studied in [14] with emphasis on the uniqueness of f (u).Furthermore, since (1.1)-( without damping term is regarded as a nonlinear eigenvalue problem, it is quite important to study (1.1)-(1.3)from a viewpoint of L 2 -theory, that is the case where q = 2.For the works in this direction, we refer to [4][5][6] and the references therein.It should be mentioned that one of the chief concern in this field is to investigate the local and global shape of the L 2 -bifurcation branch λ 2 (α), and the asymptotic behavior of λ 2 (α) as α → 0 has been studied in [4], [5].Besides, it seems important to study the asymptotic behavior of λ q (α) as α → ∞ for general 1 ≤ q < ∞ (q = 2).In particular, it is meaningful to consider this problem in L 1 -framework, since (1.1)-(1.3)without damping term comes from the equation of population dynamics.
The leading term of λ q (α) without damping term in (1.1) can be obtained easily as follows.Since it is known from [2] that EJQTDE, 2009 No. 58, p. 2 locally uniformly on I as λ → ∞, we obtain that for λ ≫ 1 (i.e.α ≫ 1), Recently, more precise asymptotic formula for λ q (α) has been obtained in [11]. where Since such a quite precise asymptotic formula for λ q (α) as (1.6) has been obtained, from the standpoint of the better understanding of the global structure of the bifurcation branch of the positive solutions, the following problem from a view point of inverse problem was proposed in [12].Let f (u, u ′ ) = f (u) in (1.1).We assume that f (u) is an unknown function, but it is known that it satisfies the following conditions (A.1) and (A.2).
The inverse problem here means whether we can reconstruct the unknown nonlinear term f (u) from the information of the asymptotic behavior of λ q (α) as α → ∞ or not. where Furthermore, assume that r 1 satisfies To consider Problem A, we note the results obtained recently.
Therefore, from a viewpoint of Theorem 1.2, it is reasonable to expect that the results like Theorem 1.2 holds for Problem A.
Another direction we would like to consider is: Problem B: Treat the case where f = f (u, u ′ ), and consider the inverse problem.
The best way for us is to consider Problems A and B in the same framework and establish unified results for the inverse problems.The results in this paper, however, gives us the difficulty how to treat Problems A and B at the same time.
Now we state our result.We put . (1.12) We note that 0 < η < 1/2 if 1 ≤ m < 2p/(p + 1).Furthermore, let Then as α → ∞, the following asymptotic formulas for λ q (α) hold: We see from (1.11) and (1.13) that we should be careful to treat the Problem A and B at the same time, since we may not determine the unknown nonlinear term from the third term Therefore, as a next step of this work, before treating the Problem A and B simultaneously, we should find reasonable conditions for f (u, u ′ ) to reconstruct f (u, u ′ ) from the asymptotic formula for λ q (α).h(m) > 0), and according to these two cases, we obtain (1.13) and (1.15), respectively.
(ii) The case 2p/(p + 1) ≤ m < 2 is worth considering by the methods developed here.
However, for instance, the case 2p/((p + 1) = m should be handled more carefully, and quite a long and complicated calculation will be necessary.From this point of view, we may go on to an more detailed study of the case 2p/(p + 1) ≤ m < 2.
2 Proof of Theorem 1.4 We begin with notations and the fundamental properties of u λ .In what follows, C and k denote various positive constants independent of λ ≫ 1 for simplicity.We write λ = λ q (α).
The proof of Theorem 1.4 is based on the following Proposition 2.1.
We accept Proposition 2.1 here tentatively.The proof will be given in Section 3. Once it is obtained, Theorem 1.4 is proved by direct calculation as follows.
Proof of Theorem 1.4.We have only to consider the case 1 ≤ m < m 0 and show (1.13), since the other cases can be treated by the same calculation as that of the case 1 ≤ m < m 0 .
By this, we obtain (1.13).Thus the proof is complete.
). (3.12) ). (3.13) If we do not have the term B λ (t), then the situation here is the same as that of [11].
Therefore, to prove Proposition 2.1, the most important part is to calculate II in Lemmas 3.2 and 3.3 precisely.Clearly, the calculation in Lemmas 3.2 and 3.3 deeply depend on the estimates of B λ (t) as λ → ∞, which is the main part of the calculation of this paper.Since it is accomplished by long and tedious calculations, we fulfill it in the next section.
On the contrary, Lemma 3.1 can be proved by applying the argument in [11] to our case.This is rather an easy part, so the proof will be given in Appendix.
We next calculate the estimate of B λ (t) from above.
Here, 0 < δ ≪ 1 is a constant.Then we apply the argument in [11,Lemma 3.2] to our case and obtain that for λ ≫ 1,