Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem

: We study the following nonlinear eigenvalue problem: − u (cid:48)(cid:48) ( t ) = λ f ( u ( t )) , u ( t ) > 0, t ∈ I : = ( − 1, 1 ) , u ( ± 1 ) = 0, where f ( u ) = log ( 1 + u ) and λ > 0 is a parameter. Then λ is a continuous function of α > 0, where α is the maximum norm α = (cid:107) u λ (cid:107) ∞ of the solution u λ associated with λ . We establish the precise asymptotic formula for L 1 -norm of the solution (cid:107) u α (cid:107) 1 as α → ∞ up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for (cid:107) u α (cid:107) 1 .


Introduction
We consider the following nonlinear eigenvalue problems −u (t) = λ f (u(t)), t ∈ I := (−1, 1), u(−1) = u(1) = 0, where λ > 0 is a parameter. In this paper, we consider the case f (u) = log(1 + u), which is motivated by the logarithmic Schroedinger equation (see [1]) and the Klein-Gordon equation with logarithmic potential, which has been introduced in the quantum field theory (see [2]). We know from [3] that, if f (u) is continuous in u ≥ 0 and positive for u > 0, then 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. Since (u α , λ(α)) is constructed explicitly by time-map method (cf. [3], Theorem 2.1), λ is a continuous for α > 0, we write as λ = λ(α) for α > 0. We introduce one of the most famous results for bifurcation curve, which was shown for the one-dimensional Gelfand problem, namely, the Equations (1)-(3) with f (u) = e u . Then it was shown in [4] that it has the exact solution where sech x = 1/ cosh x. The related results have been obtained in [5]. Unfortunately, however, such explicit solution as (4) cannot be expected in general. From this point of view, one of the standard approach for the better understanding of the asymptotic shape of u α (t) is to establish precise asymptotic expansion formula for u α (t) as α → ∞. Indeed, in some cases, the asymptotic expansion formulas for u α (t) up to the second term have been obtained. Regrettably, however, precise asymptotic expansion formula for u α is also difficult to obtain from technical point of view of pure mathematics.
In this paper, in order to understand the asymptotic behavior of u α , we establish the asymptotic expansion formula for u α 1 as α → ∞. The importance of this view point is that, u α p (p ≥ 1) characterizes (or is related to) the many significant properties, such as the density of the objects in quantum physics, logistic equation in biology, and so on. Moreover, by using this formula, it is possible to obtain the approximate value of u α 1 numerically as accurate as they want. If the readers observe Theorem 2 and Section 3 below, they understand immediately that it is impossible to purchase more correct approximate value by hand calculation any more . Now we state our main results.
uniformly on I The leading term 4α/3 in the right-hand side of (6) comes from (5) immediately. The most important point of (6) is to give the procedure to obtain the asymptotic expansion formula for u α 1 as correct as we want by using computer-assisted method, although we only obtain up to the second term of u α 1 , since the calculation is purchased by hand.
The proof depends on the time-map argument and the precise asymptotic formula for λ(α) as α → ∞.

Proof of Theorem 1
We We recall the asymptotic behavior of λ(α) as α → ∞.
For any arbitrary fixed small > 0, we set I := [−1 + , 1 − ]. Then it is obvious that there is a constant δ > 0 independent of α 1 such that 0 < δ ≤ v α (t) ≤ 1 for t ∈ I . Indeed, if there is a subsequence of {v α }, which is denoted by {v α } again, such that v α (−1 + ) → 0 as α → ∞, then v α (t) is not convex on [−1, 0] for α 1. By (1) and Theorem 3, for any t ∈ I , we have By this, we see that as uniformly on I . By this, we easily see that v α (t) → 2t and v α (t) → 1 − t 2 for t ∈ I , which implies (5). Thus the proof is complete.

Proof of Theorem 2
In this section, let α 1. In what follows, we denote by C the various positive constants independent of α. We modify the time-map method used in [6]. By (1), we have By this, (8) and putting t = 0, we obtain This along with (9) implies that for −1 ≤ t ≤ 0, where By this and putting u α (t) = αs 2 , we obtain g α (s) := 1 log(1 + α) For 0 ≤ s ≤ 1, we have By (16)-(20) and Taylor expansion, we obtain Proof. We have It was shown in [6] that as α → ∞, Moreover, it is clear that for 0 ≤ θ ≤ π/2 We put s = sin θ. By (25), (26) and integration by parts, we obtain By l'Hôpital's rule, we obtain This implies that J 111 = 0. By (25), we have By (26), we have By this, we have We put s = sin θ. Then by integration by parts, (25) and (26), we have By (24), (31) and (32), we obtain Thus the proof is complete.
Proof of Theorem 2. By Theorem 3, Lemma 1, (21) and direct calculation, we obtain Theorem 2. Thus the proof is complete.

Remark 1.
Unfortunately, the author is not familiar with computing an analytical formula through computer packages. All the calculations in this paper were obtained by hand. As the authors mentioned in Section 1, if the suitable computer packages will be found, then it will be possible to compute the term J n := 1 0 s 3 (1 − s 2 ) 3/2 g α (s) n ds (n = 2, 3, · · · .).
By this, (21) and computer, it will be possible to calculate the approximate value of u α 1 as correct as we want.