Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem

Tetsutaro Shibata

doi:10.3390/math8112064

Abstract

We study the following nonlinear eigenvalue problem: −u″(t)=λf(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=∥uλ∥∞ of the solution uλ associated with λ. We establish the precise asymptotic formula for L1-norm of the solution ∥uα∥1 as α→∞ up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for ∥uα∥1.

Highlights

We consider the following nonlinear eigenvalue problems−u00 (t) = λ f (u(t)), u(t) u(−1)t ∈ I := (−1, 1), (1) t ∈ I, (2) = u(1) = 0, (3)> 0, where λ > 0 is a parameter
We study the following nonlinear eigenvalue problem: −u00 (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
We consider the case f (u) = log(1 + u), which is motivated by the logarithmic Schroedinger equation and the Klein-Gordon equation with logarithmic potential, which has been introduced in the quantum field theory

Summary

Introduction

In order to understand the asymptotic behavior of uα , we establish the asymptotic expansion formula for kuα k1 as α → ∞ The importance of this view point is that, kuα k 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 By using this formula, it is possible to obtain the approximate value of kuα k1 numerically as accurate as they want. The most important point of (6) is to give the procedure to obtain the asymptotic expansion formula for kuα k1 as correct as we want by using computer-assisted method, we only obtain up to the second term of kuα k1 , 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

Proof of Theorem 2