Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem - Tetsutaro Shibata (Mathematics, 2020)
We study the following nonlinear eigenvalue problem: <semantics>−u″(t)=λf(u(t)),<mspace width="5.0pt"></mspace>u(t)>0,<mspace width="5.0pt"></mspace>t∈I:=(−1,1),<mspace width="5.0pt"></mspace>u(±1)=0,</semantics> where <semantics>f(u)=log(1+u)</semantics> and <semantics>λ>0</semantics> is a parameter. Then <semantics>λ</semantics> is a continuous function of <semantics>α>0</semantics>, where <semantics>α</semantics> is the maximum norm <semantics>α=∥uλ∥∞</semantics> of the solution <semantics>uλ</semantics> associated with <semantics>λ</semantics>. We establish the precise asymptotic formula for <semantics>L1</semantics>-norm of the solution <semantics>∥uα∥1</semantics> as <semantics>α→∞</semantics> up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for <semantics>∥uα∥1</semantics>.