Asymptotic shape of solutions to nonlinear eigenvalue problems

Abstract

We consider the nonlinear eigenvalue problem -u''(t) = ƒ(λ, u(t)), u > 0, u(0) = u(1) =0, where λ > 0 is a parameter. It is known that under some conditions on ƒ(λ, u), the shape of the solutions associated with λ is almost 'box' when λ ≫ 1. The purpose of this paper is to study precisely the asymptotic shape of the solutions as λ → ∞ from a standpoint of L1-framework. To do this, we establish the asymptotic formulas for L¹-norm of the solutions as λ → ∞.