Asymptotic shape of solutions to nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem $$ -u''(t) = f(lambda, u(t)), quad u mbox{greater than} 0, quad u(0) = u(1) = 0, $$ where $lambda > 0$ is a parameter. It is known that under some conditions on $f(lambda, u)$, the shape of the solutions associated with $lambda$ is almost `box' when $lambda gg 1$. The purpose of this paper is to study precisely the asymptotic shape of the solutions as $lambda o infty$ from a standpoint of $L^1$-framework. To do this, we establish the asymptotic formulas for $L^1$-norm of the solutions as $lambda o infty$.