Geodesic
Asymptotic shape of solutions to nonlinear eigenvalue problems
Electronic journal of differential equations, Tome 2005 (2005)
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 $\lambda \to \infty$ from a standpoint of $L^1$-framework. To do this, we establish the asymptotic formulas for $L^1$-norm of the solutions as $\lambda \to \infty$.
Classification : 34B15
Keywords: asymptotic formula, $L^1$-norm, simple pendulum, logistic equation 
@article{EJDE_2005__2005__a223,
     author = {Shibata,  Tetsutaro},
     title = {Asymptotic shape of solutions to nonlinear eigenvalue problems},
     journal = {Electronic journal of differential equations},
     year = {2005},
     volume = {2005},
     zbl = {1075.34018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a223/}
}
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Shibata,  Tetsutaro. Asymptotic shape of solutions to nonlinear eigenvalue problems. Electronic journal of differential equations, Tome 2005 (2005). http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a223/