Geodesic
Asymptotic shape of solutions to nonlinear eigenvalue problems
Electronic Journal of Differential Equations, Tome 2005 (2005).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     author = {Shibata, Tetsutaro},
     title = {Asymptotic shape of solutions to nonlinear eigenvalue problems},
     journal = {Electronic Journal of Differential Equations},
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     volume = {2005},
     year = {2005},
     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/