Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 161, pp. 1-7.

Title: Asymptotic expansion formulas for the maximum of
       solutions to diffusive logistic equations
   
Authors: Tetsutaro Shibata (Hiroshima Univ., Japan)

Abstract:
 We consider the nonlinear eigenvalue problems
 $$\displaylines{
 -u''(t) + u(t)^p =  \lambda u(t),\cr
 u(t) > 0, \quad t \in I := (0, 1), \quad u(0) = u(1) = 0,
 }$$
 where $p > 1$ is a constant and $\lambda > 0$ is a parameter.
 This equation is well known as the original logistic equation of
 population dynamics when $p=2$.
 We establish the precise asymptotic formula for
 $L^\infty$-norm of the solution $u_\lambda$ as $\lambda \to \infty$
 when $p=2$ and $p=5$.

Submitted May 07, 2008. Published December 09, 2008.
Math Subject Classifications: 34B15.
Key Words: Logistic equation; $L^\infty$-norm of solutions.