Asymptotic expansion formulas for the maximum of solutions to diffusive logistic equations

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 $lambda o infty$ when $p=2$ and $p=5$.