Global behavior of the branch of positive solutions to a logistic equation of population dynamics

We consider the nonlinear problem arising in population dynamics:

$$-u''(t) + u(t)^p = \lambda u(t), \enskip u(t) > 0, \quad t \in I := (0, 1), \enskip u(0) = u(1) = 0,$$

where $p > 1$ is a constant and $\lambda > 0$ is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions $\lambda_q(\alpha)$ in $L^q$-framework as $\alpha \to \infty$, where $\alpha := \Vert u_\lambda\Vert_q$ ( $1 \le q < \infty$). Especially, for the original logistic equation, namely the case where $p = 2$ and $q = 1$, we obtain not only the asymptotic expansion formula for $\lambda_1(\alpha)$ but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.