Asymptotic expansion of oscillatory bifurcation curves of ODEs with nonlinear diffusion

Abstract

We consider the nonlinear eigenvalue problem $$ [D(u)u']' + \lambda f(u) = 0, \ \ u(t) > 0, \ \ t \in I := (0,1), \ \ u(0) = u(1) = 0, $$ where $D(u) = u^p$, $f(u) = u^{q} + \sin (u^n)$ and $\lambda > 0$ is a bifurcation parameter. Here, $p \ge 0$, $n > 0$ and $q > 0$ are given constants and $k:=(p + q + 1)/2 \in \mathbb{N}$. This equation is motivated by the mathematical model of animal dispersal and invasion and $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(\alpha)$. We establish new precise asymptotic expansion formulas for $\lambda(\alpha)$ as $\alpha \to \infty$. In particular, we obtain the precise asymptotic behavior of $\lambda(\alpha)$ which tends to $0$ with oscillation as $\alpha \to \infty$.