Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion

We consider the nonlinear eigenvalue problem $[D(u(t))u(t)‘]’ + \lambda g(u(t)) = 0$,$u(t) > 0$,$t \in I := (0,1)$,$u(0) = u(1) = 0$, which comes from the porous media type equation. Here, $D(u)=p{u}^{2n}+\sin u$ ($n \in $\mathbb{N}$$,$p > 0$:givenconstants),$g(u) = u$or$g(u) = u + \sin u$\mathrm{.}$\lambda > 0$isabifurcationparameterwhichisacontinuousfunctionof$\alpha = \Vert u_\lambda\Vert_\infty$ofthesolution$u_\lambda$correspondingto$\lambda$,andisexpressedas$\lambda = \lambda(\alpha)$. Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to the structure of bifurcation curves $\lambda (\alpha )$. We prove that the simplest case $D(u) = u^{2n} + \sin u$and$g(u) = u$ gives us the most significant phenomena to the global behavior of $\lambda (\alpha )$.