Asymptotic expansion of oscillatory bifurcation curves of ODEs with nonlinear diffusion. (Q778585)

This work studies positive solutions of two-point boundary value problems of the form \[[u^p\cdot u']'+\lambda[u^1+\sin(u^n)],\quad u(0)=u(1)=0.\] The aim is to understand the curve of solutions in dependence on the parameter \(\lambda\). This is done by considering the norm \(\alpha=\|u\|_\infty\), and looking at the dependence of \(\lambda\) on \(\alpha\). An asymptotic formula for \(\lambda(\alpha)\) is proved, in particular showing the oscillatory behavior of this function as \(\alpha\rightarrow\infty\).