On Nonpower-Law Asymptotic Behavior of Blow-Up Solutions to Emden-Fowler Type Higher-Order Differential Equations

For the equation $$\begin{aligned} y^{(n)}= p_0\,{\mid y \mid }^{k}\,\mathrm{sgn}\,y,\,\,\, n\ge 12,\,\,\, k>1,\,\,\, p_0>0, \end{aligned}$$ (1) the existence of positive solutions with nonpower-law asymptotic behavior is proved, namely $$\begin{aligned} y(x)=(x^*-x)^{-\frac{n}{k-1}}\ h(\log \,(x^*-x)), \ \ x\rightarrow x^*-0, \end{aligned}$$ (2) where h is a positive periodic non-constant function on \(\mathbb {R}\). To prove the existence, a useful modification of the Hopf bifurcation theorem is used.