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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. more...

For the equation $$\displaystyle{ y^{(n)} + p_{ 0}\,\vert y\vert ^{k}\ \mathrm{sign}\,y = 0, }$$ in the cases n = 3, 4, p0 > 0 or p0 1 and singular nonlinearity 0 < k < 1 asymptotic classification of all solutions are given. more...