On uniform estimates for solutions to quasi-linear differential equations

Uniform estimates are obtained for positive solutions with the same domain to the equation
$$ y^{(n)}+\sum_{i=0}^{n-1}a_{i}(x)y^{(i)}+p(x)|y|^{k-1}y=0 $$
of even order $n$ with $k>1$ and continuous functions $p(x)>0$ and $a_i(x)$. In the case where $a_{0}(x)\equiv\dots\equiv a_{n-1}(x)\equiv0$, uniform estimates are obtained depending on $p_{*}=\inf p(x)>0$ and not on the function $p(x)$ itself.