Some asymptotic properties of principal solutions of the half-linear differential equation $${\left(a\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right)\Phi \left(x\right)=0,(*)$$ $\Phi \left(u\right)={\left|u\right|}^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.

Some recent results concerning properties of solutions of the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right):={\left|x\right|}^{p-2}x,p>1,(*)$$ are presented. A particular attention is paid to the oscillation theory of $(*)$. Related problems are also discussed.