The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the $n-$th order neutral differential equation $${(x\left(t\right)-px(t-\tau ))}^{\left(n\right)}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0,$$ where $\sigma \left(t\right)$ is a delayed or advanced argument.

Necessary and sufficient conditions have been found to force all solutions of the equation $${\left(r\left(t\right)y^{\text{'}}\left(t\right)\right)}^{(n-1)}+a\left(t\right)h\left(y\left(g\left(t\right)\right)\right)=f\left(t\right),$$ to behave in peculiar ways. These results are then extended to the elliptic equation $${\left|x\right|}^{p-1}\Delta y\left(\right|x\left|\right)+a\left(\right|x\left|\right)h\left(y\right(g\left(\right|x\left|\right)\left)\right)=f\left(\right|x\left|\right)$$ where $\Delta$ is the Laplace operator and $p\ge 3$ is an integer.

We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Our main intention in this regard is to expose precise mathematical conditions for critical paths to be minimum solutions in a variety of situations of interest in Physics. Our claim is that, with a few elementary techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models...