Some metric-singular properties of the graph of solutions of the one-dimensional p-Laplacian

We study the asymptotic behaviour of $epsilon$-neighbourhood of the graph of a type of rapidly oscillating continuous functions. Next, we estate necessary and sufficient conditions for rapid oscillations of solutions of the main equation. This enables us to verify some new singular properties of bounded continuous solutions of a class of nonlinear p-Laplacian by calculating lower and upper bounds for the Minkowski content and the $s$-dimensional density of the graph of each solution and its derivative.