Minkowski–Bouligand dimension of solutions of the one-dimensional p-Laplacian

The upper Minkowski–Bouligand dimension of the graph of continuous solutions, denoted by dimMy, is studied for a class of nonlinear one-dimensional p-Laplacian. Some sufficient conditions on the nonlinearities are given such that dimMy takes the prescribed fractional values. Next, a relation between dimMy and the order of growth for singular behaviour of Lp norm of derivatives of solutions is given. Finally, the change and stability of dimMy are considered by means of a double-parametric problem.