Highly oscillatory solutions of a Neumann problem for ap-laplacian equation

Abstract

We deal with a boundary value problem of the formwhere φ p (s) = |s| p−2 s for s ∈ R and p > 1, and W : [−1, 1] → R is a double-well potential. We study the limit profile of solutions of (1) when → 0 + and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when is small enough.

Numerical simulation of high sensitivity of solutions to the nonlinear singularly perturbed dynamical system on the initial conditions and parameter

“…is taken into consideration. Equations of this type have been studied in numerous topical works, such as a one dimensional Schrodinger equation of the form 2 u + V (x)u − |u| α−1 u = 0, where α > 0 in [4], an equation of the type − 2 u + a(x)W (u) = 0, where a is a positive weight function and W is a double-well potential in [5] and [13], a Neumann problem for a p-laplacian equation exhibiting highly oscillatory solutions in [3]. The equation of the same type was studied in [16], where the question of oscillation pattern control for nonlinear dynamical systems without the excitation of oscillatory inputs was answered, providing a novelty method for the partition of the space of initial states to the areas allowing active control of the stable steady-state oscillations.…”