Highly oscillatory solutions of a Neumann problem for a $p$-laplacian equation

TL;DR

This paper investigates the behavior of solutions to a Neumann boundary value problem involving a p-Laplacian with a double-well potential as the parameter approaches zero, revealing highly oscillatory solutions and their limit profiles.

Contribution

It establishes the existence of solutions with prescribed oscillatory limit profiles for small epsilon in a p-Laplacian Neumann problem with a double-well potential.

Findings

Solutions exhibit highly oscillatory behavior as epsilon approaches zero.

Existence of nodal solutions matching any admissible limit profile for small epsilon.

Characterization of the limit profile of solutions in the singular limit.

Abstract

We deal with a boundary value problem of the form $-\epsilon(\phi_p(\epsilon u'))'+a(x)W'(u)=0,\quad u'(0)=0=u'(1),$ where $\phi_p(s) = \vert s \vert^{p-2} s$ for $s \in \mathbb{R}$ and $p>1$, and $W:[-1,1] \to {\mathbb R}$ is a double-well potential. We study the limit profile of solutions when $\epsilon \to 0^+$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $\epsilon$ is small enough.