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

We deal with a boundary value problem of the form $-ε(ϕ_p(εu'))'+a(x)W'(u)=0,\quad u'(0)=0=u'(1),$ where $ϕ_p(s) = \vert s \vert^{p-2} s$ 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 $ε\to 0^+$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $ε$ is small enough.

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