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

We deal with a boundary value problem of the form equation(1){−ϵ(ϕp(ϵu′))′+a(x)W′(u)=0u′(0)=0=u′(1), where ϕp(s)=|s|p−2sϕp(s)=|s|p−2s for s∈Rs∈R and p>1p>1, and W:[−1,1]→RW:[−1,1]→R is a double-well potential. We study the limit profile of solutions of (1) when ϵ→0+ϵ→0+ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when ϵϵ is small enough.