Sign-changing solutions to a partially periodic nonlinear Schrodinger equation in domains with unbounded boundary

We consider the problem -Delta u + (V-infinity + V(x)) u = vertical bar u vertical bar(p-2) u, u is an element of H-0(1)(Omega), where Omega is either R-N or a smooth domain in R-N with unbounded boundary, N >= 3, V-infinity > 0, V is an element of C-0(R-N), inf R-N V > -V-infinity and 2 < p < 2N/N-2. We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that Omega is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that Omega and V are invariant under some suitable group of symmetries on the last N - m coordinates of R-N, we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least m + 1