We consider the problem $$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$ where $$\Omega $$ is either $$\mathbb {R}^{N}$$ or a smooth domain in $$\mathbb {R} ^{N}$$ with unbounded boundary, $$N\ge 3,$$ $$V_{\infty }>0,$$ $$V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),$$ $$\inf _{\mathbb {R}^{N}}V>-V_{\infty }$$ and $$2<p<\frac{2N}{N-2}$$ . 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 $$\mathbb {R}^{N}$$ , 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 $$m+1$$