On periodic non-trivial solutions of the equation $-\Delta u=g(u)$ in $\mathbb R^{N+1}$

The existence of non-trivial solutions of the equation $-\Delta u=g(u)$ in $\mathbb R^{N+1}$, which are periodic with large periods in one variable and rapidly decreasing in others, is proved using variational methods. The non-existence of such solutions for small periods is shown as well.