Geodesic
On periodic non-trivial solutions of the equation $-\Delta u=g(u)$ in~$\mathbb R^{N+1}$
Izvestiya. Mathematics , Tome 59 (1995) no. 1, pp. 101-119

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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. 
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     author = {Ya. Sh. Il'yasov},
     title = {On periodic non-trivial solutions of the equation $-\Delta u=g(u)$ in~$\mathbb R^{N+1}$},
     journal = {Izvestiya. Mathematics },
     pages = {101--119},
     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a3/}
}
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Ya. Sh. Il'yasov. On periodic non-trivial solutions of the equation $-\Delta u=g(u)$ in~$\mathbb R^{N+1}$. Izvestiya. Mathematics , Tome 59 (1995) no. 1, pp. 101-119. http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a3/