Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

Abstract

We study systems of elliptic equations \begin{document}$ -\Delta u(x)+F_{u}(x, u) = 0 $\end{document} with potentials \begin{document}$ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $\end{document} which are periodic and even in all their variables. We show that if \begin{document}$ F(x, u) $\end{document} has flip symmetry with respect to two of the components of \begin{document}$ x $\end{document} and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on \begin{document}$ {\mathbb{R}}^{n} $\end{document} .