We consider the Choquard system -Δu+V(x)u+|u|p-2u=λϕu,inR3,-Δϕ=u2,inR3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\Delta u+V( x) u+|u|^{p-2}u=\lambda \phi u , &{}\quad \text{ in } \mathbb {R}^{3},\\ -\Delta \phi = u^{2}, &{}\quad \text{ in } \mathbb {R}^{3}. \end{array} \right. \end{aligned}$$\end{document}where λ>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0$$\end{document} is a parameter, 3<p<6\documentclass[12pt]{minimal}
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\begin{document}$$3<p<6$$\end{document}, V∈C(R3)\documentclass[12pt]{minimal}
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\begin{document}$$V\in C( \mathbb {R}^{3}) $$\end{document} is 1\documentclass[12pt]{minimal}
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\begin{document}$$1$$\end{document}-periodic in xj\documentclass[12pt]{minimal}
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\begin{document}$$x_j$$\end{document} for j=1,2,3\documentclass[12pt]{minimal}
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\begin{document}$$j = 1,2,3$$\end{document} and 0 is in a spectral gap of the operator -Δ+V\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta +V$$\end{document}. This system is strongly indefinite, i.e., the operator -Δ+V\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta +V$$\end{document} has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities |u|p-2u\documentclass[12pt]{minimal}
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\begin{document}$$|u|^{p-2}u$$\end{document} and λϕu\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \phi u$$\end{document}. Moreover, the functional corresponding to this system does not satisfy the Palais–Smale condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small λ>0,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0,$$\end{document} this system has a nontrivial solution.
