We establish via variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for minimizers.
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Nicholas D. Alikakos, Giorgio Fusco, A maximum principle for systems with variational structure and an application to standing waves. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1547–1567DOI 10.4171/JEMS/538