Résumé en anglais

In this paper, we prove the existence of infinitely many solutions of a stationary nonlinear Dirac equation on the Schwarzschild metric, outside a massive ball. These solutions are the critical points of a strongly indefinite functional. Thanks to a concavity property, we are able to construct a reduced functional, which is no longer strongly indefinite. We find critical points of this new functional using the Symmetric Mountain Pass Lemma. Note that, as A. Bachelot-Motet conjectured, these solutions vanish as the radius of the massive ball tends to the horizon radius of the metric.