  The mobile impurity in a Bose-Einstein condensate (BEC) is a paradigmatic
many-body problem. For weak interaction between the impurity and the BEC, the
impurity deforms the BEC only slightly and it is well described within the
Fr\"ohlich model and the Bogoliubov approximation. For strong local attraction
this standard approach, however, fails to balance the local attraction with the
weak repulsion between the BEC particles and predicts an instability where an
infinite number of bosons is attracted toward the impurity. Here we present a
solution of the Bose polaron problem beyond the Bogoliubov approximation which
includes the local repulsion between bosons and thereby stabilizes the Bose
polaron even near and beyond the scattering resonance. We show that the Bose
polaron energy remains bounded from below across the resonance and the size of
the polaron dressing cloud stays finite. Our results demonstrate how the
dressing cloud replaces the attractive impurity potential with an effective
many-body potential that excludes binding. We find that at resonance, including
the effects of boson repulsion, the polaron energy depends universally on the
effective range. Moreover, while the impurity contact is strongly peaked at
positive scattering length, it remains always finite. Our solution highlights
how Bose polarons are self-stabilized by repulsion, providing a mechanism to
understand quench dynamics and nonequilibrium time evolution at strong
coupling.
