  We address stationary patterns in exciton-polariton condensates supported by
a narrow external pump beam, and we discover that even in the absence of
trapping potentials, such condensates may support stable localized stationary
dissipative solutions (quasi-compactons), whose field decays faster than
exponentially or even vanishes everywhere outside the pump spot. More general
conditions lead to dissipative solitons which may display bistability. The
bistability in exciton-polariton condensates, which manifests itself in
simultaneous existence of two stable and one unstable localized solitons with
different amplitudes, widths, and exciton-photon fractions under the same
physical conditions, strongly depends on the width of pump beam and is found to
disappear for sufficiently narrow pump beams.
