  In this work, we consider the conditions for the existence of autosolitons,
in trapped Bose-Einstein condensates with attractive atomic interactions.
First, the variational approach is employed to estimate the stationary
solutions for the three-dimensional Gross-Pitaevskii equation with trap
potential, linear atomic feeding from the thermal cloud and two- and three-body
inelastic processes. Next, by using exact numerical calculations, we show that
the variational approach gives reliable analytical results. We also discuss the
possible observation of autosolitons in experiments with Lithium-7.
