Context. Theoretical and numerical studies have shown that large-scale vortices in protoplanetary discs can result from various hydro-dynamical instabilities. Once produced, such vortices can survive nearly unchanged over a large number of rotation periods, slowly migrating towards the star. Lopsided asymmetries recently observed at sub-millimetre and millimetre wavelengths in a number of transition discs could be explained by the emission of the solid particles trapped by vortices in the outer disc. However, at such a distance from the star, disc self-gravity (SG) may affect the vortex evolution and must be included in models. Aims. Our first goal is to identify how vortex morphology is affected by its own gravity. Next, we look for conditions that a self-gravitating disc must satisfy in order to permit vortex survival at long timescales. Finally, we characterise as well as possible the persistent self-gravitating vortices we have found in isothermal and non-isothermal discs. Methods. We performed 2D hydrodynamic simulations using the RoSSBi 3.0 code. The outline of our computations was limited to Euler's equations assuming a non-homentropic and non-adiabatic flow for an ideal gas. A series of 45 runs were carried out starting from a Gaussian vortex-model; the evolution of vortices was followed during 300 orbits for various values of the vortex parameters and the Toomre parameter. Two simulations, with the highest resolution thus far for studies of vortices, were also run to better characterise the internal structure of the vortices and for the purpose of comparison with an isothermal case. Results. We find that SG tends to destabilise the injected vortices, but compact small-scale vortices seem to be more robust than large-scale oblong vortices. Vortex survival critically depends on the value of the disc's Toomre parameter, but may also depend on the disc temperature at equilibrium. Disc SG must be small enough to avoid destruction in successive splitting and an approximate 'stability' criterion is deduced for vortices. The self-gravitating vortices that we found persist during hundreds of rotation periods and look like the quasi-steady vortices obtained in the non-self-gravitating case. A number of these self-gravitating vortices are eventually accompanied by a secondary vortex with a horseshoe motion. These vortices reach a new rotational equilibrium in their core, tend to contract in the radial direction, and spin faster. Conclusions. We propose an approximate 'robustness criterion', which states that, for a given morphology, a vortex appears stable provided that the disc's Toomre parameter overcomes a fixed threshold. Global simulations with a high enough numerical resolution are required to avoid inappropriate decay and to follow the evolution of self-gravitating vortices in protoplanetary discs. Vortices reach a nearly steady-state more easily in non-isothermal discs than in isothermal discs.
