  Bose-Einstein condensation, observed in either strongly interacting liquid
helium or weakly interacting atomic Bose gases, is widely known to be a
second-order phase transition. Here, we predict a first-order Bose-Einstein
condensation in a cloud of harmonically trapped bosons interacting with both
attractive two-body interaction and repulsive three-body interaction,
characterized respectively by an $s$-wave scattering length $a<0$ and a
three-body scattering hypervolume $D>0$. It happens when the harmonic trapping
potential is weak, so with increasing temperature the system changes from a
low-temperature liquid-like quantum droplet to a normal gas, and therefore
experiences a first-order liquid-to-gas transition. At large trapping
potential, however, the quantum droplet can first turn into a superfluid gas,
rendering the condensation transition occurred later from a superfluid gas to a
normal gas smooth. We determine a rich phase diagram and show the existence of
a tri-critical point, where the three phases - quantum droplet, superfluid gas
and normal gas - meet together. We argue that an ensemble of spin-polarized
tritium atoms could be a promising candidate to observe the predicted
first-order Bose-Einstein condensation, across which the condensate fraction or
central condensate density jumps to zero and the surface-mode frequencies
diverge.
