Growing amount of observations indicate presence of intrinsically faint quasar subgroup ( a few % of known quasars ) with noncosmological quantized redshift .
Here we find an analytical solution of Einstein equations describing bubbles made from axions with periodic interaction potential .
Such particles are currently considered as one of the leading dark matter candidate .
The bubble interior possesses equal gravitational redshift which can have any value between zero and infinity .
Quantum pressure supports the bubble against collapse and yields states stable on the scale more then hundreds million years .
Our results explain the observed quantization of quasar redshift and suggest that intrinsically faint point-like quasars associated with nearby galaxies are axionic bubbles with masses 10 ^ { 8 } -10 ^ { 9 } M _ { \odot } and radii 10 ^ { 3 } -10 ^ { 4 } R _ { \odot } .
They are born in active galaxies and ejected into surrounding space .
Properties of such quasars unambiguously indicate presence of axion dark matter in the Universe and yield the axion mass m \approx 1 meV , which fits in the open axion mass window constrained by astrophysical and cosmological arguments .