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GivenB  1    (  0  )the unit ball of    R  n    (   n  ⩾  3   ), we study smooth positive singular solutions    u  ∈   C  2    (   B  1    (  0  )   ∖   {  0  }   )to    −  Δ  u  =   (   u    2  ★    (  s  )   −  1    /    |  x  |   s   )   −  μ   u  q    . Here    0  <  s  <  2   ,2  ★    (  s  )   :  =  2   (  n  −  s  )   /   (  n  −  2  )is critical for the Sobolev embeddings,    q  >  1    and    μ  >  0   . When    μ  =  0    and    s  =  0   , the profile at the singularity 0 was fully described by Caffarelli–Gidas–Spruck. We prove that when    μ  >  0    and    s  >  0   , besides this profile, two new profiles might occur. We provide a full description of all singular profiles. Special attention is given to solutions such thatlim inf   x  →  0|  x  |    (   n  −  2   )  /  2    u   (  x  )   =  0    andlim sup   x  →  0|  x  |    (   n  −  2   )  /  2    u   (  x  )   ∈   (  0  ,  +  ∞  )    . The particular case    q  =  (  n  +  2  )  /  (  n  −  2  )    requires a separate analysis, which we also perform.

Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign‐changing non‐linearity