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In this paper, we extend the result of Yan and Yang [ 16 ] on equations to an elliptic system involving critical Sobolev and Hardy–Sobolev exponents in bounded domains satisfying some geometric condition. In addition, we weaken the conditions on the dimension  N  and on the potentiala  ⁢   (  x  ){a(x)}    set in [ 16 ]. Our main result asserts, by a variational global-compactness argument, that the condition on the dimension  N  can be refined fromN  ≥  7{N\geq 7}    toN  &gt;   max  ⁡   (  4  ,    ⌊   2  ⁢  s   ⌋   +  2   ){N&gt;\max(4,\lfloor 2s\rfloor+2)}   , where0  &lt;  s  &lt;  2{0&lt;s&lt;2}    and still end up with infinitely many solutions.

A refined bound on the dimension of ℝ N for an elliptic system involving critical terms with infinitely many solutions