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In this work, we study the following sub-elliptic equations on Carnot groupwith Hardy-type singularity and critical Sobolev–Hardy exponentswherestands for the sub-Laplacian operator on Carnot group,, andis the critical Sobolev–Hardy exponent,  Q  is the homogeneous dimension with respect to the dilationnaturally associated with,  d  is the natural gauge associated with the fundamental solution ofon,andis the horizontal gradient associated with. Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions.

On the existence and multiplicity of solutions for a class of sub-Laplacian problems involving critical Sobolev–Hardy exponents on Carnot groups