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

Abstract

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