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

In this work, we study the following sub-elliptic equations on Carnot group G with Hardy-type singularity and critical Sobolev-Hardy exponents -Delta(G)u = lambda psi(alpha) vertical bar u vertical bar(2+)(alpha)(-2)u/d(Z)(alpha) + beta f(z)vertical bar u vertical bar(p-2)u, z is an element of G, where -Delta(G) stands for the sub-Laplacian operator on Carnot group G, 0 < alpha <= 2, and 2* (alpha) = 2(Q-alpha)/Q-2 is the critical Sobolev-Hardy exponent, Q is the homogeneous dimension with respect to the dilation delta(gamma) naturally associated with -Delta(G), d is the natural gauge associated with the fundamental solution of -Delta(G) on G, psi = vertical bar del(G)d vertical bar and del(G) is the horizontal gradient associated with del(G). Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions.