Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

Let G be a homogeneous group, and let X-1, X-2, ... , X-m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: {-Sigma(m)(j=1) X-j(2) u(x) - a/parallel to x parallel to(nu) u(x) = uQ+2/Q-2 (x), x is an element of Omega u(x) = 0, x is an element of partial derivative Omega, where O subset of G is a bounded domain with smooth boundary and 0 is an element of O, Q is the homogeneous dimension of G, a is an element of R, nu < 2. We boost u to L-p(Omega) for any 1 <= p <= 8 if u is an element of S-0(1,2) (Omega) is a weak solution of the problem above.