Summary: We study the regularity of weak solutions for a class of second order semilinear infinitely degenerate elliptic equations. We get the regularity of weak solutions up to the boundary for Dirichlet problem, by noting the logarithmic regularity estimate for a linear principal part. In relation to this linear part, we also show the controllability and strong maximum principle for second order hypoelliptic operators even in the case where they degenerate infinitely. Model equations naturally come from some variational problems, if one replaces the Laplace operator in Yamabe problems by degenerate elliptic operators. In the infinitely degenerate case, a permissible nonlinear term is not a fractional power, compared with the elliptic or subelliptic case. To treat this nonlinear term, nonlinear microlocal analysis is developed in the logarithmic Sobolev space.