Prescribing Webster scalar curvature on CR manifolds of negative conformal invariants

In this paper, we are interested in solving the following partial differential equation
$$-\Delta_{\theta}u+Ru =f u^{1+2/n}$$on a compact strictly pseudo-convex CR manifold $(M, \theta)$ of dimension $2n+1$ with $n \geqslant 1$. This problem naturally arises when solving the prescribing Webster scalar curvature problem on $M$ with the prescribed function $f$. Using variational techniques, we prove several non-existence, existence, and multiplicity results when the function $f$ is sign-changing.