Prescribing Webster scalar curvature on CR manifolds of negative conformal invariants

In this paper, we are interested in solving the following partial differential equation−Δθu+Ru=fu1+2/n on a compact strictly pseudo-convex CR manifold (M,θ) of dimension 2n+1 with n⩾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.