Optimal control in prescribing Webster scalar curvatures on 3-dimensional pseudo Hermitian manifolds

In this work, we give new existence and multiplicity results for the solutions of the prescription problem for the Webster scalar curvature on a 3-dimensional Pseudo Hermitian Manifold. The critical points of prescribed functions verify mixed conditions. We establish some Morse Inequalities at Infinity and a Poincaré–Hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the Morse index of such solutions.