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This paper is devoted to the existence of contact forms of prescribed Websterscalar curvature on a $3-$dimensional CR compact manifold locally conformallyCR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due toKazdan-Warner type obstructions, conditions on the function $H$ to be realizedas a Webster scalar curvature have to be given. We prove new existence resultsbased on a new type of Euler-Hopf type formula. Our argument gives an upperbound on the Morse index of the obtained solution. We also give a lower boundon the number of conformal contact forms having the same Webster scalarcurvature.

Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three $C R $ manifolds