Abstract

In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres $\mathbb{S}^3, \mathbb{S}^4$. Under generic conditions we establish some Morse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence result on $\mathbb{S}^4$ through an Euler-Hopf type formula