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In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres ${𝕊}^3,{𝕊}^4$. Under generic conditions we establish some, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its 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 ${𝕊}^4$ through an Euler-Hopf type formula.

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere $({𝕊}^n,h)$. We derive from this analysis some a priori estimates in dimension $5$ and $6$. On ${𝕊}^5$ these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On ${𝕊}^6$ we prove the existence of at least one...

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ in $A_{\epsilon }$, $u_{\epsilon }=0$ on $\partial A_{\epsilon }$, where $A_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^n$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{|}^2$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem...