We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.

In this Note we consider the following problem $$\left\{\begin{array}{cc}-\mathrm{\u25b3}u=N\left(N-2\right){u}^{{p}_{\u03f5}}-\lambda u\hfill & \text{in}\mathrm{\Omega}\hfill \\ u0\hfill & \text{in}\mathrm{\Omega}\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega}.\hfill \end{array}\right.$$ where $\mathrm{\Omega}$ is a bounded smooth starshaped domain in ${\mathbb{R}}^{N}$, $N\ge 3$, ${p}_{\u03f5}=\frac{N+2}{N-2}-\u03f5$, $\u03f5>0$, and $\lambda \ge 0$. We prove that if ${u}_{\u03f5}$ is a solution of Morse index $m>0$ than ${u}_{\u03f5}$ cannot have more than $m$ maximum points in $\mathrm{\Omega}$ for $\u03f5$ sufficiently small. Moreover if $\mathrm{\Omega}$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $\u03f5$ sufficiently small.

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 ${\mathbb{R}}^{n}$, $n\ge 3$, 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 ${\mathbb{R}}^{n}$, a half-space or an infinite strip. Our argument also involves a Liouville type
theorem...

Download Results (CSV)