Energy and Morse index of solutions of Yamabe type problems on thin annuli

In this paper we consider the following Yamabe type family of problem $(P_\e) : \quad -\D u_\e = u_\e ^{\frac{n+2}{n-2}}, \, \, u_\e > 0$ in $A_\e$, $u_\e =0$ on $\partial A_\e$, where $A_\e$ is an annulus-shaped domain of $\R^n$, $n\geq 3$, which becomes thinner when $\e\to 0$. We show that for every solution $u_{\e}$, the energy $\int_{A_{\e}} \, |\n u_{\e}|^2$, as well as the Morse index tends to infinity as $\e\to 0$. Such a result is proved through a fine blow-up analysis of some appropriate scalings of solutions whose limiting profiles are regular as well as singular solutions of some elliptic problem on $\R^n$, a half space or an infinite strip. Our argument involves also a Liouville-type theorem for regular solutions on the infinite strip.