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

Abstract

In this paper we consider the following Yamabe type family of problem $(P_{\epsilon }):-\Delta u_{\epsilon }=u_{\epsilon }^{\frac{n+2}{n-2}},u_{\epsilon }>0$ 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 when $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{\epsilon }{|}^2$, as well as the Morse index tends to infinity as $\epsilon \to 0$. 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 ${\mathbb{R}}^n$, a half space or an infinite strip. Our argument involves also a Liouville-type theorem for regular solutions on the infinite strip.