Abstract. We consider the Yamabe type family of problems (P ε ): − u ε = u (n+2)/(n−2) ε , u ε > 0 in A ε , u ε = 0 on ∂A ε , where A ε is an annulus-shaped domain of R n , n ≥ 3, which becomes thinner as ε → 0. We show that for every solution u ε , the energy A ε |∇u ε | 2 as well as the Morse index tend to infinity as ε → 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 for regular solutions on an infinite strip.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.