In this paper we consider the following Yamabe type family of problem in , on , where is an annulus-shaped domain of , , which becomes thinner when . We show that for every solution , the energy , as well as the Morse index tends to infinity as . 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 , a half space or an infinite strip. Our argument involves also a Liouville-type theorem for regular solutions on the infinite strip.
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Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella, Energy and Morse index of solutions of Yamabe type problems on thin annuli. J. Eur. Math. Soc. 7 (2005), no. 3, pp. 283–304DOI 10.4171/JEMS/29